- Email: [email protected]
A new treatment based on hybrid functions to the solution of telegraph equations of fractional order
ARTICLE IN PRESS
JID: APM
[m3Gsc;October 25, 2015;10:23]
Applied Mathematical Modelling 000 (2015) 1–11
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
A new treatment based on hybrid functions to the solution of telegraph equations of fractional order N. Mollahasani a,∗, M. Mohseni (Mohseni) Moghadam a, K. Afrooz b a b
Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran Department of Electrical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran
a r t i c l e
i n f o
Article history: Received 1 September 2013 Revised 18 January 2015 Accepted 26 August 2015 Available online xxx
a b s t r a c t In this paper, a new operational method based on hybrid functions of Legendre polynomials and Block-Pulse-Functions will be presented. The operational matrix of fractional integration is derived and used to take an acceptable approximate for the solution of a telegraph equation of fractional order. An error estimation will be presented to give an image of the goodness of the solution. Some numerical examples demonstrate the efficiency of the proposed method.
Keywords: Fractional telegraph equation Hybrid functions Fractional calculus Operational matrices
© 2015 Elsevier Inc. All rights reserved.
1. Introduction A class of hyperbolic partial differential equations which describes vibrations within objects and how waves are propagated, is called Telegraph equation [1]. Equations of the form of telegraph equations arise in the study of propagation of electrical signals in a cable of transmission line and wave phenomena. Interaction between convection and diffusion or reciprocal action of reaction and diffusion describes a number of nonlinear phenomena in physical, chemical and biological process [2–5]. In fact, the telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion for such branches of sciences. For example, biologists encounter these equations in the study of pulsatileblood flow in arteries and in one-dimensional random motion of bugs along a hedge [6]. Also the propagation of acoustic waves in Darcy-type porous media [7], and parallel flows of viscous Maxwell fluids [8] are just some of the phenomena governed by the telegraph equations [9–11]. The telegraph equation has also been employed in other areas. For example, it is used as a replacement for the diffusion equation to model transport of charged particles [12,13], high frequency transmission lines [14,15], solar cosmic rays [16], chemical diffusion, anomalous diffusion [17,18], hydrology and population dynamics [19]. It is also employed in the theory of hyperbolic heat transfer [20,21]. Finite difference methods are known as the first techniques for solving partial differential equations [22,23]. Even though these methods are very effective for solving various kinds of partial differential equations. Many researchers have used various numerical and analytical methods to solve the telegraph equation [24,25]. Unconditionally stable and parallel difference scheme, Chebyshev Tau method, Chebyshev method, Legendre multi-wavelet Galerkin method, meshless local weak– strong methods, homotopy analysis and Adomian decomposition are such methods [12,15,26–36]. The fractional calculus is one of the most accurate tools to refine the description of natural phenomena [37]. Fractional differential equations have attracted in the recent years a considerable interest due to their frequent appearance in various fields ∗
Corresponding author. Tel.: +983432475898. E-mail address: [email protected], [email protected] (N. Mollahasani).
http://dx.doi.org/10.1016/j.apm.2015.08.020 S0307-904X(15)00544-2/© 2015 Elsevier Inc. All rights reserved.
Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
ARTICLE IN PRESS
JID: APM 2
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
and their more accurate models of systems under consideration provided by fractional derivatives [38,39]. Fractional telegraph equations has been recently considered by many authors. Cascaval et al. [40] discussed the time-fractional telegraph equations, dealing with well posedness and presenting a study of their asymptotic behavior by using the Riemann– Liouville approach. Orsingher and Beghin [41] discussed the time-fractional telegraph equation and telegraph processes with Brownian time, showing that some processes are governed by time-fractional telegraph equations. Chen et al. [42] examined and derived a solution of the time-fractional telegraph equation with three kinds of nonhomogeneous boundary conditions, by the method of separation of variables. Fractional telegraph equations from the analytic point of view have been studied by many authors (see Saxena et al. [43] for equations with n time derivatives). The aim of this paper is to introduce a new method for approximating the solution of a time-fractional telegraph equation in the following form:
∂α ∂ α−1 ∂2 u(x, t ) + α −1 u(x, t ) + u(x, t ) − u(x, t ) = f (x, t ), ∂tα ∂t ∂ x2
(1.1)
with the initial and boundary conditions:
u(x, 0) = l1 (x),
u(0, t ) = g1 (t ),
u(x, 1) = l2 (x),
u(1, t ) = g2 (t ).
(1.2)
The right-hand-side function f(x, t) is given and 1 < α ≤ 2 is a real number. In this paper, first hybrid functions of Legendre polynomials and Block-Pulse-functions are introduced and a new operational matrix of fractional integration is constructed. Then a numerical method to solve Eq. (1.1) will be developed. Also some error analysis will be presented. Finally, some numerical examples are presented to confirm the applicability of the method. 2. Real-world application of telegraph equations In the following we mention some real-world applications of telegraph equations. (1) Consider the following fractional telegraph equation [41]:
∂ 2α u(x, t ) ∂ 2 u(x, t ) ∂ α u(x, t ) + 2λ = c2 , 0 < α ≤ 1, α 2 α ∂t ∂t ∂ x2
(2.1)
with the conditions:
1 : 2 ≤1:
0<α≤ 1 2
<α
u(x, 0) = δ(x), ut (x, 0) = 0.
Here, we consider the case α = 12 for which it is possible to obtain the solution. The probability density of the solution of (2.1) coincides with the distribution of the telegraph process T = T (t ), t > 0 with a Brownian time, that is:
W (t ) = T (|B(t )|). This means that the fundamental solution to the fractional Eq. (2.1) for α = 12 can be interpreted as the distribution of a particle moving back and forth on the real line with velocities ±c (switching of Poisson-paced time) for a random time interval of length |B(t)|. Clearly, B and T are independent of each other. Eq. (2.1) for α = 12 is a heat equation with a damping term depending on all 1
1
values of u in [0, t] and assigning an overwhelming weight to those close to t. The damping effect of ∂ 2 u/∂ t 2 reverberates on the probability density of the solution, where the governing term (solution to the heat equation) is weighted by the telegraph distribution (representing the impact of the fractional derivative). (2) The transmission line equation can be represented as:
∂ V (z, t ) ∂ I(z, t ) = −RI(z, t ) − L , ∂z ∂t
(2.2)
∂ I(z, t ) ∂ V (z, t ) = −GV (z, t ) − C , ∂z ∂t
(2.3)
where R, G, C, and L are the per-unit-length parameters of the transmission line [5]. The V(z, t) and I(z, t) are the line voltages (with respect to the reference line) and line currents, respectively. Fig. (1) shows a segment of a typical transmission line. Differentiating (2.2) and (2.3) with respect to z, we get:
∂ 2V (z, t ) ∂ 2V (z, t ) ∂ V (z, t ) + LC = RGV + ( RC + LG ) , ∂t ∂ z2 ∂t2
(2.4)
∂ 2 I(z, t ) ∂ 2 I(z, t ) ∂ I(z, t ) + LC = RGI + (RC + LG) , 2 ∂t ∂z ∂t2
(2.5)
which are two telegraph equations for α = 2. In [14], the authors have solved the following example. Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
ARTICLE IN PRESS
JID: APM
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
3
Fig. 1. A segment of a typical transmission line.
Example 2.1. The line considered here is a one conductor lossy transmission line whose parameters are L = 309nH, C = 144pF, R = 524 × 10−3 , G = 905 × 10−9 S. The total line length is 10cm. (3) Image denoising is often used for pre-processing images, so that subsequent image analysis is more reliable. The goal of image denoising is to remove the noise from the observed noisy image and get as close as possible to the original image. Besides noise removal ability, another important requirement for image denoising procedures is that true image structures should be preserved in the denoising process. The use of partial differential equation(PDE) has been studied as a useful tool for image denoising. In [44], Zhang, et al. have proposed a three dimensional fractional telegraph equation for image structure preserving denoising. 3. Function approximation The complete orthogonal set of the hybrid functions of Legendre polynomials and Block Pulse functions(BPFs) hpq (x), is defined on [0, 1) as follows:
h pq (x) =
Lq (2mx − 2p + 1),
p−1 m
0,
elsewhere,
≤x<
p , m
(3.1)
for p = 1, 2, . . . , m, q = 0, 1, . . . , n − 1, where p is the order of BPFs and q is the order of Legendre polynomials. Here, Legendre polynomials are defined on [−1, 1] by:
Lk (x) = 2k
k
k i
xi
i=0
k+i−1 2
k
,
(3.2)
for k = 0, 1, ... or in the recursive form:
L0 (x) = 1,
L1 (x) = x,
(k + 1)Lk+1 (x) = (2k + 1)xLk (x) − kLk−1 (x), k = 1, 2, ...
(3.3)
The set {Lk (x) : k = 0, 1, ...} in L2 [−1, 1] is a complete orthogonal system. A set of BPFs on [0, 1) is defined as the following:
bi (x) =
1,
i−1 n
≤ x < ni ;
0,
elsewhere,
(3.4)
for i = 1, 2, .... This set is also a set of orthogonal functions. Suppose that nm = M. Therefore, after renaming we have:
h(x) = [h1 (x), h2 (x), . . . , hM (x)]t , B(x) = [b1 (x), b2 (x), . . . , bM (x)]t . Each square integrable function f(x, t) in
f (x, t ) =
∞ ∞
ci j hi (x)h j (t ),
L2 ([0,
(3.5)
1) × [0, 1)) can be expanded into a hybrid series of infinite terms:
x ∈ [0, 1],
(3.6)
i=1 j=1
where the hybrid coefficients are determined as:
ci j =
(hi (x), ( f (x, t ), h j (t ))) , (hi (x), hi (x))(h j (t ), h j (t ))
for i, j = 1, 2, . . . and (., .) denotes the inner product. If f(x, t) is a piecewise constant function or may be approximated as a piecewise constant function on each subinterval, the series sum in Eq. (3.6) can be truncated, that is:
f (x, t ) ∼ =
M M
ci j hi (x)h j (t ) = fm (x, t ) = ht (x)Ch(t ),
x ∈ [0, 1],
(3.7)
i=1 j=1
where C is an M × M matrix and h(x) is defined as (3.5). Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
ARTICLE IN PRESS
JID: APM 4
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
4. Hybrid operational matrix of fractional integration In this section, we obtain an operational matrix of fractional integration, based on hybrid functions. The Riemann– Liouville fractional integral operator of order α > 0 of the function f(x) is defined as [38,39]:
Iα f (x) =
1
(α)
x 0
(x − t )α−1 f (t )dt, α > 0, x > 0,
(4.1)
where (.) is the so called Gamma function with the property: (x + 1) = x (x), x ∈ R. Some properties of the operator Iα can be found in [38]. We mention only the following. For α ≥ 0 and γ > −1, we have:
I α xγ =
(γ + 1) α+γ x . (α + γ + 1)
(4.2)
The Caputo fractional derivative of order α is defined as:
Dα f (x) =
1 (m − α)
t 0
(t − τ )m−α−1 f (m) (τ )dτ , m − 1 < α ≤ m,
(4.3)
where m is an integer. This definition of derivative satisfies in the following equality:
Iα Dα f (x) = f (x) −
m−1
f (k) (0+ )
k=0
tk , k!
m − 1 < α ≤ m.
(4.4)
The fractional integration of order α of h(x) can be expanded into hybrid series with hybrid coefficient matrix Pα as follows:
Iα h(x) =
1
(α)
x 0
(x − t )α−1 h(t )dt Pα h(x).
(4.5)
We call this M × M square matrix Pα the (generalized) operational matrix of fractional integration. Wang Chi-Hsu in [45] showed that the generalized Block Pulse operational Matrix Fα , is derived as:
Iα B(x) F α B(x), where:
Fα =
1 α M
(4.6)
⎛f
1
⎜ 1 ⎜ (α + 2) ⎝
f2 f1
... ... ... ...
f3 f2 f1
0
fM
⎞
fM−1 ⎟ fM−2 ⎟,
(4.7)
⎠
f1 in which f1 = 1, fk = kα +1 − 2(k − 1)α +1 + (k − 2)α +1 for (k = 2, 3, . . . , M − i + 1) and i = 1, 2, . . . , M. On the other hand, we can expand hybrid functions via BPFs by:
h(x) = B(x),
(4.8)
where = (φi j ) and:
φi j = n2
q−1
q k=0
1 q k+1 k
q+k−1 2
q
2 j − 2p + 1 n
k+1
−
2 n
( j − 1) − 2p + 1
k+1
,
(4.9)
for p = 1, 2, . . . , m, q = 0, 1, . . . , n − 1, j = 1, 2, . . . , M, i = ( p − 1)n + q + 1. By fractional integrating of order α of (4.8), we will have:
Iα h(x) = Iα ( B(x)) = Iα B(x).
(4.10)
Therefore by (4.6), we get:
Iα h(x) = F α B(x). Thus using (4.10) and (4.5) the operational matrix of fractional integration will be as the following:
Pα = F α −1 .
(4.11)
This matrix is in the following block form:
⎛
[A1 ]
Pα =
⎜ ⎜ ⎜ ⎜ ⎜ ⎝
[A2 ] [A1 ]
[A3 ] [A2 ] [A1 ]
0
... ... .. . .. .
⎞
[Am ] [Am−1 ]⎟ .. ⎟ ⎟ . ⎟, [A2 ] [A1 ]
⎟ ⎠
where each Ai for i = 1, 2, . . . , m is an n × n matrix. Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
ARTICLE IN PRESS
JID: APM
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
5
5. Algorithm of the method This algorithm will convert (1.1) to a linear system of algebraic equations which can be solved by using mathematical softwares such as MATLAB. α +2 ) can be approximated via hybrid functions as the following: According to Section (3), ∂ ∂ t αu∂(xx,t 2
∂ α+2 u(x, t ) = ht (x)Ch(t ), ∂ t α ∂ x2
(5.1)
and we also have:
α α ∂ α−1 ∂ ∂ t t t t = T1 h(t ), t = T2 h(t ), (x − 1) g (t ) = h (x)G1 h(t ), −x g (t ) = ht (x)G2 h(t ), ∂tα 1 ∂tα 2 ∂ t α−1 α−1 ∂ − T (x, t ) = ht (x)T3 h(t ), −T (x, t )) = ht (x)T4 h(t ), x = ht (x)X, (1 − t )l1 (x) = ht (x)L1 h(t ), ∂ t α−1
tl2 (x) = ht (x)L2 h(t ),
f (x, t ) = ht (x)F h(t ).
(5.2)
Now we do the following: 1. Use (4.4) and fractional integration of order α with respect to t for (5.1) to get:
∂ 2 u(x, t ) = ht (x)C (Iα h(t )) + l1 (x) + t[l2 (x) − l1 (x) − ht (x)C (Iα h(1))]. ∂ x2
(5.3)
2. Integrate (5.1) twice with respect to x and use (4.4) to obtain:
∂ α u(x, t ) ∂α ∂α = ht (x)(P2 )t Ch(t ) + α g1 (t ) + x[ α (g2 (t ) − g1 (t )) − ht (1)(P2 )t Ch(t )]. α ∂t ∂t ∂t
(5.4)
3. Integrate (5.3) twice with respect to x and apply (4.4) to have:
u(x, t ) = ht (x)(P2 )t CPα h(t ) − tht (x)(P2 )t CPα h(1) − xht (1)(P2 )t CPα h(t ) + txht (1)(P2 )t CPα h(1) + T (x, t ),
(5.5)
where:
T (x, t ) = g1 (t ) + x[g2 (t ) − g1 (t ) − l1 (1) + l1 (0)] + xt[l2 (0) − l2 (1) + l1 (1) − l1 (0)] + t[l2 (x) − l2 (0) −l1 (x) + l1 (0)] + l1 (x) − l1 (0). 4. Now differentiate (5.5) fractionally of order α − 1 with respect to t, to get:
∂ α−1 u(x, t ) ∂ α−1 = ht (x)(P2 )t CP1 h(t ) − α −1 tht (x)(P2 )t CPα h(1) − xht (1)(P2 )t CP1 h(t ) α −1 ∂t ∂t ∂ α−1 t ∂ α−1 2 t + x α −1 th (1)(P ) CPα h(1) + α −1 T (x, t ). ∂t ∂t
(5.6)
5. Apply (5.2) for (5.3), (5.4), (5.5), (5.6). 6. Substitute (5.3), (5.4), (5.5), (5.6) into (1.1) and simplify, to have:
ht (x)[(P2 )t C + (P2 )t CP1 + (P2 )t CPα − CPα ]h(t ) − ht (x)Xht (1)[(P2 )t C + (P2 )t CP1 + (P2 )t CPα ]h(t )
− ht (x)(P2 )t CPα T1t h(t ) − ht (x)[(P2 )t − I]CPα Pα h(1)T2t h(t ) + ht (x)Xht (1)(P2 )t CPα h(1)(T1t + T2t )h(t ) = ht (x)G1 h(t ) + ht (x)G2 h(t ) + ht (x)T3 h(t ) + ht (x)T4 h(t ) + ht (x)L1 h(t ) + ht (x)L2 h(t ) + ht (x)F h(t ).
7. Solve the following resulted system to obtain the unknown C:
(P2 )t C[I + P1 + Pα − Pα h(1)T1t − Pα h(1)T2t ] − Xht (1)(P2 )t C[I + P1 + Pα − Pα h(1)(T1t + T2t )] − CPα [I − h(1)T2t ] = G1 + G2 + T3 + T4 + L1 + L2 + F. 8. Substitute C in (5.5) to get the approximate solution of (1.1). 6. Error analysis In real problems, we often tend to solve some equations with unknown exact solutions. Hence, when we apply our method to these kinds of problems, it is necessary to introduce a process for estimating the error function. Since um (x, t) is considered as an approximate solution of Eq. (1.1), it satisfies the following problem:
∂α ∂ α−1 ∂2 um (x, t ) + α −1 um (x, t ) + um (x, t ) − 2 um (x, t ) = f (x, t ) + Rm (x, t ). α ∂t ∂t ∂x
(6.1)
Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
JID: APM 6
ARTICLE IN PRESS
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
Fig. 2. The error function for t = 0.1 and some M, with α = 2.
Fig. 3. The error function for t = 0.3 and some M, with α = 2.
The perturbation term Rm (x, t) can be obtained by substituting the estimated solution um (x, t) into the equation:
Rm (x, t ) =
∂α ∂ α−1 ∂2 u (x, t ) + α −1 um (x, t ) + um (x, t ) − 2 um (x, t ) − f (x, t ). ∂tα m ∂t ∂x
(6.2)
Subtracting Eq. (6.1) from Eq. (1.1), we get the following equation:
∂α ∂ α−1 ∂2 e ( x, t ) + e ( x, t ) + e ( x, t ) − e (x, t ) = −Rm (x, t ). m m m ∂tα ∂ t α−1 ∂ x2 m
(6.3)
Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
ARTICLE IN PRESS
JID: APM
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
7
Fig. 4. The error function for t = 0.7 and some M, with α = 2.
Fig. 5. The comparison between approximate solutions of Example 7.1 for t = 0.1 and M = 6, with α = 1.25, 1.5, 1.75, 2.
Obviously the above equation is a time-fractional telegraph equation in which the error function, em (x), is the unknown function. We can easily apply our method to the above equation to find an approximation of the error function em (x).
7. Numerical examples To show the applicability of the suggested method we will employ our method to obtain the approximate solution of the following examples. All of the computations have been done by MATLAB. Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
ARTICLE IN PRESS
JID: APM 8
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
Fig. 6. The comparison between approximate and exact solutions of Example 7.2 for t = 0.6 and M = 6, with α = 2.
Fig. 7. The comparison between approximate and exact solutions of Example 7.2 for t = 0.7 and M = 6, with α = 2.
Note that:
em (x) 2 =
1 0
1/2 e2m
(x)dx
∼ =
1/2
N 1 2 em (xi ) N
,
i=0
where em (x) is the error function as in Section (5). Example 7.1. Consider the following time-fractional telegraph equation:
∂α ∂ α−1 ∂2 u ( x, t ) + u ( x, t ) + u ( x, t ) − u(x, t ) = x2 + t − 1, ∂tα ∂ t α−1 ∂ x2 Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
JID: APM
ARTICLE IN PRESS
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
9
Fig. 8. The comparison between approximate and exact solutions of Example 7.2 for t = 0.8 and M = 6, with α = 2.
Fig. 9. The comparison between approximate and exact solutions of Example 7.2 for t = 1 and M = 6, with α = 2.
with the initial and boundary conditions:
u(x, 0) = x2 , u(0, t ) = t, u(x, 1) = 1 + x2 , u(1, t ) = 1 + t. The exact solution of the above equation when α = 2 is u(x, t ) = x2 + t. The 2-norm of the error, em (x) 2 , for different values of t and M are presented in Table 1 and also there is a comparison between our method and the method of [33]. Fig. 2–4 show the error function for different values of M and t = 0.1, 0.3, 0.7, respectiv ely. There is a comparison between the approximate solutions of 7.1 for different values of α in Fig. 5. Example 7.2. Consider the following telegraph equation of fractional order [46]:
∂α ∂ α−1 ∂2 u ( x, t ) + u ( x, t ) + u ( x, t ) − u(x, t ) = (2 − 2t + t 2 )(x − x2 )e−t + 2t 2 e−t , ∂tα ∂ t α−1 ∂ x2 Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
ARTICLE IN PRESS
JID: APM 10
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
Fig. 10. The comparison between approximate and exact solutions of Example 7.2 for t = 1.1 and M = 6, with α = 2. Table 1 Approximate 2-norm of absolute error, em (x) 2 , for M = 3, 6, 10 and the comparison with [33]. t, M
M=3
M=6
M = 10
M = 12 [33]
t = 0.1 t = 0.3 t = 0.7 t=1
6.8395 ×10−4 6.1598 ×10−4 9.1357 ×10−4 0
3.588 ×10−4 3.6785 ×10−4 4.1076 ×10−4 0
1.4312 ×10−4 2.8533 ×10−4 1.1572 ×10−4 0
8.64 ×10−4 8.06 ×10−4 7.50 ×10−4 0
Table 2 Approximate 2-norm of absolute error, em (x) 2 , for M = 3, 6, 10. t, M
M=3
M=6
M = 10
t = 0.6 t = 0.7 t = 0.8 t=1 t = 1.1
9.6191 ×10−3 1.998 ×10−2 1.2885 ×10−2 0 7.7667 ×10−3
4.4345 ×10−3 2.6738 ×10−3 2.1412 ×10−3 0 4.3717 ×10−3
2.1875 ×10−3 1.4545 ×10−3 1.6187 ×10−3 0 2.3711 ×10−3
with the initial and boundary conditions:
u(x, 0) = 0, u(0, t ) = 0, u(x, 1) = (x − x2 )e−1 , u(1, t ) = 0. The exact solution of the above equation when α = 2 is u(x, t ) = (x − x2 )t 2 e−t . The 2-norm of the error for different values of t and M are presented in Table 2. Figs. 6–10 show the exact and the approximate solutions for different values of M and t. 8. Conclusions Here a new operational method to approximate the solution of time-fractional telegraph equations have been introduced. To this end, the new operational matrix of fractional integration is obtained. It appears that using hybrid functions will give more accurate solutions than other existing methods. A comparison with another method for solving telegraph equations confirms this, i.e. our method, hybrid functions of Legendre polynomials and BPFs, is more accurate. As it can be seen in tables and figures as M increases, the 2-norm of absolute error decreases. Applying operational methods simplifies the problem and reduces the amount of computations (e. g. see [47]). Acknowledgments The authors are highly grateful to editors and reviewers, for their valuable comments and suggestions to improve the paper. Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
JID: APM
ARTICLE IN PRESS N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
[m3Gsc;October 25, 2015;10:23] 11
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47]
W. Hachbusch, Integral equations theory and numerical treatments, ISNM 120, 1995. A. Jeffrey, Applied partial differential equations, Academic Press, New York, 2002. A. Jeffrey, Advanced engineering mathematics, Harcourt Academic Press, 2002. A. Mohebbi, M. Dehghan, High order compact solution of the one space- dimensional linear hyperbolic equation, Numer. Methods Partial Differ. Equ. 24 (2008) 1222–1235. D.M. Pozar, Microwave engineering, Addison-Wesley, New York, 1990. R.K. Mohanty, New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations, Comput. Math. 86 (2008) 2061– 2071. H. Pascal, Pressure wave propagation in a fluid flowing through a porous medium and problems related to interpretation of Stoneley wave attenuation in acoustical well logging, Eng. Sci. 24 (1986) 1553–1570. G. Bohme, Non-Newtonian fluid mechanics, North-Holland, New York, 1987. D.J. Evans, H. Bulut, The numerical solution of the telegraph equation by the alternating group explicit method, Comput. Math. 80 (2003) 1289–1297. P.M. Jordan, M.R. Meyer, A. Puri, Causal implications of viscous damping in compressible fluid flows, Phys. Rev. 62 (2000) 7918–7926. Q. Liu, F. Liu, I. Turner, V. Anh, Y.T. Gu, An RBF meshless approach for modelling a fractal mobile/immobile transport model, Appl. Math. Comput. 226 (2014) 336–347. M.A. Abdou, Adomian decomposition method for solving the telegraph equation in charged particle transport, J. Quant. Spectrosc. Radiat. Transf. 95 (2005) 407–414. T.I. Gombosi, J.R. Jokipii, J. Kota, K. Lorencz, L.L. Williams, The telegraph equation in charged particle transport, Astrophys. J. 403 (1993) 377–384. K. Afrooz, A. Abdipour, J. Rashed-Mohassel, Time domain analysis of transmission line using FDTD excited By modulated signal, in: First Conference on Millimeter-Wave and Terahertz Technologies, 2009. K. Afrooz, A. Abdipour, Efficient method for time domain analysis of Lossy nonuniform multiconductor transmission line driven by modulated signal using FDTD method, IEEE Trans. Electromagn. Compat. 54 (2) (2012) 482–494. Y. Fedorov, B.A. Shakhov, Description of non-diffusive solar cosmic ray propagation in a homogeneous regular magnetic field, Astron. Astrophys. 402 (2003) 805–817. Y. Zhang, J. Qian, C. Papelis, P. Sun, Z. Yu, Improved understanding of bimolecular reactions in deceptively simple homogeneous media: from laboratory experiments to Lagrangian quantification, Water Resour. Res. 50 (2014) 1704–1715. H.G. Sun, W. Chen, C. Li, Y.Q. Chen, Fractional differential models for anomalous diffusion, Phys. A: Stat. Mech. Appl. 389 (2010) 2719–2724. E. Holmes, Are diffusion models too simple? a comparison with telegraph models of invasion, Am. Nat. 142 (5) (1993) 779–795. A. Barletta, E. Zanchini, A thermal potential formulation of hyperbolic heat conduction, Am. Soc. Mech. Eng. J. Heat Transf. 121 (1999) 166–169. K.J. Baumeister, T.D. Hamill, Hyperbolic heat-conduction equation–A solution for the semi-infinite body problem, Am. Soc. Mech. Eng. J. Heat Transf. 91 (1969) 543–548. M. Dehghan, Parameter determination in a partial differential equation from the overspecified data, Math. Comput. Model. 41 (2005) 196–213. M. Dehghan, Implicit collocation technique for heat equation with nonclassic initial condition, Int. J. Nonlinear Sci. Numer. Simulat. 7 (2006) 447–450. M. Dehghan, S.A. Yousefi, A. Lotfi, The use of He’s variational iteration method for solving the telegraph and fractional telegraph equations, Int. J. Numer. Methods Biomed. Eng. 27 (2011) 219–231. M. Dehghan, R. Salehi, A method based on meshless approach for the numerical solution of the two-space dimensional hyperbolic telegraph equation, Math. Methods Appl. Sci. 35 (2012) 1220–1233. F. Gao, C. Chi, Unconditionally stable difference scheme for a one-space dimensional linear hyperbolic equation, Appl. Math. Comput. 187 (2007) 1272–1276. A. Saadatmandi, M. Dehghan, Numerical solution of hyperbolic telegraph equation using the Chebyshev Tau method, Numer. Methods Partial Differ. Equ. 26 (1) (2010) 239–252. S.A. Yousefi, Legendre multi wavelet Galerkin method for solving the hyperbolic telegraph equation, Numer. Methods Partial Differ. Equ. 26 (3) (2010) 535–543. M. Dehghan, A. Ghesmati, Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method, Eng. Anal. Boundary Elem. 34 (2010) 51–59. S. Das, P.K. Gupta, Homotopy analysis method for solving fractional hyperbolic partial differential equations, Int. J. Comput. Math. 88 (2011) 578–588. M. Lakestani, B.N. Saray, Numerical solution of telegraph equation using interpolating scaling functions, Comput. Math. Appl. 60 (2010) 1964–1972. A. Borhanifar, R. Abazari, An unconditionally stable parallel difference scheme for telegraph equation scheme for telegraph equation, Math. Prob. Eng. 2009 (2009) 1–17. A. Sadeghian, S. Karbassi, M. Hushmandasl, M. Heydari, Numerical solution of time-fractional telegraph equation by Chebyshev wavelet method, Int. J. Theoret. Appl. Phys. 2 (II) (2012) 163–181. F. Liu, P. Zhuang, I. Turner, K. Burrage, V. Anh, A new fractional finite volume method for solving the fractional diffusion equation, Appl. Math. Model. 38 (2014) 3871–3878. H. Jiang, F. Liu, I. Turner, K. Burrage, Analytical solution for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain, Comput. Math. Appl. 64 (2012) 3377–3388. A. Saadatmandi, M. Dehghan, A Tau approach for solution of the space fractional diffusion equation, Comput. Math. Appl. 62 (2011) 1135–1142. A. Mohebbi, M. Abbaszadeh, M. Dehghan, Compact finite difference schem and RBF meshless approach for solving 2D Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives, Comput. Methods Appl. Mech. Eng. 264 (2013) 163–177. I. Podlubny, Fractional differential equations, Academic press, New York, 1999. K. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, 1993. R.C. Cascaval, E.C. Eckstein, C.L. Frota, J.A. Goldstein, Fractional telegraph equations, J.Math. Anal. Appl. 276 (1) (2002) 145–159. E. Orsingher, L. Beghin, Time-fractional telegraph equations and telegraph processes with Brownian time, Probab. Theory Rel. Fields 128 (1) (2004) 141–160. J. Chen, F. Liu, V. Anh, Analytical solution for the time-fractional telegraph equation by the method of separating variables, J. Math. Anal. Appl. 338 (2) (2008) 1364–1377. R.K. Saxena, A.M. Mathai, H.J. Haubold, Reaction-diffusion systems and nonlinear waves, Astrophys. Space Sci. 305 (2006) 297–303. W. Zhang, J. Li, Y. Yang, Spatial fractional telegraph equation for image structure preserving denoising, Signal Process. 107 (2015) 368–377. C. Wang, On the generalization of Block Pulse operational matrices for fractional and operational calculus, J. Franklin Inst. 315 (2) (1983) 91–102. M. Dehghan, A. Shokri, A numerical method for solving the hyperbolic telegraph equation, Numer. Methods Partial Differ. Equ. 24 (2008) 1080–1093. M.R. Eslahchi, M. Dehghan, Application of Taylor series in obtaining the orthogonal operational matrix, Comput. Math. Appl. 61 (2011) 2596–2604.
Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
JID: APM
[m3Gsc;October 25, 2015;10:23]
Applied Mathematical Modelling 000 (2015) 1–11
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
A new treatment based on hybrid functions to the solution of telegraph equations of fractional order N. Mollahasani a,∗, M. Mohseni (Mohseni) Moghadam a, K. Afrooz b a b
Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran Department of Electrical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran
a r t i c l e
i n f o
Article history: Received 1 September 2013 Revised 18 January 2015 Accepted 26 August 2015 Available online xxx
a b s t r a c t In this paper, a new operational method based on hybrid functions of Legendre polynomials and Block-Pulse-Functions will be presented. The operational matrix of fractional integration is derived and used to take an acceptable approximate for the solution of a telegraph equation of fractional order. An error estimation will be presented to give an image of the goodness of the solution. Some numerical examples demonstrate the efficiency of the proposed method.
Keywords: Fractional telegraph equation Hybrid functions Fractional calculus Operational matrices
© 2015 Elsevier Inc. All rights reserved.
1. Introduction A class of hyperbolic partial differential equations which describes vibrations within objects and how waves are propagated, is called Telegraph equation [1]. Equations of the form of telegraph equations arise in the study of propagation of electrical signals in a cable of transmission line and wave phenomena. Interaction between convection and diffusion or reciprocal action of reaction and diffusion describes a number of nonlinear phenomena in physical, chemical and biological process [2–5]. In fact, the telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion for such branches of sciences. For example, biologists encounter these equations in the study of pulsatileblood flow in arteries and in one-dimensional random motion of bugs along a hedge [6]. Also the propagation of acoustic waves in Darcy-type porous media [7], and parallel flows of viscous Maxwell fluids [8] are just some of the phenomena governed by the telegraph equations [9–11]. The telegraph equation has also been employed in other areas. For example, it is used as a replacement for the diffusion equation to model transport of charged particles [12,13], high frequency transmission lines [14,15], solar cosmic rays [16], chemical diffusion, anomalous diffusion [17,18], hydrology and population dynamics [19]. It is also employed in the theory of hyperbolic heat transfer [20,21]. Finite difference methods are known as the first techniques for solving partial differential equations [22,23]. Even though these methods are very effective for solving various kinds of partial differential equations. Many researchers have used various numerical and analytical methods to solve the telegraph equation [24,25]. Unconditionally stable and parallel difference scheme, Chebyshev Tau method, Chebyshev method, Legendre multi-wavelet Galerkin method, meshless local weak– strong methods, homotopy analysis and Adomian decomposition are such methods [12,15,26–36]. The fractional calculus is one of the most accurate tools to refine the description of natural phenomena [37]. Fractional differential equations have attracted in the recent years a considerable interest due to their frequent appearance in various fields ∗
Corresponding author. Tel.: +983432475898. E-mail address: [email protected], [email protected] (N. Mollahasani).
http://dx.doi.org/10.1016/j.apm.2015.08.020 S0307-904X(15)00544-2/© 2015 Elsevier Inc. All rights reserved.
Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
ARTICLE IN PRESS
JID: APM 2
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
and their more accurate models of systems under consideration provided by fractional derivatives [38,39]. Fractional telegraph equations has been recently considered by many authors. Cascaval et al. [40] discussed the time-fractional telegraph equations, dealing with well posedness and presenting a study of their asymptotic behavior by using the Riemann– Liouville approach. Orsingher and Beghin [41] discussed the time-fractional telegraph equation and telegraph processes with Brownian time, showing that some processes are governed by time-fractional telegraph equations. Chen et al. [42] examined and derived a solution of the time-fractional telegraph equation with three kinds of nonhomogeneous boundary conditions, by the method of separation of variables. Fractional telegraph equations from the analytic point of view have been studied by many authors (see Saxena et al. [43] for equations with n time derivatives). The aim of this paper is to introduce a new method for approximating the solution of a time-fractional telegraph equation in the following form:
∂α ∂ α−1 ∂2 u(x, t ) + α −1 u(x, t ) + u(x, t ) − u(x, t ) = f (x, t ), ∂tα ∂t ∂ x2
(1.1)
with the initial and boundary conditions:
u(x, 0) = l1 (x),
u(0, t ) = g1 (t ),
u(x, 1) = l2 (x),
u(1, t ) = g2 (t ).
(1.2)
The right-hand-side function f(x, t) is given and 1 < α ≤ 2 is a real number. In this paper, first hybrid functions of Legendre polynomials and Block-Pulse-functions are introduced and a new operational matrix of fractional integration is constructed. Then a numerical method to solve Eq. (1.1) will be developed. Also some error analysis will be presented. Finally, some numerical examples are presented to confirm the applicability of the method. 2. Real-world application of telegraph equations In the following we mention some real-world applications of telegraph equations. (1) Consider the following fractional telegraph equation [41]:
∂ 2α u(x, t ) ∂ 2 u(x, t ) ∂ α u(x, t ) + 2λ = c2 , 0 < α ≤ 1, α 2 α ∂t ∂t ∂ x2
(2.1)
with the conditions:
1 : 2 ≤1:
0<α≤ 1 2
<α
u(x, 0) = δ(x), ut (x, 0) = 0.
Here, we consider the case α = 12 for which it is possible to obtain the solution. The probability density of the solution of (2.1) coincides with the distribution of the telegraph process T = T (t ), t > 0 with a Brownian time, that is:
W (t ) = T (|B(t )|). This means that the fundamental solution to the fractional Eq. (2.1) for α = 12 can be interpreted as the distribution of a particle moving back and forth on the real line with velocities ±c (switching of Poisson-paced time) for a random time interval of length |B(t)|. Clearly, B and T are independent of each other. Eq. (2.1) for α = 12 is a heat equation with a damping term depending on all 1
1
values of u in [0, t] and assigning an overwhelming weight to those close to t. The damping effect of ∂ 2 u/∂ t 2 reverberates on the probability density of the solution, where the governing term (solution to the heat equation) is weighted by the telegraph distribution (representing the impact of the fractional derivative). (2) The transmission line equation can be represented as:
∂ V (z, t ) ∂ I(z, t ) = −RI(z, t ) − L , ∂z ∂t
(2.2)
∂ I(z, t ) ∂ V (z, t ) = −GV (z, t ) − C , ∂z ∂t
(2.3)
where R, G, C, and L are the per-unit-length parameters of the transmission line [5]. The V(z, t) and I(z, t) are the line voltages (with respect to the reference line) and line currents, respectively. Fig. (1) shows a segment of a typical transmission line. Differentiating (2.2) and (2.3) with respect to z, we get:
∂ 2V (z, t ) ∂ 2V (z, t ) ∂ V (z, t ) + LC = RGV + ( RC + LG ) , ∂t ∂ z2 ∂t2
(2.4)
∂ 2 I(z, t ) ∂ 2 I(z, t ) ∂ I(z, t ) + LC = RGI + (RC + LG) , 2 ∂t ∂z ∂t2
(2.5)
which are two telegraph equations for α = 2. In [14], the authors have solved the following example. Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
ARTICLE IN PRESS
JID: APM
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
3
Fig. 1. A segment of a typical transmission line.
Example 2.1. The line considered here is a one conductor lossy transmission line whose parameters are L = 309nH, C = 144pF, R = 524 × 10−3 , G = 905 × 10−9 S. The total line length is 10cm. (3) Image denoising is often used for pre-processing images, so that subsequent image analysis is more reliable. The goal of image denoising is to remove the noise from the observed noisy image and get as close as possible to the original image. Besides noise removal ability, another important requirement for image denoising procedures is that true image structures should be preserved in the denoising process. The use of partial differential equation(PDE) has been studied as a useful tool for image denoising. In [44], Zhang, et al. have proposed a three dimensional fractional telegraph equation for image structure preserving denoising. 3. Function approximation The complete orthogonal set of the hybrid functions of Legendre polynomials and Block Pulse functions(BPFs) hpq (x), is defined on [0, 1) as follows:
h pq (x) =
Lq (2mx − 2p + 1),
p−1 m
0,
elsewhere,
≤x<
p , m
(3.1)
for p = 1, 2, . . . , m, q = 0, 1, . . . , n − 1, where p is the order of BPFs and q is the order of Legendre polynomials. Here, Legendre polynomials are defined on [−1, 1] by:
Lk (x) = 2k
k
k i
xi
i=0
k+i−1 2
k
,
(3.2)
for k = 0, 1, ... or in the recursive form:
L0 (x) = 1,
L1 (x) = x,
(k + 1)Lk+1 (x) = (2k + 1)xLk (x) − kLk−1 (x), k = 1, 2, ...
(3.3)
The set {Lk (x) : k = 0, 1, ...} in L2 [−1, 1] is a complete orthogonal system. A set of BPFs on [0, 1) is defined as the following:
bi (x) =
1,
i−1 n
≤ x < ni ;
0,
elsewhere,
(3.4)
for i = 1, 2, .... This set is also a set of orthogonal functions. Suppose that nm = M. Therefore, after renaming we have:
h(x) = [h1 (x), h2 (x), . . . , hM (x)]t , B(x) = [b1 (x), b2 (x), . . . , bM (x)]t . Each square integrable function f(x, t) in
f (x, t ) =
∞ ∞
ci j hi (x)h j (t ),
L2 ([0,
(3.5)
1) × [0, 1)) can be expanded into a hybrid series of infinite terms:
x ∈ [0, 1],
(3.6)
i=1 j=1
where the hybrid coefficients are determined as:
ci j =
(hi (x), ( f (x, t ), h j (t ))) , (hi (x), hi (x))(h j (t ), h j (t ))
for i, j = 1, 2, . . . and (., .) denotes the inner product. If f(x, t) is a piecewise constant function or may be approximated as a piecewise constant function on each subinterval, the series sum in Eq. (3.6) can be truncated, that is:
f (x, t ) ∼ =
M M
ci j hi (x)h j (t ) = fm (x, t ) = ht (x)Ch(t ),
x ∈ [0, 1],
(3.7)
i=1 j=1
where C is an M × M matrix and h(x) is defined as (3.5). Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
ARTICLE IN PRESS
JID: APM 4
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
4. Hybrid operational matrix of fractional integration In this section, we obtain an operational matrix of fractional integration, based on hybrid functions. The Riemann– Liouville fractional integral operator of order α > 0 of the function f(x) is defined as [38,39]:
Iα f (x) =
1
(α)
x 0
(x − t )α−1 f (t )dt, α > 0, x > 0,
(4.1)
where (.) is the so called Gamma function with the property: (x + 1) = x (x), x ∈ R. Some properties of the operator Iα can be found in [38]. We mention only the following. For α ≥ 0 and γ > −1, we have:
I α xγ =
(γ + 1) α+γ x . (α + γ + 1)
(4.2)
The Caputo fractional derivative of order α is defined as:
Dα f (x) =
1 (m − α)
t 0
(t − τ )m−α−1 f (m) (τ )dτ , m − 1 < α ≤ m,
(4.3)
where m is an integer. This definition of derivative satisfies in the following equality:
Iα Dα f (x) = f (x) −
m−1
f (k) (0+ )
k=0
tk , k!
m − 1 < α ≤ m.
(4.4)
The fractional integration of order α of h(x) can be expanded into hybrid series with hybrid coefficient matrix Pα as follows:
Iα h(x) =
1
(α)
x 0
(x − t )α−1 h(t )dt Pα h(x).
(4.5)
We call this M × M square matrix Pα the (generalized) operational matrix of fractional integration. Wang Chi-Hsu in [45] showed that the generalized Block Pulse operational Matrix Fα , is derived as:
Iα B(x) F α B(x), where:
Fα =
1 α M
(4.6)
⎛f
1
⎜ 1 ⎜ (α + 2) ⎝
f2 f1
... ... ... ...
f3 f2 f1
0
fM
⎞
fM−1 ⎟ fM−2 ⎟,
(4.7)
⎠
f1 in which f1 = 1, fk = kα +1 − 2(k − 1)α +1 + (k − 2)α +1 for (k = 2, 3, . . . , M − i + 1) and i = 1, 2, . . . , M. On the other hand, we can expand hybrid functions via BPFs by:
h(x) = B(x),
(4.8)
where = (φi j ) and:
φi j = n2
q−1
q k=0
1 q k+1 k
q+k−1 2
q
2 j − 2p + 1 n
k+1
−
2 n
( j − 1) − 2p + 1
k+1
,
(4.9)
for p = 1, 2, . . . , m, q = 0, 1, . . . , n − 1, j = 1, 2, . . . , M, i = ( p − 1)n + q + 1. By fractional integrating of order α of (4.8), we will have:
Iα h(x) = Iα ( B(x)) = Iα B(x).
(4.10)
Therefore by (4.6), we get:
Iα h(x) = F α B(x). Thus using (4.10) and (4.5) the operational matrix of fractional integration will be as the following:
Pα = F α −1 .
(4.11)
This matrix is in the following block form:
⎛
[A1 ]
Pα =
⎜ ⎜ ⎜ ⎜ ⎜ ⎝
[A2 ] [A1 ]
[A3 ] [A2 ] [A1 ]
0
... ... .. . .. .
⎞
[Am ] [Am−1 ]⎟ .. ⎟ ⎟ . ⎟, [A2 ] [A1 ]
⎟ ⎠
where each Ai for i = 1, 2, . . . , m is an n × n matrix. Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
ARTICLE IN PRESS
JID: APM
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
5
5. Algorithm of the method This algorithm will convert (1.1) to a linear system of algebraic equations which can be solved by using mathematical softwares such as MATLAB. α +2 ) can be approximated via hybrid functions as the following: According to Section (3), ∂ ∂ t αu∂(xx,t 2
∂ α+2 u(x, t ) = ht (x)Ch(t ), ∂ t α ∂ x2
(5.1)
and we also have:
α α ∂ α−1 ∂ ∂ t t t t = T1 h(t ), t = T2 h(t ), (x − 1) g (t ) = h (x)G1 h(t ), −x g (t ) = ht (x)G2 h(t ), ∂tα 1 ∂tα 2 ∂ t α−1 α−1 ∂ − T (x, t ) = ht (x)T3 h(t ), −T (x, t )) = ht (x)T4 h(t ), x = ht (x)X, (1 − t )l1 (x) = ht (x)L1 h(t ), ∂ t α−1
tl2 (x) = ht (x)L2 h(t ),
f (x, t ) = ht (x)F h(t ).
(5.2)
Now we do the following: 1. Use (4.4) and fractional integration of order α with respect to t for (5.1) to get:
∂ 2 u(x, t ) = ht (x)C (Iα h(t )) + l1 (x) + t[l2 (x) − l1 (x) − ht (x)C (Iα h(1))]. ∂ x2
(5.3)
2. Integrate (5.1) twice with respect to x and use (4.4) to obtain:
∂ α u(x, t ) ∂α ∂α = ht (x)(P2 )t Ch(t ) + α g1 (t ) + x[ α (g2 (t ) − g1 (t )) − ht (1)(P2 )t Ch(t )]. α ∂t ∂t ∂t
(5.4)
3. Integrate (5.3) twice with respect to x and apply (4.4) to have:
u(x, t ) = ht (x)(P2 )t CPα h(t ) − tht (x)(P2 )t CPα h(1) − xht (1)(P2 )t CPα h(t ) + txht (1)(P2 )t CPα h(1) + T (x, t ),
(5.5)
where:
T (x, t ) = g1 (t ) + x[g2 (t ) − g1 (t ) − l1 (1) + l1 (0)] + xt[l2 (0) − l2 (1) + l1 (1) − l1 (0)] + t[l2 (x) − l2 (0) −l1 (x) + l1 (0)] + l1 (x) − l1 (0). 4. Now differentiate (5.5) fractionally of order α − 1 with respect to t, to get:
∂ α−1 u(x, t ) ∂ α−1 = ht (x)(P2 )t CP1 h(t ) − α −1 tht (x)(P2 )t CPα h(1) − xht (1)(P2 )t CP1 h(t ) α −1 ∂t ∂t ∂ α−1 t ∂ α−1 2 t + x α −1 th (1)(P ) CPα h(1) + α −1 T (x, t ). ∂t ∂t
(5.6)
5. Apply (5.2) for (5.3), (5.4), (5.5), (5.6). 6. Substitute (5.3), (5.4), (5.5), (5.6) into (1.1) and simplify, to have:
ht (x)[(P2 )t C + (P2 )t CP1 + (P2 )t CPα − CPα ]h(t ) − ht (x)Xht (1)[(P2 )t C + (P2 )t CP1 + (P2 )t CPα ]h(t )
− ht (x)(P2 )t CPα T1t h(t ) − ht (x)[(P2 )t − I]CPα Pα h(1)T2t h(t ) + ht (x)Xht (1)(P2 )t CPα h(1)(T1t + T2t )h(t ) = ht (x)G1 h(t ) + ht (x)G2 h(t ) + ht (x)T3 h(t ) + ht (x)T4 h(t ) + ht (x)L1 h(t ) + ht (x)L2 h(t ) + ht (x)F h(t ).
7. Solve the following resulted system to obtain the unknown C:
(P2 )t C[I + P1 + Pα − Pα h(1)T1t − Pα h(1)T2t ] − Xht (1)(P2 )t C[I + P1 + Pα − Pα h(1)(T1t + T2t )] − CPα [I − h(1)T2t ] = G1 + G2 + T3 + T4 + L1 + L2 + F. 8. Substitute C in (5.5) to get the approximate solution of (1.1). 6. Error analysis In real problems, we often tend to solve some equations with unknown exact solutions. Hence, when we apply our method to these kinds of problems, it is necessary to introduce a process for estimating the error function. Since um (x, t) is considered as an approximate solution of Eq. (1.1), it satisfies the following problem:
∂α ∂ α−1 ∂2 um (x, t ) + α −1 um (x, t ) + um (x, t ) − 2 um (x, t ) = f (x, t ) + Rm (x, t ). α ∂t ∂t ∂x
(6.1)
Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
JID: APM 6
ARTICLE IN PRESS
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
Fig. 2. The error function for t = 0.1 and some M, with α = 2.
Fig. 3. The error function for t = 0.3 and some M, with α = 2.
The perturbation term Rm (x, t) can be obtained by substituting the estimated solution um (x, t) into the equation:
Rm (x, t ) =
∂α ∂ α−1 ∂2 u (x, t ) + α −1 um (x, t ) + um (x, t ) − 2 um (x, t ) − f (x, t ). ∂tα m ∂t ∂x
(6.2)
Subtracting Eq. (6.1) from Eq. (1.1), we get the following equation:
∂α ∂ α−1 ∂2 e ( x, t ) + e ( x, t ) + e ( x, t ) − e (x, t ) = −Rm (x, t ). m m m ∂tα ∂ t α−1 ∂ x2 m
(6.3)
Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
ARTICLE IN PRESS
JID: APM
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
7
Fig. 4. The error function for t = 0.7 and some M, with α = 2.
Fig. 5. The comparison between approximate solutions of Example 7.1 for t = 0.1 and M = 6, with α = 1.25, 1.5, 1.75, 2.
Obviously the above equation is a time-fractional telegraph equation in which the error function, em (x), is the unknown function. We can easily apply our method to the above equation to find an approximation of the error function em (x).
7. Numerical examples To show the applicability of the suggested method we will employ our method to obtain the approximate solution of the following examples. All of the computations have been done by MATLAB. Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
ARTICLE IN PRESS
JID: APM 8
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
Fig. 6. The comparison between approximate and exact solutions of Example 7.2 for t = 0.6 and M = 6, with α = 2.
Fig. 7. The comparison between approximate and exact solutions of Example 7.2 for t = 0.7 and M = 6, with α = 2.
Note that:
em (x) 2 =
1 0
1/2 e2m
(x)dx
∼ =
1/2
N 1 2 em (xi ) N
,
i=0
where em (x) is the error function as in Section (5). Example 7.1. Consider the following time-fractional telegraph equation:
∂α ∂ α−1 ∂2 u ( x, t ) + u ( x, t ) + u ( x, t ) − u(x, t ) = x2 + t − 1, ∂tα ∂ t α−1 ∂ x2 Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
JID: APM
ARTICLE IN PRESS
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
9
Fig. 8. The comparison between approximate and exact solutions of Example 7.2 for t = 0.8 and M = 6, with α = 2.
Fig. 9. The comparison between approximate and exact solutions of Example 7.2 for t = 1 and M = 6, with α = 2.
with the initial and boundary conditions:
u(x, 0) = x2 , u(0, t ) = t, u(x, 1) = 1 + x2 , u(1, t ) = 1 + t. The exact solution of the above equation when α = 2 is u(x, t ) = x2 + t. The 2-norm of the error, em (x) 2 , for different values of t and M are presented in Table 1 and also there is a comparison between our method and the method of [33]. Fig. 2–4 show the error function for different values of M and t = 0.1, 0.3, 0.7, respectiv ely. There is a comparison between the approximate solutions of 7.1 for different values of α in Fig. 5. Example 7.2. Consider the following telegraph equation of fractional order [46]:
∂α ∂ α−1 ∂2 u ( x, t ) + u ( x, t ) + u ( x, t ) − u(x, t ) = (2 − 2t + t 2 )(x − x2 )e−t + 2t 2 e−t , ∂tα ∂ t α−1 ∂ x2 Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
ARTICLE IN PRESS
JID: APM 10
[m3Gsc;October 25, 2015;10:23]
N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
Fig. 10. The comparison between approximate and exact solutions of Example 7.2 for t = 1.1 and M = 6, with α = 2. Table 1 Approximate 2-norm of absolute error, em (x) 2 , for M = 3, 6, 10 and the comparison with [33]. t, M
M=3
M=6
M = 10
M = 12 [33]
t = 0.1 t = 0.3 t = 0.7 t=1
6.8395 ×10−4 6.1598 ×10−4 9.1357 ×10−4 0
3.588 ×10−4 3.6785 ×10−4 4.1076 ×10−4 0
1.4312 ×10−4 2.8533 ×10−4 1.1572 ×10−4 0
8.64 ×10−4 8.06 ×10−4 7.50 ×10−4 0
Table 2 Approximate 2-norm of absolute error, em (x) 2 , for M = 3, 6, 10. t, M
M=3
M=6
M = 10
t = 0.6 t = 0.7 t = 0.8 t=1 t = 1.1
9.6191 ×10−3 1.998 ×10−2 1.2885 ×10−2 0 7.7667 ×10−3
4.4345 ×10−3 2.6738 ×10−3 2.1412 ×10−3 0 4.3717 ×10−3
2.1875 ×10−3 1.4545 ×10−3 1.6187 ×10−3 0 2.3711 ×10−3
with the initial and boundary conditions:
u(x, 0) = 0, u(0, t ) = 0, u(x, 1) = (x − x2 )e−1 , u(1, t ) = 0. The exact solution of the above equation when α = 2 is u(x, t ) = (x − x2 )t 2 e−t . The 2-norm of the error for different values of t and M are presented in Table 2. Figs. 6–10 show the exact and the approximate solutions for different values of M and t. 8. Conclusions Here a new operational method to approximate the solution of time-fractional telegraph equations have been introduced. To this end, the new operational matrix of fractional integration is obtained. It appears that using hybrid functions will give more accurate solutions than other existing methods. A comparison with another method for solving telegraph equations confirms this, i.e. our method, hybrid functions of Legendre polynomials and BPFs, is more accurate. As it can be seen in tables and figures as M increases, the 2-norm of absolute error decreases. Applying operational methods simplifies the problem and reduces the amount of computations (e. g. see [47]). Acknowledgments The authors are highly grateful to editors and reviewers, for their valuable comments and suggestions to improve the paper. Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020
JID: APM
ARTICLE IN PRESS N. Mollahasani et al. / Applied Mathematical Modelling 000 (2015) 1–11
[m3Gsc;October 25, 2015;10:23] 11
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47]
W. Hachbusch, Integral equations theory and numerical treatments, ISNM 120, 1995. A. Jeffrey, Applied partial differential equations, Academic Press, New York, 2002. A. Jeffrey, Advanced engineering mathematics, Harcourt Academic Press, 2002. A. Mohebbi, M. Dehghan, High order compact solution of the one space- dimensional linear hyperbolic equation, Numer. Methods Partial Differ. Equ. 24 (2008) 1222–1235. D.M. Pozar, Microwave engineering, Addison-Wesley, New York, 1990. R.K. Mohanty, New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations, Comput. Math. 86 (2008) 2061– 2071. H. Pascal, Pressure wave propagation in a fluid flowing through a porous medium and problems related to interpretation of Stoneley wave attenuation in acoustical well logging, Eng. Sci. 24 (1986) 1553–1570. G. Bohme, Non-Newtonian fluid mechanics, North-Holland, New York, 1987. D.J. Evans, H. Bulut, The numerical solution of the telegraph equation by the alternating group explicit method, Comput. Math. 80 (2003) 1289–1297. P.M. Jordan, M.R. Meyer, A. Puri, Causal implications of viscous damping in compressible fluid flows, Phys. Rev. 62 (2000) 7918–7926. Q. Liu, F. Liu, I. Turner, V. Anh, Y.T. Gu, An RBF meshless approach for modelling a fractal mobile/immobile transport model, Appl. Math. Comput. 226 (2014) 336–347. M.A. Abdou, Adomian decomposition method for solving the telegraph equation in charged particle transport, J. Quant. Spectrosc. Radiat. Transf. 95 (2005) 407–414. T.I. Gombosi, J.R. Jokipii, J. Kota, K. Lorencz, L.L. Williams, The telegraph equation in charged particle transport, Astrophys. J. 403 (1993) 377–384. K. Afrooz, A. Abdipour, J. Rashed-Mohassel, Time domain analysis of transmission line using FDTD excited By modulated signal, in: First Conference on Millimeter-Wave and Terahertz Technologies, 2009. K. Afrooz, A. Abdipour, Efficient method for time domain analysis of Lossy nonuniform multiconductor transmission line driven by modulated signal using FDTD method, IEEE Trans. Electromagn. Compat. 54 (2) (2012) 482–494. Y. Fedorov, B.A. Shakhov, Description of non-diffusive solar cosmic ray propagation in a homogeneous regular magnetic field, Astron. Astrophys. 402 (2003) 805–817. Y. Zhang, J. Qian, C. Papelis, P. Sun, Z. Yu, Improved understanding of bimolecular reactions in deceptively simple homogeneous media: from laboratory experiments to Lagrangian quantification, Water Resour. Res. 50 (2014) 1704–1715. H.G. Sun, W. Chen, C. Li, Y.Q. Chen, Fractional differential models for anomalous diffusion, Phys. A: Stat. Mech. Appl. 389 (2010) 2719–2724. E. Holmes, Are diffusion models too simple? a comparison with telegraph models of invasion, Am. Nat. 142 (5) (1993) 779–795. A. Barletta, E. Zanchini, A thermal potential formulation of hyperbolic heat conduction, Am. Soc. Mech. Eng. J. Heat Transf. 121 (1999) 166–169. K.J. Baumeister, T.D. Hamill, Hyperbolic heat-conduction equation–A solution for the semi-infinite body problem, Am. Soc. Mech. Eng. J. Heat Transf. 91 (1969) 543–548. M. Dehghan, Parameter determination in a partial differential equation from the overspecified data, Math. Comput. Model. 41 (2005) 196–213. M. Dehghan, Implicit collocation technique for heat equation with nonclassic initial condition, Int. J. Nonlinear Sci. Numer. Simulat. 7 (2006) 447–450. M. Dehghan, S.A. Yousefi, A. Lotfi, The use of He’s variational iteration method for solving the telegraph and fractional telegraph equations, Int. J. Numer. Methods Biomed. Eng. 27 (2011) 219–231. M. Dehghan, R. Salehi, A method based on meshless approach for the numerical solution of the two-space dimensional hyperbolic telegraph equation, Math. Methods Appl. Sci. 35 (2012) 1220–1233. F. Gao, C. Chi, Unconditionally stable difference scheme for a one-space dimensional linear hyperbolic equation, Appl. Math. Comput. 187 (2007) 1272–1276. A. Saadatmandi, M. Dehghan, Numerical solution of hyperbolic telegraph equation using the Chebyshev Tau method, Numer. Methods Partial Differ. Equ. 26 (1) (2010) 239–252. S.A. Yousefi, Legendre multi wavelet Galerkin method for solving the hyperbolic telegraph equation, Numer. Methods Partial Differ. Equ. 26 (3) (2010) 535–543. M. Dehghan, A. Ghesmati, Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method, Eng. Anal. Boundary Elem. 34 (2010) 51–59. S. Das, P.K. Gupta, Homotopy analysis method for solving fractional hyperbolic partial differential equations, Int. J. Comput. Math. 88 (2011) 578–588. M. Lakestani, B.N. Saray, Numerical solution of telegraph equation using interpolating scaling functions, Comput. Math. Appl. 60 (2010) 1964–1972. A. Borhanifar, R. Abazari, An unconditionally stable parallel difference scheme for telegraph equation scheme for telegraph equation, Math. Prob. Eng. 2009 (2009) 1–17. A. Sadeghian, S. Karbassi, M. Hushmandasl, M. Heydari, Numerical solution of time-fractional telegraph equation by Chebyshev wavelet method, Int. J. Theoret. Appl. Phys. 2 (II) (2012) 163–181. F. Liu, P. Zhuang, I. Turner, K. Burrage, V. Anh, A new fractional finite volume method for solving the fractional diffusion equation, Appl. Math. Model. 38 (2014) 3871–3878. H. Jiang, F. Liu, I. Turner, K. Burrage, Analytical solution for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain, Comput. Math. Appl. 64 (2012) 3377–3388. A. Saadatmandi, M. Dehghan, A Tau approach for solution of the space fractional diffusion equation, Comput. Math. Appl. 62 (2011) 1135–1142. A. Mohebbi, M. Abbaszadeh, M. Dehghan, Compact finite difference schem and RBF meshless approach for solving 2D Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives, Comput. Methods Appl. Mech. Eng. 264 (2013) 163–177. I. Podlubny, Fractional differential equations, Academic press, New York, 1999. K. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, 1993. R.C. Cascaval, E.C. Eckstein, C.L. Frota, J.A. Goldstein, Fractional telegraph equations, J.Math. Anal. Appl. 276 (1) (2002) 145–159. E. Orsingher, L. Beghin, Time-fractional telegraph equations and telegraph processes with Brownian time, Probab. Theory Rel. Fields 128 (1) (2004) 141–160. J. Chen, F. Liu, V. Anh, Analytical solution for the time-fractional telegraph equation by the method of separating variables, J. Math. Anal. Appl. 338 (2) (2008) 1364–1377. R.K. Saxena, A.M. Mathai, H.J. Haubold, Reaction-diffusion systems and nonlinear waves, Astrophys. Space Sci. 305 (2006) 297–303. W. Zhang, J. Li, Y. Yang, Spatial fractional telegraph equation for image structure preserving denoising, Signal Process. 107 (2015) 368–377. C. Wang, On the generalization of Block Pulse operational matrices for fractional and operational calculus, J. Franklin Inst. 315 (2) (1983) 91–102. M. Dehghan, A. Shokri, A numerical method for solving the hyperbolic telegraph equation, Numer. Methods Partial Differ. Equ. 24 (2008) 1080–1093. M.R. Eslahchi, M. Dehghan, Application of Taylor series in obtaining the orthogonal operational matrix, Comput. Math. Appl. 61 (2011) 2596–2604.
Please cite this article as: N. Mollahasani et al., A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.08.020