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Application of Bessel functions for solving differential and integro-differential equations of the fractional order
Applied Mathematical Modelling xxx (2014) xxx–xxx
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Application of Bessel functions for solving differential and integro-differential equations of the fractional order q K. Parand ⇑, M. Nikarya Department of Computer Sciences, Shahid Beheshti University, G.C., Tehran, Iran
a r t i c l e
i n f o
Article history: Received 15 October 2012 Received in revised form 18 June 2013 Accepted 3 February 2014 Available online xxxx Keywords: Fractional differential equation Fractional integro-differential equation Bessel functions Collocation algorithm Non-linear
a b s t r a c t In this paper, a new numerical algorithm to solve the linear and nonlinear fractional differential equations (FDE) is introduced. Fractional calculus and fractional differential equations have many applications in physics, chemistry, engineering, finance, and other sciences. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, which is convergent for any x 2 R. In this method, we reduce the solution of a nonlinear fractional problem to the solution of a system of the nonlinear algebraic equations. To illustrate the reliability of this method, we solve some important equations of fractional order, and present numerical results of the present method to show convergence rate, applicability and reliability of this method. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Although Fractional calculus is an ancient mathematical topic, in the last few decades fractional calculus and fractional differential equations have found applications in different branches of sciences, and scientists are attracted to study fractional differential equations in physics, chemistry, engineering, finance, and other sciences more than ever; for instance, see [1–5]. Most results about solving fractional differential equations are obtained by using numerical methods, because only some particular fractional differential equations can be solved analytically [6–11]. Due to the growing applications, considerable attention has been given to the exact and numerical solutions of fractional differential equations (FDE). In this paper, we attempt to introduce a new method, based on Bessel functions of the first kind for solving FDEs. Previously, many of researchers study FDEs and attempt to solve them by utilizing several techniques of spectral methods, for example, Wang and Fan by using Chebyshev wavelet method[9], Rehman and Khan by using Legendre wavelet method[10], Doha and Bhrawy by using tau method chebyshev [12], Esmaeili and Shamsi by using pseudo-spectral method [13], Pedas and Tamme by using spline collocation methods [14] as well as many other researchers [15–20]. In this paper, we aim to solve FDEs by applying the Bessel functions and spectral collocation methods. Spectral methods are among the strongest methods for solving differential and integral equations.formerly, Bessel functions and Bessel polynomials collocation method were used for solving several type of problems, such as Blasius equation, Lane-Emden type equations, etc. [21–25] and the researchers obtained the excellent results, therefore, we decided to use the Bessel function collocation (BFC) method to solve linear and nonlinear differential and integro-differential equation of fractional order.
q
Member of research group of Scientific Computing.
⇑ Corresponding author. Tel.: +98 21 22431653; fax: +98 21 22431650. E-mail addresses: [email protected] (K. Parand), [email protected] (M. Nikarya). http://dx.doi.org/10.1016/j.apm.2014.02.001 0307-904X/Ó 2014 Elsevier Inc. All rights reserved.
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The remainder of this paper is organized as follows. In Section 1.1, some essential information about the fractional calculus theory and FDEs are described. The basic information of the first kind of Bessel functions and its properties are presented in Section 1.2. In Section 2, we have described how to approximate the functions, as well as the Collocation algorithm. In Section 3, a number of important problems in several fields have been considered then, we applied the proposed algorithm to solve them in order to demonstrate the accuracy of our method. Finally, in the last section, we have described several concluding remarks. 1.1. Basic definitions of fractional In this section, we present some notations, definitions and preliminary facts of the fractional calculus theory which will be used further in this work. Definition 1. The Riemann–Liouville fractional integral operator Ia of order a on a usual Lebesgue space L1 ½a; b is given by
Ia f ðtÞ ¼
1 CðaÞ
Z
t
ðt sÞa1 f ðsÞds:
ð1Þ
0
Some properties of this definition are:
I0 f ðtÞ ¼ f ðtÞ Ia Ib f ðtÞ ¼ Iaþb f ðtÞ Ia Ib f ðtÞ ¼ Ib Ia f ðtÞ Cðm þ 1Þ Ia ðt aÞm ¼ ðt aÞaþm Cða þ m þ 1Þ where f 2 L1 ½a; b; b; a P 0 and m > 1. Riemann–Liouville fractional derivative of order a > 0 is normally used
Da f ðtÞ ¼
n d ðIna f ðtÞÞ; dt
ð2Þ
where n is an integer, which is satisfied in n 1 < a 6 n. However, its derivative has certain disadvantages when trying to model real-world phenomena with fractional differential equations. Therefore, we shall introduce a modified fractional differential operator Da proposed by Caputo. Definition 2. The Caputo’s definition of fractional differential operator is given by:
Da f ðtÞ ¼
1 Cðn aÞ
Z
t
ðt sÞna1 f ðnÞ ðsÞds;
ð3Þ
0
where t > 0; n is an integer which satisfy in the relation n 1 < a 6 n and f 2 L1 ½a; b The two basic features of theses definitions for f 2 L1 ½a; b are:
Da Ia f ðtÞ ¼ f ðtÞ; Ia Da f ðtÞ ¼ f ðtÞ
ð4Þ n1 X ðt aÞk f ðkÞ ð0þ Þ : k! k¼0
ð5Þ
For more details and explanation about fractional derivatives and integrals, you can see [3,4] 1.2. Bessel functions and their properties In this section, we describe the first kind of Bessel functions and their properties which are used to construct the Bessel functions collocation (BFC) method. The Bessel’s equation of order n, is [26,27]:
x2 y00 ðxÞ þ xy0 ðxÞ þ ðx2 n2 ÞyðxÞ ¼ 0;
for x 2 ð1; 1Þ;
ðn 2 RÞ:
ð6Þ
An obtained solution of this equation is [27]: 1 X ð1Þr Cðn þ 1Þ x 2rþn a0 2r ; 2 r!Cðn þ r þ 1Þ 2 r¼0
for any value of a0 ; where CðkÞ is the gamma function, which is defined as follows:
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CðkÞ ¼
Z
3
1
et t k1 dt: 0
1 Let us choose a0 ¼ 2n Cðnþ1Þ . Accordingly, we obtain the solution which we shall denote by J n ðxÞ and we call it the Bessel func-
tion of the first kind of order n:
J n ðxÞ ¼
1 X r¼0
x 2rþn ð1Þr ; r!Cðn þ r þ 1Þ 2
ð7Þ
where series (7) is convergent for all 1 < x < 1. Some recursive relations of derivation are as follows [27]:
d n ðx J n ðxÞÞ ¼ xn J n1 ðxÞ; dx n J 0n ðxÞ ¼ J n1 ðxÞ Jn ðxÞ; nx n J 0n ðxÞ ¼ J n ðxÞ J nþ1 ðxÞ: x 2. Function approximation Spectral methods, in the context of numerical schemes for differential equations, generically belong to the family of weighted residual methods (WRMs) [28]. WRMs represent a particular group of approximation techniques, in which the residuals (or errors) are minimized in a certain way and thereby leading to specific methods including Galerkin, Petrov–Galerkin, collocation and tau formulations. WRMs are traditionally regarded as the foundation and cornerstone of the finite element, spectral, finite volume, boundary element and some other methods. Now in this section, we will describe WRMs and algorithm of collocation and tau method for solving differential equations [29,28]. Before introducing spectral methods, we try first to give a brief introduction to the WRM. Consider the approximation of the following problem:
LuðxÞ þ N uðxÞ ¼ f ðxÞ;
x 2 X;
ð8Þ
where L is differential or integral operation, and N is a lower-order linear and/or nonlinear operator involving only derivatives (if exist), and f ðxÞ is a function of variablex, with enough initial conditions. The starting point of the WRM is to approximate the solution u by a finite sum N X ai /i ðxÞ;
uðxÞ uN ðxÞ ¼
x 2 X;
ð9Þ
i¼0
where /i ðxÞ are the basis functions, and the expansion coefficients must be determined. Replacing u by uN in Eq. (8) leads to the residual function:
RN ðxÞ ¼ LuN ðxÞ þ N uN ðxÞ f ðxÞ:
ð10Þ
The notion of the WRM is to force the residual to zero by requiring:
< RN ; w>x ¼
Z
RN ðxÞwj ðxÞxðxÞdx ¼ 0;
0 6 j 6 N;
ð11Þ
X
where fwj ðxÞg are test functions, and x is positive weight function. If, we choose the Lagrange basis polynomials as test functions in (11), such that wj ðxk Þ ¼ djk , where fxk g are preassigned collocation points, Hence the residual is forced to zero at xj , i.e., RN ðxj Þ ¼ 0, the name of this method is collocation [23,28,30,31]. In this paper we suggest L in (8) is a fractional operation, introduced in (3). 2.1. Collocation algorithm for solving FDEs A method for forcing the residual function (10) to zero, is Collocation algorithm. In this method by substituting the finite series (9) to residual function (10), and collocating it on fxk g, we have N þ 1 equations and N þ 1 unknown coefficients (spectral coefficients), in all of spectral methods, the purpose is to findthese coefficients. Now, we aim to use these methods for solving FDEs, by substituting the fractional operator (3) into Eq. 8, instead of L. In shape of algorithmic, we do for solving Eq. (8): BEGIN. 1. Input N. 2. Construct the series (9) by using Bessel functions (7) as follows:
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uN ðxÞ ¼
N X
ai J i ðxÞ:
ð12Þ
i¼0
3. Insert the constructed series of step 2, into Eq. (8). 4. Construct the Residual function as follows:
Resðx; a0 ; a1 ; . . . ; an Þ ¼ LuN ðxÞ þ N uN ðxÞ f ðxÞ: ¼
1 Cðn aÞ
Z 0
t
ð13Þ
ðt sÞna1 uN ðsÞds þ N uN ðxÞ f ðxÞ ðnÞ
ð14Þ
5. Substitute the conditions of problem into set of equations (or change the basis function to satisfy the conditions). Now, we have N þ 1 unknown fan gNn¼0 . To obtain these unknown coefficients, we need N þ 1 equations, thus: 6. By choosing N þ 1 points fxi g; i ¼ 0; 1; . . . ; N, in the domain of the Eq. (8) as collocation points and substituting them in Resðx; a0 ; a1 ; . . . ; an Þ ¼ LuN ðxÞ þ N uN ðxÞ f ðxÞ, we construct a system containing N þ 1 equations. 7. By solving obtained system of equations in step 4, via Newton’s method and gain the an ; n ¼ 0; 1; . . . ; N. The main difficulty with such a system is to find the way we can choose an initial guess to handle the Newtons method; in other words, to find out how many solutions the system of nonlinear equations admits. We think the best way to discover the proper initial guess (or initial guesses) is to solve the system analytically for the very small N (by means of symbolic softwares programs such as Mathematica, Matlab or Maple) then, we can find proper initial guesses and particularly, the multiplicity of solutions of such system. This action has been done by starting from proper initial guesses with the maximum number of ten iterations. The Collocation method has been used increasingly for solving differential or integro-differentialequations [23–25,30,32,33]. Also, it is very useful in providing highly accurate solutions for differential equations. This method is easy to implement and it yields the desired accuracy. It reduces the solution of a nonlinear problem to the solution of a system of nonlinear algebraic equations. The important concerns of the collocation approach are the selection of the basis functions and collocation points. The basis functions have three different properties: easy computation, rapid convergence and completeness, which means that any solution can be represented with arbitrary high accuracy by taking the truncation N to be sufficiently large. Therefore, we used the first kind of Bessel functions as the basis functions. Often, in spectral methods specially collocation methods the error in beginning of domain is worse than the other place of domain, therefore, to rectify this problem, we have used the roots of shifted Chebyshev polynomials into the domain of uðxÞ as collocation points. As known, the root of the Chebyshev, Legendre, Jacobi polynomials are compact around of 1 and 1 [27].
3. Solving Some well-known FDE problems In this section, we apply the BFC algorithm which is described in Section 2.1,, to solve some differential equations of the fractional order. Example 1. Consider the nonlinear fractional Abel differential equation of the first kind [8]:
Da yðxÞ ¼ yðxÞ3 sin x xyðxÞ2 þ x2 yðxÞ x3 ;
x 2 ð0; 1;
a 2 ð0; 1Þ
ð15Þ
with initial condition:
yð0Þ ¼ 0:
ð16Þ
The Figs. 1 and 2 show graphs of solutions and residual function of the Eq. (15), respectively, for a ¼ 0:8; 0:85; 0:9. To show the convergency of present method to solve this example for a ¼ 0:8; 0:85; 0:9 in the Fig. 3, in which in these graphs, we showed that, by increasing the N the residual function decreases, to show the convergency of presented method. Example 2. Consider the fractional differential equation:
y00 ðxÞ þ Da yðxÞ þ yðxÞ ¼ 1 þ x;
x 2 ½0; 1;
a 2 ð0; 2Þ;
ð17Þ
with conditions
y0 ð0Þ ¼ yð0Þ ¼ 1;
ð18Þ
the exact solution of (17) is yðxÞ ¼ 1 þ x [9]. The Table 2 shows the absolute error, and convergence rate of BFC method’s solution for Eq. (17). Please cite this article in press as: K. Parand, M. Nikarya, Application of Bessel functions for solving differential and integro-differential equations of the fractional order, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.001
K. Parand, M. Nikarya / Applied Mathematical Modelling xxx (2014) xxx–xxx
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Fig. 1. Graph of the approximation solutions of Example 1 for N ¼ 10 and a ¼ 0:8; 0:85; 0:9.
Fig. 2. Graph of the residual functions of Example 1 for N ¼ 10 and a ¼ 0:8; 0:85; 0:9.
Example 3. Consider the fractional differential equation:
Da yðxÞ þ yðxÞ ¼ 0
ð19Þ
with conditions:
yð0Þ ¼ 1;
y0 ð0Þ ¼ 0ðfor a 2 ð1; 2Þ:
ð20Þ
The exact solution for (19) is:
yðxÞ ¼
1 X ðxa Þk k¼0
ak þ 1
The Fig. 4 shows the obtained solution and compares it with exact solution of Eq. (19) [9]. To show the convergency of proposed method to exact solution we present Table 3 Example 4. Consider the fractional differential equation:
Da yðxÞ þ 3yðxÞ ¼ 3x3 þ
8
Cð0:5Þ
x1:5
ð21Þ
with condition: Please cite this article in press as: K. Parand, M. Nikarya, Application of Bessel functions for solving differential and integro-differential equations of the fractional order, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.001
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Fig. 3. Residual functions graphs of Example 1 for N ¼ 5; 8; 10; 12 and a ¼ 0:8; 0:85; 0:9, to show the convergence rate of BFC method.
Table 1 The obtained values of Example 1 for a ¼ 0:8; 0:85; 0:9 for N ¼ 10. x
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
a ¼ 0:8; N ¼ 10
a ¼ 0:85; N ¼ 10
a ¼ 0:9; N ¼ 10
yN ðxÞ
ResðxÞ
yN ðxÞ
ResðxÞ
yN ðxÞ
ResðxÞ
0.000053344 0.000744347 0.003492701 0.010522097 0.024954200 0.051085888 0.094999267 0.165925912 0.279684833 0.469341539
0.5062e7 0.2341e7 0.1785e7 0.8256e8 0.5850e8 0.4234e7 0.1591e7 0.1001e6 0.6652e5 0.8708e5
0.000044 0.000637 0.003053 0.009316 0.022301 0.045937 0.085707 0.149708 0.251066 0.414919
2.8356e6 5.2135e6 3.5297e6 7.1609e6 1.9642e6 9.4394e6 1.4910e5 5.1897e5 9.0014e5 4.3630e4
0.0000365 0.0005465 0.0026670 0.0082482 0.0199300 0.0413235 0.0774071 0.1353508 0.2263126 0.3701646
9.8461e7 3.2642e6 2.1555e6 4.2884e6 1.1563e6 8.3369e6 8.4920e6 2.9012e5 4.9196e5 2.3136e4
Table 2 The absolute error of Example 2 for a ¼ 1:5 and several N. x
N¼4
N¼6
N¼8
N ¼ 10
0.2 0.4 0.6 0.8 1.0
2.5190e5 5.2834e5 7.4812e5 8.6195e5 1.7146e5
4.69776e8 9.10242e8 1.26860e7 1.56436e7 4.49094e9
3.40510e11 6.43686e11 8.91707e11 1.09824e10 4.22735e11
1.185961e15 2.240080e15 3.103329e15 3.814391e16 4.400919e13
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yð0Þ ¼ 0;
y0 ð0Þ ¼ 0:
7
ð22Þ
the exact solution of (21) is x3 . The Table 4 shows the absolute error, and convergence rate of BFC method’s solution for Eq. (21) (See Table 1). Example 5. Consider the fractional differential equation:
16 y00 ðxÞ 2y0 ðxÞ þ Da yðxÞ þ yðxÞ ¼ x3 6x2 þ 6x þ pffiffiffiffi x2:5 5 p
ð23Þ
Fig. 4. Graph of the approximation solution of Example 3 for N ¼ 8 and a ¼ 1:5.
Table 3 The absolute error of Example 3 for a ¼ 1:5 and several N x
N¼5
N¼8
N ¼ 10
0.2 0.4 0.6 0.8 1.0
2.69776e6 9.54542e7 2.59760e8 9.56745e7 3.25896e8
8.93561e10 6.36286e10 1.25691e11 8.02257e12 2.58935e10
3.235871e16 9.321574e15 1.987452e15 7.325849e14 3.217951e13
Table 4 The absolute error of Example 4 for a ¼ 1:5 and several N. x
N¼3
N¼5
N¼7
N¼9
0.1 0.3 0.5 0.7 0.9
2.2655e5 3.4118e4 6.3397e4 1.6705e4 5.5043e4
8.6896e10 1.60361e6 9.63817e7 1.71558e7 8.98678e7
1.67093e10 1.67093e10 1.062120e9 7.03957e10 1.92731e10
3.65474e14 4.21589e14 9.14555e13 8.21475e14 1.32512e13
Table 5 The absolute error of Example 5 for a ¼ 0:5 and several N. x
N¼3
N¼5
N¼7
0.2 0.4 0.6 0.8 1.0
1.8769e4 9.7022e4 2.6184e3 3.4247e3 6.9465e3
2.98543e7 3.84222e6 7.27991e6 1.24040e6 1.75728e5
8.02961e10 3.057123e9 6.018712e9 9.156311e9 9.754413e9
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Fig. 5. Graph of the approximation solution of Example 6 for N ¼ 8 and several a;
j ¼ 0:5.
Fig. 6. Graph of the approximation solution of Example 6 for N ¼ 8 and several a;
j ¼ 0:2.
Table 6 The values of BFC method and residual of Example 4 for x
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00
a ¼ 0:5; N ¼ 5
j ¼ 0:5 and several a.
a ¼ 0:75; N ¼ 7
a ¼ 0:8; N ¼ 8
a ¼ 0:9; N ¼ 7
yN ðxÞ
ResðxÞ
yN ðxÞ
ResðxÞ
yN ðxÞ
ResðxÞ
yN ðxÞ
ResðxÞ
0.2822 0.3771 0.4149 0.4171 0.3984 0.3690 0.3354 0.3017 0.2702 0.2420 0.2173 0.1957
2.015e3 6.246e3 2.959e3 1.054e3 2.871e3 2.595e3 1.294e3 1.001e5 7.401e4 8.179e4 4.621e4 6.883e5
0.1957 0.2985 0.3791 0.4276 0.4448 0.4367 0.4113 0.3759 0.3368 0.2983 0.2628 0.2311
9.810e3 1.720e3 5.058e3 3.994e3 3.410e5 2.545e3 2.391e3 7.157e4 8.522e4 1.351e4 7.843e4 1.727e4
0.1856 0.2783 0.3631 0.4238 0.4525 0.4506 0.4261 0.3894 0.3491 0.3106 0.2749 0.2410
2.345e3 1.323e3 2.176e3 7.128e4 3.069e3 1.789e3 1.618e3 3.395e3 1.649e3 1.891e3 3.476e3 9.068e4
0.1661 0.2526 0.3374 0.4059 0.4497 0.4662 0.4575 0.4286 0.3865 0.3387 0.2918 0.2505
9.852e3 3.386e3 7.152e2 8.543e3 1.922e3 5.715e3 8.536e3 5.217e3 1.353e3 6.576e3 7.050e3 2.460e3
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K. Parand, M. Nikarya / Applied Mathematical Modelling xxx (2014) xxx–xxx Table 7 The values of BFC method and residual of Example 4 for x
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
a ¼ 0:5; N ¼ 6
j ¼ 0:2 and several a.
a ¼ 0:75 N ¼ 6
yN ðxÞ
ResðxÞ
yN ðxÞ
ResðxÞ
0.47355 0.60855 0.63838 0.62828 0.60428 0.57411 0.54054 0.50719 0.47694 0.44283
4.954e3 1.504e3 1.306e3 1.124e4 2.216e4 1.749e4 9.374e4 6.787e4 1.760e3 4.891e3
0.23445 0.39109 0.51755 0.59655 0.63118 0.63365 0.61739 0.59223 0.56263 0.52902
9.529e3 2.917e4 3.337e3 9.044e4 2.768e5 9.916e4 8.125e4 1.415e4 2.755e3 1.613e4
Fig. 7. Residual functions graphs of Example 6 for
j ¼ 0:5; N ¼ 5; 10; 15 and a ¼ 0:5; 0:75; 0:8, and 0.9 to show the convergence rate of BFC method.
with conditions:
yð0Þ ¼ y0 ð0Þ ¼ 0:
ð24Þ
The exact solution of Eq. (23) with conditions (24) is x3 . The Table 5 shows the absolute error, and convergence rate of BFC method’s solution for Eq. (23). Example 6. Volterra’s population equation of fractional order:
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Fig. 8. Residual functions graphs of Example 6 for
jDa yðxÞ ¼ yðxÞ þ y2 ðxÞ yðxÞ
Z
j ¼ 0:2; N ¼ 5; 10; 15 and a ¼ 0:5 and ; 0:75 to show the convergence rate of BFC method.
x
yðtÞdt;
ð25Þ
0
with condition:
yð0Þ ¼ 0:1:
ð26Þ
The obtained solution of Eq. (25) for j ¼ 0:5 and 0:2 for several a are shown in Fig. 5 and Fig. 6, respectively. This example has been solved by Momani, Yousabsi [7,34]. In Tables 6 and 7 have been shown the values of approximation solutions for several a and j ¼ 0:5 and 0:2, respectively. Also, convergency of BFC method for solving Example 6 is showed in Figs. 7 and 8.
4. Conclusion The fractional calculus and fractional differential equations have found application in different sciences. Therefore, solving and approaching the fractional differential equations have become a field of mathematics and computer science. In this article, we solve some differential equations of fractional order to show the application of BFC method in solving FDE. In Example 1 (Abel FDE of first kind) and Example 6 (Volterra population FDE), we have used BFC algorithm to solve nonlinear FDE. Other examples are linear FDEs. The obtained values for these examples are shown in different graphs and tables and the ability of BFC method in solving FDEs is presented. Acknowledgment The corresponding author would like to thank Shahid Beheshti University for the awarded Grant. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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[16] A. Saadatmandi, M. Dehghan, M.R. Azizi, The Sinc–Legendre collocation method for a class of fractional convection–diffusion equations with variable coefficients, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 4125–4136. [17] D. Baleanu, A. Saadatmandi, A. Kadem, M. Dehghan, The fractional linear systems of equations within an operational approach, J. Comput. Nonlinear Dyn. 8 (2013), http://dx.doi.org/10.1115/1.4007192. [18] A. Lotfi, S.A. Yousefi, A numerical technique for solving a class of fractional variational problems, J. Comput. Appl. Math. 237 (2013) 633–643. [19] R. Caponetto, S. Fazzino, A semi-analytical method for the computation of the Lyapunov exponents of fractional-order systems, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 22–27. [20] M. El-Shahed, M. Gaber, M. Al-Yami, The fractional q-differential transformation and its application, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 42–55. [21] K. Parand, M. Nikarya, J.A. Rad, F. Baharifard, A new reliable numerical algorithm based on the first kind of bessel functions to solve prandtlblasius laminar viscous flow over a semi-infinite flat plate, Zeitsc. fr Naturf. A 67a (2012) 665–673. [22] K. Pranad, M. Nikarya, J.A. Rad, Solving the non-linear lane-emden type equations by using bessel orthogonal functions collocation method, Cele. Mech. Day. Astron. 116 (2013) 97–107. [23] S. Yuzbasi, M. Sezer, A collocation method to find solutions of linear complex differential equations in circular domains, Appl. Math. Comput. 219 (2013) 10610–10626. [24] S. Yuzbasi, M. Sezer, An improved bessel collocation method with a residual error function to solve a class of lane-emden differential equations, Math. Comput. Model. 57 (2013) 1298–1311. [25] S. Yuzbasi, Numerical solution of the bagley-torvik equation by the bessel collocation method, Math. Methods Appl. Sci. 36 (2013) 300–312. [26] G. Watson, A Treatise on the Theory of Bessel Functions, 2nd edition., Cambridge University Press, Cambridge, England, 1967. [27] W.W. Bell, Special Functions For Scientists And Engineers, D. Van Nostrand Company Ltd, Canada, 1967. [28] J. Shen, T. Tang, L.L. Wang, Spectral Methods Algorithms, Analysics And Applicattions, first ed., Springer, 2001. [29] J.P. Boyd, Chebyshev and Fourier Spectral Methods, second ed., Dover, New York, 2000. [30] K. Parand, M. Dehghan, A. Pirkhedri, Modified deneralized Laguerre function collocation method for solving the Blasius equation, J. Commput. Appl. Math. 233 (2009) 980–989. [31] K. Parand, S. Abbasbandy, S. Kazem, J.A. Rad, A novel appliction of radial basis functions for solving a model of first-order integro-ordinary differential equation, Commun. Nonlinear Sci. Num. Simul. 16 (2001) 4250–4258. [32] K. Parand, A.R. Rezaei, A. Taghavi, Lagrangian method for solving Lane-Emden type equation arising in astrophysics on semi-infinite domains, Acta Astoraut. 67 (2010) 673–680. [33] K. Parand, M. Dehghan, M. Shahini, Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type, J. Comput. Phys. 228 (2009) 8830–8840. [34] S. Yuzbasi, A numerical approximation for volterras population growth model with fractional order, Appl. Math. Model. 37 (2013) 3216–3227.
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Application of Bessel functions for solving differential and integro-differential equations of the fractional order q K. Parand ⇑, M. Nikarya Department of Computer Sciences, Shahid Beheshti University, G.C., Tehran, Iran
a r t i c l e
i n f o
Article history: Received 15 October 2012 Received in revised form 18 June 2013 Accepted 3 February 2014 Available online xxxx Keywords: Fractional differential equation Fractional integro-differential equation Bessel functions Collocation algorithm Non-linear
a b s t r a c t In this paper, a new numerical algorithm to solve the linear and nonlinear fractional differential equations (FDE) is introduced. Fractional calculus and fractional differential equations have many applications in physics, chemistry, engineering, finance, and other sciences. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, which is convergent for any x 2 R. In this method, we reduce the solution of a nonlinear fractional problem to the solution of a system of the nonlinear algebraic equations. To illustrate the reliability of this method, we solve some important equations of fractional order, and present numerical results of the present method to show convergence rate, applicability and reliability of this method. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Although Fractional calculus is an ancient mathematical topic, in the last few decades fractional calculus and fractional differential equations have found applications in different branches of sciences, and scientists are attracted to study fractional differential equations in physics, chemistry, engineering, finance, and other sciences more than ever; for instance, see [1–5]. Most results about solving fractional differential equations are obtained by using numerical methods, because only some particular fractional differential equations can be solved analytically [6–11]. Due to the growing applications, considerable attention has been given to the exact and numerical solutions of fractional differential equations (FDE). In this paper, we attempt to introduce a new method, based on Bessel functions of the first kind for solving FDEs. Previously, many of researchers study FDEs and attempt to solve them by utilizing several techniques of spectral methods, for example, Wang and Fan by using Chebyshev wavelet method[9], Rehman and Khan by using Legendre wavelet method[10], Doha and Bhrawy by using tau method chebyshev [12], Esmaeili and Shamsi by using pseudo-spectral method [13], Pedas and Tamme by using spline collocation methods [14] as well as many other researchers [15–20]. In this paper, we aim to solve FDEs by applying the Bessel functions and spectral collocation methods. Spectral methods are among the strongest methods for solving differential and integral equations.formerly, Bessel functions and Bessel polynomials collocation method were used for solving several type of problems, such as Blasius equation, Lane-Emden type equations, etc. [21–25] and the researchers obtained the excellent results, therefore, we decided to use the Bessel function collocation (BFC) method to solve linear and nonlinear differential and integro-differential equation of fractional order.
q
Member of research group of Scientific Computing.
⇑ Corresponding author. Tel.: +98 21 22431653; fax: +98 21 22431650. E-mail addresses: [email protected] (K. Parand), [email protected] (M. Nikarya). http://dx.doi.org/10.1016/j.apm.2014.02.001 0307-904X/Ó 2014 Elsevier Inc. All rights reserved.
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The remainder of this paper is organized as follows. In Section 1.1, some essential information about the fractional calculus theory and FDEs are described. The basic information of the first kind of Bessel functions and its properties are presented in Section 1.2. In Section 2, we have described how to approximate the functions, as well as the Collocation algorithm. In Section 3, a number of important problems in several fields have been considered then, we applied the proposed algorithm to solve them in order to demonstrate the accuracy of our method. Finally, in the last section, we have described several concluding remarks. 1.1. Basic definitions of fractional In this section, we present some notations, definitions and preliminary facts of the fractional calculus theory which will be used further in this work. Definition 1. The Riemann–Liouville fractional integral operator Ia of order a on a usual Lebesgue space L1 ½a; b is given by
Ia f ðtÞ ¼
1 CðaÞ
Z
t
ðt sÞa1 f ðsÞds:
ð1Þ
0
Some properties of this definition are:
I0 f ðtÞ ¼ f ðtÞ Ia Ib f ðtÞ ¼ Iaþb f ðtÞ Ia Ib f ðtÞ ¼ Ib Ia f ðtÞ Cðm þ 1Þ Ia ðt aÞm ¼ ðt aÞaþm Cða þ m þ 1Þ where f 2 L1 ½a; b; b; a P 0 and m > 1. Riemann–Liouville fractional derivative of order a > 0 is normally used
Da f ðtÞ ¼
n d ðIna f ðtÞÞ; dt
ð2Þ
where n is an integer, which is satisfied in n 1 < a 6 n. However, its derivative has certain disadvantages when trying to model real-world phenomena with fractional differential equations. Therefore, we shall introduce a modified fractional differential operator Da proposed by Caputo. Definition 2. The Caputo’s definition of fractional differential operator is given by:
Da f ðtÞ ¼
1 Cðn aÞ
Z
t
ðt sÞna1 f ðnÞ ðsÞds;
ð3Þ
0
where t > 0; n is an integer which satisfy in the relation n 1 < a 6 n and f 2 L1 ½a; b The two basic features of theses definitions for f 2 L1 ½a; b are:
Da Ia f ðtÞ ¼ f ðtÞ; Ia Da f ðtÞ ¼ f ðtÞ
ð4Þ n1 X ðt aÞk f ðkÞ ð0þ Þ : k! k¼0
ð5Þ
For more details and explanation about fractional derivatives and integrals, you can see [3,4] 1.2. Bessel functions and their properties In this section, we describe the first kind of Bessel functions and their properties which are used to construct the Bessel functions collocation (BFC) method. The Bessel’s equation of order n, is [26,27]:
x2 y00 ðxÞ þ xy0 ðxÞ þ ðx2 n2 ÞyðxÞ ¼ 0;
for x 2 ð1; 1Þ;
ðn 2 RÞ:
ð6Þ
An obtained solution of this equation is [27]: 1 X ð1Þr Cðn þ 1Þ x 2rþn a0 2r ; 2 r!Cðn þ r þ 1Þ 2 r¼0
for any value of a0 ; where CðkÞ is the gamma function, which is defined as follows:
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CðkÞ ¼
Z
3
1
et t k1 dt: 0
1 Let us choose a0 ¼ 2n Cðnþ1Þ . Accordingly, we obtain the solution which we shall denote by J n ðxÞ and we call it the Bessel func-
tion of the first kind of order n:
J n ðxÞ ¼
1 X r¼0
x 2rþn ð1Þr ; r!Cðn þ r þ 1Þ 2
ð7Þ
where series (7) is convergent for all 1 < x < 1. Some recursive relations of derivation are as follows [27]:
d n ðx J n ðxÞÞ ¼ xn J n1 ðxÞ; dx n J 0n ðxÞ ¼ J n1 ðxÞ Jn ðxÞ; nx n J 0n ðxÞ ¼ J n ðxÞ J nþ1 ðxÞ: x 2. Function approximation Spectral methods, in the context of numerical schemes for differential equations, generically belong to the family of weighted residual methods (WRMs) [28]. WRMs represent a particular group of approximation techniques, in which the residuals (or errors) are minimized in a certain way and thereby leading to specific methods including Galerkin, Petrov–Galerkin, collocation and tau formulations. WRMs are traditionally regarded as the foundation and cornerstone of the finite element, spectral, finite volume, boundary element and some other methods. Now in this section, we will describe WRMs and algorithm of collocation and tau method for solving differential equations [29,28]. Before introducing spectral methods, we try first to give a brief introduction to the WRM. Consider the approximation of the following problem:
LuðxÞ þ N uðxÞ ¼ f ðxÞ;
x 2 X;
ð8Þ
where L is differential or integral operation, and N is a lower-order linear and/or nonlinear operator involving only derivatives (if exist), and f ðxÞ is a function of variablex, with enough initial conditions. The starting point of the WRM is to approximate the solution u by a finite sum N X ai /i ðxÞ;
uðxÞ uN ðxÞ ¼
x 2 X;
ð9Þ
i¼0
where /i ðxÞ are the basis functions, and the expansion coefficients must be determined. Replacing u by uN in Eq. (8) leads to the residual function:
RN ðxÞ ¼ LuN ðxÞ þ N uN ðxÞ f ðxÞ:
ð10Þ
The notion of the WRM is to force the residual to zero by requiring:
< RN ; w>x ¼
Z
RN ðxÞwj ðxÞxðxÞdx ¼ 0;
0 6 j 6 N;
ð11Þ
X
where fwj ðxÞg are test functions, and x is positive weight function. If, we choose the Lagrange basis polynomials as test functions in (11), such that wj ðxk Þ ¼ djk , where fxk g are preassigned collocation points, Hence the residual is forced to zero at xj , i.e., RN ðxj Þ ¼ 0, the name of this method is collocation [23,28,30,31]. In this paper we suggest L in (8) is a fractional operation, introduced in (3). 2.1. Collocation algorithm for solving FDEs A method for forcing the residual function (10) to zero, is Collocation algorithm. In this method by substituting the finite series (9) to residual function (10), and collocating it on fxk g, we have N þ 1 equations and N þ 1 unknown coefficients (spectral coefficients), in all of spectral methods, the purpose is to findthese coefficients. Now, we aim to use these methods for solving FDEs, by substituting the fractional operator (3) into Eq. 8, instead of L. In shape of algorithmic, we do for solving Eq. (8): BEGIN. 1. Input N. 2. Construct the series (9) by using Bessel functions (7) as follows:
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uN ðxÞ ¼
N X
ai J i ðxÞ:
ð12Þ
i¼0
3. Insert the constructed series of step 2, into Eq. (8). 4. Construct the Residual function as follows:
Resðx; a0 ; a1 ; . . . ; an Þ ¼ LuN ðxÞ þ N uN ðxÞ f ðxÞ: ¼
1 Cðn aÞ
Z 0
t
ð13Þ
ðt sÞna1 uN ðsÞds þ N uN ðxÞ f ðxÞ ðnÞ
ð14Þ
5. Substitute the conditions of problem into set of equations (or change the basis function to satisfy the conditions). Now, we have N þ 1 unknown fan gNn¼0 . To obtain these unknown coefficients, we need N þ 1 equations, thus: 6. By choosing N þ 1 points fxi g; i ¼ 0; 1; . . . ; N, in the domain of the Eq. (8) as collocation points and substituting them in Resðx; a0 ; a1 ; . . . ; an Þ ¼ LuN ðxÞ þ N uN ðxÞ f ðxÞ, we construct a system containing N þ 1 equations. 7. By solving obtained system of equations in step 4, via Newton’s method and gain the an ; n ¼ 0; 1; . . . ; N. The main difficulty with such a system is to find the way we can choose an initial guess to handle the Newtons method; in other words, to find out how many solutions the system of nonlinear equations admits. We think the best way to discover the proper initial guess (or initial guesses) is to solve the system analytically for the very small N (by means of symbolic softwares programs such as Mathematica, Matlab or Maple) then, we can find proper initial guesses and particularly, the multiplicity of solutions of such system. This action has been done by starting from proper initial guesses with the maximum number of ten iterations. The Collocation method has been used increasingly for solving differential or integro-differentialequations [23–25,30,32,33]. Also, it is very useful in providing highly accurate solutions for differential equations. This method is easy to implement and it yields the desired accuracy. It reduces the solution of a nonlinear problem to the solution of a system of nonlinear algebraic equations. The important concerns of the collocation approach are the selection of the basis functions and collocation points. The basis functions have three different properties: easy computation, rapid convergence and completeness, which means that any solution can be represented with arbitrary high accuracy by taking the truncation N to be sufficiently large. Therefore, we used the first kind of Bessel functions as the basis functions. Often, in spectral methods specially collocation methods the error in beginning of domain is worse than the other place of domain, therefore, to rectify this problem, we have used the roots of shifted Chebyshev polynomials into the domain of uðxÞ as collocation points. As known, the root of the Chebyshev, Legendre, Jacobi polynomials are compact around of 1 and 1 [27].
3. Solving Some well-known FDE problems In this section, we apply the BFC algorithm which is described in Section 2.1,, to solve some differential equations of the fractional order. Example 1. Consider the nonlinear fractional Abel differential equation of the first kind [8]:
Da yðxÞ ¼ yðxÞ3 sin x xyðxÞ2 þ x2 yðxÞ x3 ;
x 2 ð0; 1;
a 2 ð0; 1Þ
ð15Þ
with initial condition:
yð0Þ ¼ 0:
ð16Þ
The Figs. 1 and 2 show graphs of solutions and residual function of the Eq. (15), respectively, for a ¼ 0:8; 0:85; 0:9. To show the convergency of present method to solve this example for a ¼ 0:8; 0:85; 0:9 in the Fig. 3, in which in these graphs, we showed that, by increasing the N the residual function decreases, to show the convergency of presented method. Example 2. Consider the fractional differential equation:
y00 ðxÞ þ Da yðxÞ þ yðxÞ ¼ 1 þ x;
x 2 ½0; 1;
a 2 ð0; 2Þ;
ð17Þ
with conditions
y0 ð0Þ ¼ yð0Þ ¼ 1;
ð18Þ
the exact solution of (17) is yðxÞ ¼ 1 þ x [9]. The Table 2 shows the absolute error, and convergence rate of BFC method’s solution for Eq. (17). Please cite this article in press as: K. Parand, M. Nikarya, Application of Bessel functions for solving differential and integro-differential equations of the fractional order, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.001
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Fig. 1. Graph of the approximation solutions of Example 1 for N ¼ 10 and a ¼ 0:8; 0:85; 0:9.
Fig. 2. Graph of the residual functions of Example 1 for N ¼ 10 and a ¼ 0:8; 0:85; 0:9.
Example 3. Consider the fractional differential equation:
Da yðxÞ þ yðxÞ ¼ 0
ð19Þ
with conditions:
yð0Þ ¼ 1;
y0 ð0Þ ¼ 0ðfor a 2 ð1; 2Þ:
ð20Þ
The exact solution for (19) is:
yðxÞ ¼
1 X ðxa Þk k¼0
ak þ 1
The Fig. 4 shows the obtained solution and compares it with exact solution of Eq. (19) [9]. To show the convergency of proposed method to exact solution we present Table 3 Example 4. Consider the fractional differential equation:
Da yðxÞ þ 3yðxÞ ¼ 3x3 þ
8
Cð0:5Þ
x1:5
ð21Þ
with condition: Please cite this article in press as: K. Parand, M. Nikarya, Application of Bessel functions for solving differential and integro-differential equations of the fractional order, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.001
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Fig. 3. Residual functions graphs of Example 1 for N ¼ 5; 8; 10; 12 and a ¼ 0:8; 0:85; 0:9, to show the convergence rate of BFC method.
Table 1 The obtained values of Example 1 for a ¼ 0:8; 0:85; 0:9 for N ¼ 10. x
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
a ¼ 0:8; N ¼ 10
a ¼ 0:85; N ¼ 10
a ¼ 0:9; N ¼ 10
yN ðxÞ
ResðxÞ
yN ðxÞ
ResðxÞ
yN ðxÞ
ResðxÞ
0.000053344 0.000744347 0.003492701 0.010522097 0.024954200 0.051085888 0.094999267 0.165925912 0.279684833 0.469341539
0.5062e7 0.2341e7 0.1785e7 0.8256e8 0.5850e8 0.4234e7 0.1591e7 0.1001e6 0.6652e5 0.8708e5
0.000044 0.000637 0.003053 0.009316 0.022301 0.045937 0.085707 0.149708 0.251066 0.414919
2.8356e6 5.2135e6 3.5297e6 7.1609e6 1.9642e6 9.4394e6 1.4910e5 5.1897e5 9.0014e5 4.3630e4
0.0000365 0.0005465 0.0026670 0.0082482 0.0199300 0.0413235 0.0774071 0.1353508 0.2263126 0.3701646
9.8461e7 3.2642e6 2.1555e6 4.2884e6 1.1563e6 8.3369e6 8.4920e6 2.9012e5 4.9196e5 2.3136e4
Table 2 The absolute error of Example 2 for a ¼ 1:5 and several N. x
N¼4
N¼6
N¼8
N ¼ 10
0.2 0.4 0.6 0.8 1.0
2.5190e5 5.2834e5 7.4812e5 8.6195e5 1.7146e5
4.69776e8 9.10242e8 1.26860e7 1.56436e7 4.49094e9
3.40510e11 6.43686e11 8.91707e11 1.09824e10 4.22735e11
1.185961e15 2.240080e15 3.103329e15 3.814391e16 4.400919e13
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yð0Þ ¼ 0;
y0 ð0Þ ¼ 0:
7
ð22Þ
the exact solution of (21) is x3 . The Table 4 shows the absolute error, and convergence rate of BFC method’s solution for Eq. (21) (See Table 1). Example 5. Consider the fractional differential equation:
16 y00 ðxÞ 2y0 ðxÞ þ Da yðxÞ þ yðxÞ ¼ x3 6x2 þ 6x þ pffiffiffiffi x2:5 5 p
ð23Þ
Fig. 4. Graph of the approximation solution of Example 3 for N ¼ 8 and a ¼ 1:5.
Table 3 The absolute error of Example 3 for a ¼ 1:5 and several N x
N¼5
N¼8
N ¼ 10
0.2 0.4 0.6 0.8 1.0
2.69776e6 9.54542e7 2.59760e8 9.56745e7 3.25896e8
8.93561e10 6.36286e10 1.25691e11 8.02257e12 2.58935e10
3.235871e16 9.321574e15 1.987452e15 7.325849e14 3.217951e13
Table 4 The absolute error of Example 4 for a ¼ 1:5 and several N. x
N¼3
N¼5
N¼7
N¼9
0.1 0.3 0.5 0.7 0.9
2.2655e5 3.4118e4 6.3397e4 1.6705e4 5.5043e4
8.6896e10 1.60361e6 9.63817e7 1.71558e7 8.98678e7
1.67093e10 1.67093e10 1.062120e9 7.03957e10 1.92731e10
3.65474e14 4.21589e14 9.14555e13 8.21475e14 1.32512e13
Table 5 The absolute error of Example 5 for a ¼ 0:5 and several N. x
N¼3
N¼5
N¼7
0.2 0.4 0.6 0.8 1.0
1.8769e4 9.7022e4 2.6184e3 3.4247e3 6.9465e3
2.98543e7 3.84222e6 7.27991e6 1.24040e6 1.75728e5
8.02961e10 3.057123e9 6.018712e9 9.156311e9 9.754413e9
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Fig. 5. Graph of the approximation solution of Example 6 for N ¼ 8 and several a;
j ¼ 0:5.
Fig. 6. Graph of the approximation solution of Example 6 for N ¼ 8 and several a;
j ¼ 0:2.
Table 6 The values of BFC method and residual of Example 4 for x
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00
a ¼ 0:5; N ¼ 5
j ¼ 0:5 and several a.
a ¼ 0:75; N ¼ 7
a ¼ 0:8; N ¼ 8
a ¼ 0:9; N ¼ 7
yN ðxÞ
ResðxÞ
yN ðxÞ
ResðxÞ
yN ðxÞ
ResðxÞ
yN ðxÞ
ResðxÞ
0.2822 0.3771 0.4149 0.4171 0.3984 0.3690 0.3354 0.3017 0.2702 0.2420 0.2173 0.1957
2.015e3 6.246e3 2.959e3 1.054e3 2.871e3 2.595e3 1.294e3 1.001e5 7.401e4 8.179e4 4.621e4 6.883e5
0.1957 0.2985 0.3791 0.4276 0.4448 0.4367 0.4113 0.3759 0.3368 0.2983 0.2628 0.2311
9.810e3 1.720e3 5.058e3 3.994e3 3.410e5 2.545e3 2.391e3 7.157e4 8.522e4 1.351e4 7.843e4 1.727e4
0.1856 0.2783 0.3631 0.4238 0.4525 0.4506 0.4261 0.3894 0.3491 0.3106 0.2749 0.2410
2.345e3 1.323e3 2.176e3 7.128e4 3.069e3 1.789e3 1.618e3 3.395e3 1.649e3 1.891e3 3.476e3 9.068e4
0.1661 0.2526 0.3374 0.4059 0.4497 0.4662 0.4575 0.4286 0.3865 0.3387 0.2918 0.2505
9.852e3 3.386e3 7.152e2 8.543e3 1.922e3 5.715e3 8.536e3 5.217e3 1.353e3 6.576e3 7.050e3 2.460e3
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K. Parand, M. Nikarya / Applied Mathematical Modelling xxx (2014) xxx–xxx Table 7 The values of BFC method and residual of Example 4 for x
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
a ¼ 0:5; N ¼ 6
j ¼ 0:2 and several a.
a ¼ 0:75 N ¼ 6
yN ðxÞ
ResðxÞ
yN ðxÞ
ResðxÞ
0.47355 0.60855 0.63838 0.62828 0.60428 0.57411 0.54054 0.50719 0.47694 0.44283
4.954e3 1.504e3 1.306e3 1.124e4 2.216e4 1.749e4 9.374e4 6.787e4 1.760e3 4.891e3
0.23445 0.39109 0.51755 0.59655 0.63118 0.63365 0.61739 0.59223 0.56263 0.52902
9.529e3 2.917e4 3.337e3 9.044e4 2.768e5 9.916e4 8.125e4 1.415e4 2.755e3 1.613e4
Fig. 7. Residual functions graphs of Example 6 for
j ¼ 0:5; N ¼ 5; 10; 15 and a ¼ 0:5; 0:75; 0:8, and 0.9 to show the convergence rate of BFC method.
with conditions:
yð0Þ ¼ y0 ð0Þ ¼ 0:
ð24Þ
The exact solution of Eq. (23) with conditions (24) is x3 . The Table 5 shows the absolute error, and convergence rate of BFC method’s solution for Eq. (23). Example 6. Volterra’s population equation of fractional order:
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Fig. 8. Residual functions graphs of Example 6 for
jDa yðxÞ ¼ yðxÞ þ y2 ðxÞ yðxÞ
Z
j ¼ 0:2; N ¼ 5; 10; 15 and a ¼ 0:5 and ; 0:75 to show the convergence rate of BFC method.
x
yðtÞdt;
ð25Þ
0
with condition:
yð0Þ ¼ 0:1:
ð26Þ
The obtained solution of Eq. (25) for j ¼ 0:5 and 0:2 for several a are shown in Fig. 5 and Fig. 6, respectively. This example has been solved by Momani, Yousabsi [7,34]. In Tables 6 and 7 have been shown the values of approximation solutions for several a and j ¼ 0:5 and 0:2, respectively. Also, convergency of BFC method for solving Example 6 is showed in Figs. 7 and 8.
4. Conclusion The fractional calculus and fractional differential equations have found application in different sciences. Therefore, solving and approaching the fractional differential equations have become a field of mathematics and computer science. In this article, we solve some differential equations of fractional order to show the application of BFC method in solving FDE. In Example 1 (Abel FDE of first kind) and Example 6 (Volterra population FDE), we have used BFC algorithm to solve nonlinear FDE. Other examples are linear FDEs. The obtained values for these examples are shown in different graphs and tables and the ability of BFC method in solving FDEs is presented. Acknowledgment The corresponding author would like to thank Shahid Beheshti University for the awarded Grant. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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Please cite this article in press as: K. Parand, M. Nikarya, Application of Bessel functions for solving differential and integro-differential equations of the fractional order, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.02.001