Numerical solution of nonlinear mixed Volterra-Fredholm integral equations in complex plane via PQWs

Applied Mathematics and Computation 369 (2020) 124828

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Numerical solution of nonlinear mixed Volterra-Fredholm integral equations in complex plane via PQWs H. Beiglo, M. Gachpazan∗ Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

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Article history: Received 20 July 2018 Revised 24 September 2019 Accepted 6 October 2019

Keywords: Nonlinear mixed Volterra-Fredholm integral equations Periodic quasi-wavelet Complex plane Fixed point theorem

This paper presents an efficient numerical method for solving nonlinear mixed Volterraintegral equations in the complex plane. The method is based on the fixed point iteration procedure. In each iteration of this method, periodic quasi-wavelets are used as basis functions to approximate the solution. Also, using the Banach fixed point theorem, some results concerning the error analysis are obtained. Finally, some numerical examples show the implementation and accuracy of this method. © 2019 Elsevier Inc. All rights reserved.

1. Introduction The nonlinear mixed Volterra-Fredholm integral equations arise from parabolic boundary value problems, the mathematical modeling of the spatio-temporal development of an epidemic, and various physical and biological models. We are interested in the numerical solution of nonlinear mixed Volterra-Fredholm integral equations of the form,

u (x ) = f (x ) + α



T 0

k1 (x, t )F1 (u(t ))dt + β



x 0

k2 (x, t )F2 (u(t ))dt, 0 ≤ x ≤ T,

(1)

where k1 , k2 : [0, T ] × [0, T ] → C, f : [0, T ] → C and F1 , F2 : R → C are continuous complex periodic functions, α and β are complex numbers, and u is an unknown complex function to be found; see [1]. There is a considerable amount of literature that discusses the approximation of the solution of nonlinear mixed VolterraFredholm integral equations [2–13]. Recent contributions in this regard include Legendre wavelets [14], rationalized Haar functions [15,16], collocation method [17–19], the homotopy perturbation method [20], radial basis functions (RBFs) [21], Bernstein’s approximation [22], least square method [23,24], block-pulse functions [25,26] and B-spline functions [27]. Our developed method in this paper is fundamentally different from the above existing methods. The new method can be applied easily and accurately to nonlinear equations in the complex plane. While several numerical methods for approximating the solution of real integral equations with high accuracy are known, for complex one, only a few methods are considered [28–33]. In this paper, we describe a numerical scheme using periodic quasi-wavelets (PQWs) for solving Eq. (1) in the complex plane. Also, we provide the convergence conditions and determine the order of convergence. The

∗

Corresponding author. E-mail addresses: [email protected] (H. Beiglo), [email protected] (M. Gachpazan).

https://doi.org/10.1016/j.amc.2019.124828 0096-3003/© 2019 Elsevier Inc. All rights reserved.

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H. Beiglo and M. Gachpazan / Applied Mathematics and Computation 369 (2020) 124828

theory of PQWs based on B-spline functions is found in [28,34–37]. The pioneering work in integral equations via periodic quasi-wavelets was led by Chen and co-workers [34,35]. This paper is organized as follows: in Section 2, we briefly review some concepts and properties about periodic quasiwavelet functions on[0, T]. Section 3 states the fixed point method related to periodic quasi-wavelet functions for nonlinear mixed Volterra-Fredholm integral equations in the complex plane. Section 4 is dedicated to the convergence analysis of the proposed approximation method. Finally, in Section 5, numerical examples are given to demonstrate the efficiency and accuracy of the proposed numerical scheme. 2. Periodic quasi-wavelets In this section, we review periodic quasi-wavelets based on B-spline functions construction which were given in [34,35]. In order to achieve this aim, we must introduce the periodic spline functions. First, let L2p [0, T ] denote the periodic L2 [0, T] space functions with period T by the following inner product:

 f, g =

1 T



T 0

f (x )g(x )dx,

for all f, g ∈ L2p [0, T ].

Let n, k ∈ N such that k ≥ n + 1, h = follows: n Bn,m p (x ) = km





l∈Z

sin(l π /km ) lπ

T k

and hm =



n+1 exp

T km

(2)

where km = 2m k. Now, the periodic B-spline function is defined as



2π il x . T

(3)

Next, we consider Bn,m p (x − jhm ), j = 0, . . . , km − 1, as a basis of Vm , that is, the space of periodic spline functions. Note that the basis of Vm is not orthogonal, but we will have an orthogonal basis function for Vm by introducing Arn,m (x ) as follows (see [28]): n,m An,m r (x ) = Cr

k m −1



exp

l=0

where

 Crn,m

=

t0 + 2

k 



2π il r Bn,m p (x − lhm ), x ∈ R, km

(4)

−1/2 ts cos(srhm )

(5)

s=1

and

tλ = B2k+1,m (λhm ). p By using properties of

Dn,m r

(x ) :=

(6)

Arn,m (x ),

bn,m+1 Dn,m+1 r r

we define

(x ) − an,m+1 Dn,m+1 (x ) x ∈ R, r r+km

(7)

where arn,m+1 and bn,m+1 are defined in [28]. r r = 0, . . . , km − 1}, then Wm has an orthogonal basis and moreover Vm+1 = Vm Wm ; Similar to Vm , if Wm := span{Dk,m r see [1]. 3. Solution of nonlinear mixed Volterra-Fredholm integral equation In this section, the periodic quasi-wavelets provided in Section 2 are used to solve the nonlinear mixed Volterra-Fredholm integral equation. Consider the nonlinear mixed Volterra-Fredholm integral equation given in Eq. (1). Let S be a operator defined by

S(u(x )) = f (x ) + α



T 0

k1 (x, t )F1 (u(t ))dt + β



x 0

k2 (x, t )F2 (u(t ))dt, 0 ≤ x ≤ T,

(8)

As a mathematical foundation, we employ Banach’s fixed point theory in the following. The Banach fixed point theorem guarantees a unique fixed point for S if we can show that S is a contraction map. Furthermore, under appropriate assumptions on α and β , assume that S is contractive with contraction number L. Then, by the mapping contraction theorem, S has a unique fixed point u0 , which is the unique solution Eq. (1). In addition, u0 = limn→∞ Sn (u ), where u is any continuous function on L2p [0, T ]. Since, in general, it is not possible to calculate explicitly u0 from the sequence of functions {Sn (u}n∈N , we define a new sequence of functions, denoted by {ξn }n∈N , obtained recursively making use of PQWs basis. More concretely, we get ξn+1 from ξ n , approximating S(ξ n ) by using the sequence of projections of such PQWs basis.

H. Beiglo and M. Gachpazan / Applied Mathematics and Computation 369 (2020) 124828

3

We now describe the idea of the proposed numerical method. Since in general we cannot calculate explicitly S(ξ 0 ), we approximate this function in the following way: For each x, t ∈ [0, T], n ≥ 1, we define recursively

ψn (t ) := F1 (ξn (t )), ϕn (t ) := F2 (ξn (t )), 

ξn (x ) := f (x ) + α

T

0

k1 (x, t )Pn (ψn−1 )dt + β

 0

x

k2 (x, t )Pn (ϕn−1 )dt, 0 ≤ x ≤ T,

(9)

where Pn is an orthogonal projection operator of L2p [0, T ] to space of periodic quasi wavelets Vn . 4. Convergence analysis In this section, we study the existence and uniqueness of the solution of nonlinear mixed Volterra-Fredholm integral equations (1) in the complex plane. The result is obtained by using some extensions of Banachs contraction principle in complete metric spaces. We list the following hypotheses for our convenience: (H1 ) F1 , F2 : L2p [0, T ] → C are continues and Lipschitz with Lipschitz constants N1 , N2 , respectively, that is,

for all u, v ∈ L2p [0, T ] : Fi (u ) − Fi (v ) ≤ Ni u − v ,

i = 1, 2,

and N = max{N1 , N2 }. (H2 ) k1 , k2 : [0, T ] × [0, T ] → C are continues and M = max{M1 , M2 } that M1 and M2 are finite numbers, where

M1 = (H3 )



  T

|α|2 + |β|2 <

T

|k1 (x, t )|

2 dxdt,

0

0

M2 =

  T 0

T 0

|k2 (x, t )|2 dxdt,

1 , where α , and β are complex numbers. MN

Theorem 4.1. Suppose that the hypotheses (H1 ) − (H3 ) hold. Then there exists a unique solution u0 (t) of Eq. (1) on [0, T] and for each u ∈ L2p [0, T ], we have limn→∞ Sn (u ) = u0 . Proof. For u, v ∈ L2p [0, T ], we have

S(u ) − S(v ) 22 = α 1 T

=



T 0 T

 0

k1 (s, t )(F1 (u(t )) − F1 (v(t )))dt + β

 0

x

k2 (s, t )(F2 (u(t )) − F2 (v(t )))dt 22

 x  T 2   k1 (s, t )(F1 (u(t )) − F1 (v(t )))dt + β k2 (s, t )(F2 (u(t )) − F2 (v(t )))dt  ds. α 0

0

Using the Cauchy–Schwarz inequality, We have

 T  2 2     k1 (s, t ) dsdt (F1 (u(t )) − F1 (v(t ))) dt 0 0 0  T  T T 2 2 1     + |β|2 k2 (s, t ) dsdt (F2 (u(t )) − F2 (v(t ))) dt   T 0 0 0  T  T 2 2 1 1     ≤ |α|2 M12 (F1 (u(t )) − F1 (v(t ))) dt + |β|2 M22 (F2 (u(t )) − F2 (v(t ))) dt. 1 |α|2 T

S(u ) − S(v ) 22 ≤

T



T



T

0

Hence by using (H1 ), we get

1 |α|2 M12 N12 T

S(u ) − S(v ) 22 ≤

T



T

0

0

 T  2 2 1     u(t ) − v(t ) dt + T |β|2 M22 N22 u(t ) − v(t ) dt 0

|α|2 M12 N12 u − v 22 + |β|2 M22 N22 u − v 22 ≤ (|α|2 + |β|2 )N2 M2 u − v 22 ; ≤

therefore

S ( u ) − S ( v ) 2 ≤





|α|2 + |β|2 MN u − v 2 .

(H3 ) shows that S is a |α|2 + |β|2 MN-contraction. By the assumption, S is a contraction. Then S has a unique fixed point, which is the unique solution of Eq. (1). Furthermore for u ∈ L2p [0, T ], we have

S(u ) − S2 (u ) 2 ≤ L u − S(u ) 2 .

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H. Beiglo and M. Gachpazan / Applied Mathematics and Computation 369 (2020) 124828

Adding u − S(u ) 2 , it can be written as

u − S(u ) 2 ≤ (1 − L )−1 ( u − S(u ) 2 − S(u ) − S2 (u ) 2 , where L =



|α|2 + |β|2 MN. We define χ : L2p [0, T ] → R+ by

χ (u ) = (1 − L )−1 u − S(u ) 2 ; thus for u ∈ L2p [0, T ],

u − S(u ) 2 ≤ χ (u ) − χ S(u ). So for m < n and u ∈ L2p [0, T ], we obtain

Sm (u ) − Sn+1 (u ) 2 ≤

n 

S j (u ) − S j+1 (u ) 2 ≤ χ Sm (u ) − χ Sn+1 (u ).

j=m

By Considering m = 1 and letting n → ∞, we have ∞ 

S j (u ) − S j+1 (u ) 2 ≤ χ S(u ) < ∞.

j=1

Therefore {Sn (u)} is a Cauchy sequence. Since L2p [0, T ] is complete, there exists u0 ∈ L2p [0, T ] such that

lim Sn (u ) = u0 .

n→∞

 In (9), we define a sequence of functions, {ξn }n∈N , obtained recursively, making use of PQWs basis. A natural question arises under what conditions it will still be the case limn→∞ ξn = u0 , if we replace the sequence {ξn }n∈N with {Sn }n∈N . Lemma 4.2. Suppose that S is the operator given by (8), and let {ξ n } be defined by (9), where ξ0 ∈ L2p [0, T ]. Then for all n ≥ 1, it follows that

ξn+1 − S(ξn ) 2 ≤



|α|2 + |β|2 MO(hd+1 n+1 ).

(10)

Proof. Let Pm be a bounded projection operator of L2p [0, T ] to Vm . Using the Jackson type theorem for splines [38], we can conclude that, for f ∈ L2p [0, T ], there exists a constant Cj such that

f − Pm f 2 ≤ C j hmj ω ( f ( j ) , hm ), with 0 ≤ j ≤ d,

(11)

where ω is the modulus of continuity of f(j) . Moreover, if f has the continuous derivative of order d + 1, then

f − Pm f 2 = O(hd+1 m ). Also,

ξn+1 − S(ξn ) 22 = α =

1 T

(12)



T 0 T

 0

k1 (s, t )(Pn+1 (ψn )(t ) − ψn (t ))dt + β



x 0

k2 (s, t )(Pn+1 (ϕn )(t ) − ϕn (t ))dt 22

 x  T 2   k1 (s, t )(Pn+1 (ψn )(t ) − ψn (t ))dt + β k2 (s, t )(Pn+1 (ϕn )(t ) − ϕn (t ))dt  ds. α 0

0

Using the Cauchy–Schwarz inequality implies

 T  2 2     k1 (s, t ) dsdt (Pn+1 (ψn )(t ) − ψn (t )) dt 0 0 0  T  T T 2 2 1     2 + |β| k2 (s, t ) dsdt (Pn+1 (ϕn )(t ) − ϕn (t )) dt   T 0 0 0  T  T 2 2 1 1     ≤ |α|2 M12 (Pn+1 (ψn )(t ) − ψn (t )) dt + |β|2 M22 (Pn+1 (ϕn )(t ) − ϕn (t )) dt

ξn+1 − S(ξn ) 22 ≤

1 |α|2 T

T



T



T

0

T

0

≤ M2 (|α|2 (Pn+1 (ψn ) − ψn ) 22 + |β|2 (Pn+1 (ϕn ) − ϕn ) 22 ). Hence by using (12), we have Pn+1 ( f ) − f 2 = O(hd+1 ). Then we obtain n+1

ξn+1 − S(ξn ) 2 ≤



|α|2 + |β|2 MO(hd+1 n+1 ). 

Theorem 4.3. Under the hypotheses (H1 ) − (H3 ) of Theorem 4.1, the iterative sequence ξ n converges to the unique fixed point S.

H. Beiglo and M. Gachpazan / Applied Mathematics and Computation 369 (2020) 124828

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Table 1 Numerical results for modulus of the error for Example 1.. x

m=2

m=4

m=8

m=16

1 2 3 4 5 6

1.48448E−2 1.49837E−2 1.49303E−2 1.50434E−2 1.47709E−2 1.48454E−2

3.50512E−3 3.51046E−3 3.48362E−3 3.50182E−3 3.51107E−3 3.48636E−3

2.50607E−4 2.5084E−4 2.49044E−4 2.50233E−4 2.51104E−4 2.49258E−4

1.28571E−6 1.2869E−6 1.27768E−6 1.28378E−6 1.28826E−6 1.27879E−6

Proof. For u ∈ L2p [0, T ], we have

Sm (u ) − ξm 2 ≤ S(Sm−1 )(u ) − S(ξm−1 ) 2 + S(ξm−1 ) − ξm 2  ≤ L Sm−1 (u ) − ξm−1 2 + |α|2 + |β|2 MO(hd+1 m ), which implies that



S m ( u ) − ξ m 2 ≤

|α|2 + |β|2 M

m 

Lm− j O(hd+1 ). j

j=0

So

ξm − u0 2 ≤ ξm − Sm (u ) 2 + Sm (u ) − u0 2 m   ≤ |α|2 + |β|2 M Lm− j O(hd+1 ) + S m ( u ) − u 0 2 . j j=0

For any  > 0, there exists n0 ∈ N such that for m ≥ n0 , O(hd+1 m ) ≤  . Thus m 

Lm− j O(hd+1 )= j

j=0

n0 

Lm− j O(hd+1 )+ j n0 

Ln0 − j O(hd+1 )+ j

lim

m→∞

|α| + |β| 2

2M

m 

 L

m− j

m 

Lm− j .

j=n0 +1

j=0

Therefore



Lm− j O(hd+1 ) j

j=n0 +1

j=0

≤ Lm−n0

m 

O(

hd+1 j

j=0

)≤

Ln0 +1 1−L



|α|2 + |β|2 M.

By Theorem 4.1. and Since  is arbitrary, we conclude that

lim

n→∞

ξm = u 0 . 

5. Numerical examples In this section, for showing the accuracy and efficiency of the described method, we present some examples. The computations associated with the examples were performed by using Mathematica 9. Example 1. For α = i and β = 1, consider the following complex nonlinear integral equation:

u (x ) = f (x ) + i

 2π 0

sin (t ) u2 (t )dt + 13 + ln(2 + i cos(x )) 2



x 0

sin(t ) u(t )dt, (11 + cos(x ))2

where f(x) is chosen such that the exact solution is u(x ) = ecos(x ) + i sin(x ). The solution for u(x) is obtained by the method in Section 3. The results of Example 2 are shown in Figs. 1, 4 and Table 1. Example 2. For α = i and β = i, consider the following complex nonlinear integral equation:

u (x ) = f (x ) + i

 2π 0

sin(t ) Re2 (u(t ))dt + i √ 5 e1+sin(3x)



x 0

cos(t ) Im(u(t ))dt, √ 3 e(1+cos(2x))5

where f(x) is chosen such that the exact solution to be

u(x ) = (5 cos(x ) + 2 sin(4x ))2 + i(5 sin(x ) + 2 cos(4x )).

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H. Beiglo and M. Gachpazan / Applied Mathematics and Computation 369 (2020) 124828 Table 2 Numerical results for modulus of the error for Example 2. x

m=2

m=4

m=8

m=16

1 2 3 4 5 6

6.91291E−1 8.56557E−1 7.24133E−1 3.44205E−1 7.58578E−1 4.40253E−1

8.90943E−3 1.05847E−2 1.44903E−2 2.02964E−2 1.40952E−2 2.03169E−2

3.61749E−6 3.80426E−6 5.48609E−6 2.54469E−6 1.46173E−5 6.63863E−6

5.23683E−11 5.81264E−11 3.68705E−11 3.00658E−11 3.30559E−11 4.30663E−11

Fig. 1. Comparison between approximate and exact solution of Example 1 for m = 2, 16

approximate solution,

exact solution.

Fig. 2. Comparison between approximate and exact solution of Example 2 for m = 2, 16

approximate solution,

exact solution.

The solution for u(x) is obtained by the method in Section 3. The results for Example 3 are shown in Figs. 2, 5 and Table 2. Example 3. For α = i and β = 0, consider the following complex nonlinear integral equation:

u (x ) = f (x ) + i

 2π 0



tan(

t −4 ) 3

4i + 5 cot

3 x

3 u (t )dt, 3

where f(x) is chosen such that the exact solution to be

u(x ) = 3 cos(x ) + cos(4x ) + i(3 sin(x ) + sin(4x )). The solution for u(x) is obtained by the method in Section 3. The results for Example 1 are shown in Figs. 3, 6 and Table 3.

H. Beiglo and M. Gachpazan / Applied Mathematics and Computation 369 (2020) 124828

Fig. 3. Comparison between approximate and exact solution of Example 3 for m = 2, 16

approximate solution,

7

exact solution.

Fig. 4. The error of real and imaginary parts for m = 16, Example 1.

Fig. 5. The error of real and imaginary parts for m = 16, Example 2.

Example 4. Consider the following nonlinear integral equation:

u(x ) = 1 + sin (x ) − 2



x 0

3 sin(x − t )u2 (t )dt

with the exact solution u(x ) = cos(x ). The above example has been solved with the Chebyshev collocation method [39] and Chebyshev operational vector [40]. Table 4 demonstrates the comparison between the Chebyshev collocation method [39], Chebyshev operational vector, [40] and the present scheme.

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H. Beiglo and M. Gachpazan / Applied Mathematics and Computation 369 (2020) 124828

Fig. 6. The error of real and imaginary parts for m = 16, Example 3. Table 3 Numerical results for modulus of the error for Example 3. x

m=2

m=4

m=8

m=16

1 2 3 4 5 6

2.18266E−2 1.73456E−1 5.44208E−1 1.0061E 0 1.12292E 0 7.50338E−1

4.17676E−3 3.31927E−2 1.0414E−2 1.92529E−2 2.14883E−2 1.43586E−2

1.52002E−5 1.20796E−4 3.78989E−4 7.00653E−4 7.82006E−4 5.22539E−4

1.21647E−8 9.66726E−8 3.03305E−7 5.60733E−7 6.2584E−7 4.18188E−7

Table 4 A comparison between the PQWs, Chebyshev operational and Chebyshev collocation. m

PQWs

Chebyshev operational

Chebyshev collocation

4 8 12 16

1.83E−04 3.67E−06 1.33E−10 5.92E−12

2.74E−04 2.28E−08 1.71E−10 1.30E−10

2.04E−02 2.45E−05 4.59E−07 2.48E−07

6. Conclusion In this paper, we used an iterative method based on the fixed point technique to find the approximate solution of nonlinear mixed Volterra integral equations in the complex plane. In each iteration of this method, periodic quasi-wavelets were used as basis functions to approximate the solution. Using this method, a sequence of functions was obtained, which was proved to uniformly converge to the exact solution. Then, we have presented some numerical examples, confirming the approximation properties of the proposed method. Although, our discussion was restricted to nonlinear mixed Volterra integral equations in the complex plane of the second kind, it can also apply for to solve linear and nonlinear integro-differential equations and functional integral equation in the complex plane arising in physics and other fields of applied mathematics. References [1] H.L. Chen, Complex Harmonic Splines, Periodic Quasi-wavelets, Theory and Applications, Kluwer Academic Publishers, 1999. [2] P. Das, G. Nelakanti, Convergence analysis of discrete Legendre spectral projection methods for Hammerstein integral equations of mixed type, Appl. Math. Comput. 265 (2015) 574–601. [3] Z. Chen, W. Jiang, An approximate solution for a mixed linear Volterra-Fredholm integral equation, Appl. Math. Lett. 25 (8) (2012) 1131–1134. [4] K. Maleknejad, K. Nouri, M.N. Sahlan, Convergence of approximate solution of nonlinear Fredholm-Hammerstein integral equations, Commun. Nonlinear Sci. Numer. Simul. 15 (6) (2010) 1432–1443. [5] M. Ganesh, M.C. Joshi, Numerical solvability of Hammerstein integral equations of mixed type, IMA J. Numer. Anal. 11 (1) (1991) 21–31. [6] A.M. Wazwaz, A reliable treatment for mixed Volterra-Fredholm integral equations, Appl. Math. Comput. 127 (23) (2002) 405–414. [7] M. Hadizadeh, Posteriori error estimates for the nonlinear Volterra-Fredholm integral equations, Comput. Math. Appl. 45 (45) (2003) 677–687. [8] M.A. Abdou, K.I. Mohamed, A.S. Ismail, On the numerical solutions of Fredholm-Volterra integral equation, Appl. Math. Comput. 146 (23) (2003) 713–728. [9] N. Bildik, M. Inc, Modified decomposition method for nonlinear Volterra-Fredholm integral equations, Chaos Solitons Fractals 33 (1) (2007) 308–313. [10] F. Caliò, M.V.F. Muñoz, E. Marchetti, Direct and iterative methods for the numerical solution of mixed integral equations, Appl. Math. Comput. 216 (12) (2010) 3739–3746. [11] K. Wang, Q. Wang, Taylor polynomial method and error estimation for a kind of mixed Volterra-Fredholm integral equations, Appl. Math. Comput. 229 (11301397) (2014) 53–59. [12] F. Caliò, I. Garralda-Guillem, E. Marchetti, M.R. Galán, About some numerical approaches for mixed integral equations, Appl. Math. Comput. 219 (2) (2012) 464–474. [13] P. Das, G. Nelakanti, Convergence analysis of discrete Legendre spectral projection methods for Hammerstein integral equations of mixed type, Appl. Math. Comput. 265 (2015) 574–601.

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