NUMERICAL SOLUTIONS OF TWO-DIMENSIONAL MIXED VOLTERRA-FREDHOLM INTEGRAL EQUATIONS VIA BERNOULLI COLLOCATION METHOD

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1 NUMERICAL SOLUTIONS OF TWO-DIMENSIONAL MIXED VOLTERRA-FREDHOLM INTEGRAL EQUATIONS VIA BERNOULLI COLLOCATION METHOD R. M. HAFEZ 1,2,a, E. H. DOHA 3,b, A. H. BHRAWY 4,c, D. BALEANU 5,6,d 1 Department of Mathematics, Alwagjh University College, Tabuk University, Tabuk, 71491, Saudi Arabia, a : r mhafez@yahoo.com 2 Department of Basic Science, Institute of Information Technology, Modern Academy, Cairo, 11434, Egypt 3 Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt, b : eiddoha@sci.cu.edu.eg 4 Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt, c : alibhrawy@yahoo.co.uk 5 Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Ankara, Turkey, d : dumitru@cankaya.edu.tr 6 Institute of Space Sciences, Magurele-Bucharest, Romania Received December 13, 216 Abstract. The mixed Volterra-Fredholm integral equations (VFIEs) arise in various physical and biological models. The main purpose of this article is to propose and analyze efficient Bernoulli collocation techniques for numerically solving classes of two-dimensional linear and nonlinear mixed VFIEs. The novel aspect of the technique is that it reduces the problem under consideration to a system of algebraic equations by using the Gauss-Bernoulli nodes. One of the main advantages of the present approach is its superior accuracy. Consequently, good results can be obtained even by using a relatively small number of collocation nodes. In addition, several numerical results are given to illustrate the features of the proposed technique. Key words: Volterra-Fredholm integral equations; error analysis; collocation schemes; Bernoulli-Gauss nodes. 1. INTRODUCTION In different areas of sciences like physics, biology, economics, engineering and others, several high-accuracy numerical methods are introduced to deal with the related problems. In the forefront of these methods, spectral methods are widely used as powerful algorithms in the construction of numerical solutions for various kinds of integral and differential equations (see, for instance, Refs. [1 1]). Recently, numerous approaches have been used to nonlinear computation of the two-dimensional Volterra-Fredholm integral equations (VFIEs) such as the matrix based method [11], the homotopy perturbation method [12] and the modified Romanian Journal of Physics 62, 111 (217) v.2.* #93926d62

2 Article no. 111 R. M. Hafez et al. 2 homotopy perturbation method [13], the spline collocation method [14] and the iterative method [15]. Behzadi [16] proposed a technique based on Homotopy analysis method for solving nonlinear VFIEs of the first kind. Shekarabi et al. [17] used the two-dimensional Bernstein operational matrices method to solve the linear mixed VFIEs, while in [18], Dastjerdi et al. used the radial basis function approximation for solving the mixed VFIEs. Also, the Taylor polynomial method have been proposed for approximating the solution of the mixed VFIEs in Ref. [19]. On the other hand, Paripour and Kamyar [2] applied new basis functions for solving the nonlinear VFIEs via direct method, while in Ref. [21], Bhrawy et al. applied Legendre-Gauss- Lobatto collocation method for approximating the solution of the multi-dimensional Fredholm integral equations. It is well known that, Bernoulli polynomials play a vital role as basis functions for the spectral solutions of integral, differential, and integro-differential equations, (see, for example, Refs. [22 26]). In this work, we aim to develop some effective and efficient collocation schemes to solve two-dimensional linear and nonlinear mixed VFIEs. One great advantage of such schemes is that it reduces the problems under confederation to systems of algebraic equations by using the the Bernoulli polynomials as basis functions and the Gauss-Bernoulli nodes as the collocation points. The collocation method has successfully been applied to many situations [27 39]. The remaining sections of this paper are organized as follows. In the second Section we present a few relevant properties of Bernoulli polynomials. Sections 3 and 4 are devoted to the theoretical derivation of the proposed approaches for linear and nonlinear two-dimensional mixed Volterra-Fredholm integral equations, respectively. Moreover, in Sec. 5 the proposed schemes are applied to two different test problems. Finally, the conclusion is given in Sec DEFINITION AND PROPERTIES OF BERNOULLI POLYNOMIALS The generating function of the Bernoulli polynomials is given by te xt e t 1 = etb(x) = k= B k (x) tk k!, (1) with the usual convention about replacing B k (x) by B k (x). In the special case, x =, B k () = B k are called the k-th Bernoulli numbers. From (1), we note that (see Refs. [4 47]) where δ m,n is the Kronecker symbol. B = 1, (B + 1) k B k = B k (1) B k = δ 1,k, (2) (c) RJP 62(Nos. 3-4), id:111-1 (217) v.2.* #93926d62

3 3 Numerical solutions of 2D mixed Volterra-Fredholm integral equations Article no. 111 From (1), we have (see [22, 48]) k ( k B k (x) = l l= l= ) B k x k l. (3) The Bernoulli basis polynomials B k (x) of degree k are constructed from the following relation [49] k ( ) k + 1 B l l (x) = (1 k)x k, k =,1,... (4) The first five Bernoulli polynomials are B (x) = 1, B 1 (x) = x 1 2, B 2 (x) = x 2 x + 1 6, B 3 (x) = x 3 3x2 2 + x 2, B 4 (x) = x 4 2x 3 + x From (2), and (3), we note that 1 Using (4), we have the Bernoulli vector B k (x)dx = δ,k k + 1. (5) B(x) = [B (x),b 1 (x),...,b N (x)] T. 3. LINEAR INTEGRAL EQUATIONS Here, we are interested in using the Bernoulli collocation (BC) method for solving linear two-dimensional mixed VFIEs: v(x,y) = K(x,s,y,t)v(s,t)dsdt + g(x,y), (6) where v(x,y) is an unknown function, g(x,y) and K(x,s,y,t) are analytical functions on [,1] [,1] and [,1] [,1] [,1] [,1], respectively. The aim of our method is to get solution using the way mentioned in the previous Section as follows: N M v(x,y) c ij B i (x)b j (y) = ψ(x,y)c, (7) i= j= (c) RJP 62(Nos. 3-4), id:111-1 (217) v.2.* #93926d62

4 Article no. 111 R. M. Hafez et al. 4 where c ij, i =,1,...,N, j =,1,...,M are the unknown coefficients, C = [c,c 1,...,c N ;c 1,c 11,...,c N1 ;c M,c 1M,...,c NM ] T, while N and M are any arbitrary positive integers, B i (x), i =,1,...,N and B j (y), j =,1,...,M are Bernoulli polynomials defined in Eq. (3). Also ψ(x,y) is a 1 (N + 1)(M + 1) matrix introduced as follows where ψ(x,y) =[r (x,y),r 1 (x,y),...,r N (x,y);r 1 (x,y),r 11 (x,y),...,r N1 (x,y) ;r M (x,y),r 1M (x,y),...,r NM (x,y)], r ij (x,y) = B i (x)b j (y), i =,1,...,N, j =,1,...,M. Substituting Eq. (7) into Eq. (6) yields: N i= j= M c ij B i (x)b j (y)= Suppose that f ij (x,y) = B i (x)b j (y) then, Eq. (8) can be rewritten as N i= j= K(x,s,y,t) N i= j= M c ij B i (s)b j (t)dsdt + g(x,y). (8) K(x,s,y,t)B i (s)b j (t)dsdt, M c ij f ij (x,y) = g(x,y). (9) Collocating Eq. (9) at N +1 and M +1 roots of the Bernoulli polynomials B N+1 (x), the Gauss-Bernoulli nodes, we obtain: N M c ij f ij (x n,y m ) = g(x n,y m ), for n =,1,...,N, m =,1,...,M, (1) i= j= which can be rewritten in the matrix form: where and F T C = G, G =[g(x,y ),g(x 1,y ),...,g(x N,y );g(x,y 1 ),g(x 1,y 1 ),...,g(x N,y 1 ); g(x,y M ),g(x 1,y M ),...,g(x N,y M )] T, F = (f ijnm ), i,n =,1,...,N, j,m =,1,...,M, f ijnm = f ij (x n,y m ), i,n =,1,...,N, j,m =,1,...,M. (c) RJP 62(Nos. 3-4), id:111-1 (217) v.2.* #93926d62

5 5 Numerical solutions of 2D mixed Volterra-Fredholm integral equations Article no. 111 Finally, the unknown vector C can be given by: C = (F T ) 1 G, and the approximate solution of Eq. (6) is given by v(x,y) = ψ(x,y)c. 4. NONLINEAR INTEGRAL EQUATIONS The two-dimensional nonlinear mixed VFIE is given by: x v(x,y) = M(x,y,s,t,v(s,t))dtds + g(x,y), (11) where v(x,y) is an unknown function defined on Ω D = [,T ] Ω, and Ω is a closed subset of R n, n = 1,2,3. The functions M(x,y,s,t,v) and g(x,y) are given functions defined on D and S = {(x,y,s,t,u) : s x T, y Ω, t Ω}, respectively [5]. Note that any finite interval [,T ] can be transformed to [,1] by a linear map, so we suppose that [,T ] = [,1] and Ω = [,1] without loss of generality. We first approximate the solution v(x,y) using Eq. (7); then we get ψ(x,y)c = g(x,y) + M(x, y, s, t, ψ(s, t)c)dtds. (12) Now, we collocate Eq. (12) at points (x n,y m ) for n =,1,...,N, m =,1,...,M. Hence, we have ψ(x n,y m )C = xn 1 M(x n,y m,s,t,ψ(s,t)c)dtds + g(x n,y m ), (13) for n =,1,...,N, m =,1,...,M. From (13), we obtain a nonlinear system of algebraic equations that can be solved by Newton iteration method to obtain the unknown vector C SOME USEFUL LEMMAS Lemma 4.1 Assume that v(x,y) H H = L 2 [,1] L 2 [,1] is a smooth function, then the coefficients c ij for all i,j =,1,..., of the approximate solution (7) can be calculated from c i,j = i+j v(x,y) i!j! x i y j dxdy. (14) (c) RJP 62(Nos. 3-4), id:111-1 (217) v.2.* #93926d62

6 Article no. 111 R. M. Hafez et al. 6 Lemma 4.2 (see [23]) Suppose that v(x,y) is a smooth function and v N (x,y) is the approximate solution of v(x, y). Thanks to lemma 4.1, the error bound can be obtained as follows: E(v N (x,y)) = v(x,y) v N (x,y) CÛN(2π) N, (15) where Û is a positive constant independent of N and is a bound for the partial derivative of v(x,y) ERROR ANALYSIS In this Subsection, we provide the error analysis for the numerical scheme of the linear two-dimensional mixed VFIEs (6), which indicates that the errors decay exponentially provided that the source and the kernel functions are sufficiently smooth. To do so, the Sobolev inequality and some properties of Banach algebras are considered. Theorem 4.3 Let v(x, y) be the exact solution of linear two-dimensional mixed VFIEs (6) and assume that u N (x,y) is the approximate solution defined by (7), then, for sufficiently smooth function g(x,y) and K(x,y,s,t) in (6) we have Proof. Consider v(x,y) v N (x,y) λcûn(2π) N (16) v(x,y) = K(x, y, s, t)v(s, t)dsdtg(x, y), (17) and v(x,y) is a continuous function on [,1]. According to the proposed method, the approximate solution is given by v N (x,y) = and subtracting (17) from (18), we get v(x,y) v N (x,y)= e N (x,y) = K(x,y,s,t)v N (s,t)dsdt + g(x,y), (18) K(x,y,s,t)v(s,t)dsdt K(x,y,s,t)v N (s,t)dsdt, (19) K(x,y,s,t)(v(s,t) v N (s,t))dsdt, (2) (c) RJP 62(Nos. 3-4), id:111-1 (217) v.2.* #93926d62

7 7 Numerical solutions of 2D mixed Volterra-Fredholm integral equations Article no. 111 From Gronwall inequality we have e N (x) = where the value of λ is Therefore, from (15) K(x,y,s,t)(v(s,t) v N (s,t))dsdt K(x,y,s,t)(v(s,t) v N (s,t))dsdt, λ (v(s,t) v N (s,t)), λ = max K(x,y,s,t). a x,y b e N (x) L 2 w α,β (I) λcûn(2π) N. (21) 5. NUMERICAL EXAMPLES Based on the two previous algorithms, we give in this Section some numerical results. The effectiveness, appropriateness, and high accuracy of our method are evident when we compare it with other methods. Our results are compared with the exact solutions by calculating the following error function: E(x,y) = v(x,y) ũ(x,y), (22) where v(x,y) and ũ(x,y) are the exact solution and the numerical solution at the point (x, y), respectively. Moreover, the maximum absolute errors (MAEs) are given by MAEs = Max{E(x,t) : (x,t) [,1] [,1]} (23) Example 1 Consider the linear VFIEs ([51]): v(x,y) = g(x,y) (2t 1)e s v(s,t)dtds, x,y [,1), (24) where g(x,y) = sinx+y 1 6 ex + 1 6, which has the exact solution v(x,y) = sinx+y, for x,y < 1. In Table 1, we give the MAEs for different values of x, y and N = M = 8. Moreover, the results obtained by our method are compared with those obtained by operational matrix (OM) method [52] and multiquadric (MQs) radial basis functions method [51]. It is clear that our method gives more accurate results. (c) RJP 62(Nos. 3-4), id:111-1 (217) v.2.* #93926d62

8 Article no. 111 R. M. Hafez et al. 8 Table 1 Comparison of the absolute errors at various choices of x,y, for Example 1. (x,y) OM [52] MQs [51] BC method (.,.) (.1,.1) (.2,.2) (.3,.3) (.4,.4) (.5,.5) (.6,.6) (.7,.7) (.8,.8) (.9,.9) Example 2 Consider the linear VFIE ([51]): v(x,y) = g(x,y) y 2 e t v(s,t)dtds, x,y [,1), (25) where g(x,y) = 1 3 x2 (3e y + xy 2 ), which has the exact solution v(x,y) = x 2 e y, for x < 1. In this example (see, Table 2), we compare our results obtained by the BC method for the different choices x,y, with the method of operational matrix (OM) [52] and the method of multiquadric (MQs) radial basis functions [51]. The numerical results in this Table demonstrates that the absolute errors obtained by the BC method is significantly better than those obtained by the OM method [52] and the multiquadric radial basis functions method [51]. Table 2 Comparison of the absolute errors with various choices of x,y, for Example 2. (x,y) OM [52] MQs [51] BC method (.,.) (.1,.1) (.2,.2) (.3,.3) (.4,.4) (.5,.5) (.6,.6) (.7,.7) (.8,.8) (.9,.9) (c) RJP 62(Nos. 3-4), id:111-1 (217) v.2.* #93926d62

9 9 Numerical solutions of 2D mixed Volterra-Fredholm integral equations Article no. 111 Example 3 Consider the nonlinear VFIE ([51]): v(x,y) = g(x,y) + y 2 e 4s [v(s,t)] 2 dtds, x,y [,1), (26) where g(x,y) = e 2y x e 4x ( 1+e 4 )( 3+3e 4x 4x(3+2x(3+4x(1+x))))y 2, which has the exact solution v(x,y) = x 2 e y, for x < 1. Almasieh and Meleh [51] introduced this problem and applied the MQs method, using Legendre-Gauss-Lobatto nodes and weights for obtaining its numerical solution. In order to show that our algorithm is more accurate than the MQs method [51], in Table 3, we list the absolute errors with several choices of x, y andn = M = 4 and compare the achieved results with those obtained using the MQs method [51]. Table 3 Comparison of the absolute errors at various choices of x,y, for Example 3. (x,y) MQs [51] BC method (.,.) (.1,.1) (.2,.2) (.3,.3) (.4,.4) (.5,.5) (.6,.6) (.7,.7) (.8,.8) (.9,.9) Table 4 MAEs at various choices of x,y, for Example 4 at N = M = 2. (x,y) BC method (x,y) BC method (.,.) (.5,.5) (.1,.1) (.6,.6) (.2,.2) (.7,.7) (.3,.3) (.8,.8) (.4,.4) (.9,.9) Example 4 Consider the following nonlinear VFIE ([52]): v(x,y) = 16 e (x+y+s+t) [v(s,t)] 3 dtds + g(x,y), x,y [,1), (27) where g(x,y) = e 5x+y + e 4+x+y e 4+5x+y, which has the exact solution v(x,y) = e x+y, for x < 1. (c) RJP 62(Nos. 3-4), id:111-1 (217) v.2.* #93926d62

10 Article no. 111 R. M. Hafez et al. 1 Table 4 lists the absolute errors for different values of x,y and N = M = 2. It is observed that the new method present a good accuracy for the nonlinear VFIE. 6. CONCLUSIONS We have applied a Bernoulli collocation method to solve linear and nonlinear two-dimensional mixed VFIEs. This method uses Bernoulli-Gauss nodes to reduce the considered problem to the solution of a simpler algebraic system. In the examples given, by selecting relatively few Bernoulli-Gauss nodes, we are able to get very accurate approximation, and we are thus able to demonstrate the utility of our approach over other analytical or numerical methods such as other collocation methods. We provided illustrative examples to check the validity and applicability of the method. The results demonstrated that the new collection method is extremely effective. Based on the listed comparisons with other methods, we observe the effectiveness, appropriateness, and high accuracy of our method. REFERENCES 1. I. Aziz, J. Comput. Appl. Math. 239, 333 (213). 2. A. H. Bhrawy and M. A. Zaky, Appl. Numer. Math. 111, 197 (217). 3. A. H. Bhrawy and M. A. Zaky, Nonl. Dyn. 85, 1815 (216). 4. A. H. Bhrawy and M. A. Zaky, Math. Meth. Appl. Sci. 39, 1765 (216). 5. A. H. Bhrawy, J. Vib. Contr. 22, 2288 (216). 6. A. H. Bhrawy, M. A. Zaky, and D. Baleanu, Rom. Rep. Phys. 67, 34 (215). 7. A. H. Bhrawy and M. A. Abdelkawy, J. Comput. Phys. 294, 462 (215). 8. H. Khalil, M. Al-Smadi, K. Moaddy, R. A. Khan, and I. Hashim, Disc. Dyn. Nature. Soc. 216, (216). 9. R. Salehi, Numer. Algorithms, DOI:1.17/s z (216). 1. M. A. Abdelkawy, M. A. Zaky, A. H. Bhrawy, and D. Baleanu, Rom. Rep. Phys. 67, 773 (215). 11. S. A. Hosseini, S. Shahmorad, F. Talati, Numer. Algor., DOI:1.17/s (214). 12. M. Ghasemi, M. T. Kajani, and A. Davari, Appl. Math. Comput. 189, 341 (27). 13. C. Dong, Z. Chen, and W. Jiang, J. Comput. Appl. Math. 239, 359 (213). 14. H. Brunner, Siam J. Numer. Anal. 27, 987 (199). 15. K. Wang, Q. Wang, and K. Guan, Appl. Math. Comput. 225, 631 (213). 16. Sh. S. Behzadi, Int. J. Ind. Math. 6, 124 (214). 17. F. H. Shekarabi, K. Maleknejad, and R. Ezzati, Afr. Math., DOI:1.17/s (214). 18. H. L. Dastjerdia, F. M. M. Ghainia, and M. Hadizadeh, Inter. J. Comput. Math. 9, 527 (213). 19. K. Wang and Q. Wang, Appl. Math. Comput. 229, 53 (214). 2. M. Paripour and M. Kamyar, Commun. Numer. Anal. 213, 1 (213). 21. A. H. Bhrawy, M. A. Abdelkawy, J. Tenreiro Machado, and A. Z. M. Amin, Comput. Math. Appl., DOI:1.116/j.camwa (216). 22. S. Bazm, J. Comput. Appl. Math. 275, 44 (215). (c) RJP 62(Nos. 3-4), id:111-1 (217) v.2.* #93926d62

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