Hybrid Bessel polynomials and Block-Pulse function approach for nonlinear Volterra-Fredholm integral equations

M.Moradi, E.Hashemizadeh

ABSTRACT

In this paper, an efficient numerical method is developed for solving the Volterra'Fredholm integral equations by Hybrid Bessel polynomials and Block-Pulse function, this method is based on replacement of unknown function by truncated series of well Known Hybrid Bessel expansion of functions. This is a grate advantage of this method which has lowest operation count in contrast to other early methods which use operational matrices (with huge number of operations) or involve intermediate numerical techniques for evaluating intermediate integrals which presented in integral equation or solve special case of nonlinear integral equations, . Numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results. The results show the efficiency and accuracy of the present work.

Keywords:

Hybrid Bessel polynomials and Block-Pulse, Nonlinear integral equations, Volterra'Fredholm integral, Operational matrix

1. Introduction

One of the fundamental classes of equations is integral equations. Such equations arise in many areas of science and engineering [1'6] and play an important role in the modeling of real-life phenomena in other fields of science [7'8]. For these reasons, integral equations have received much attention in the last decades. Some of integral equations cannot be solved by the well-known exact methods. Hence, it is desirable to introduce numerical methods with high accuracy to solve these equations numerically.

These equations are usually di'cult to solve analytically, so it is required to obtain the approximate solutions. Many numerical methods have been studied for approximating the solution of Volterra'Fredholm integral equations, such as Taylor polynomials method [9'13], homotopy perturbation method [14], Legendre wavelets method [15], interpolation method [16], Chebyshev polynomials method [17], rationalized Haar functions method [18] and spectral method [19].

Consider the Volterra-Fredholm integral equations of the form

where the functions are known kernel function on the interval and the functions and are known functions defined on the interval and , is the unknown function and are real constants such that When is first-order polynomial, the Eq. Is functional integral equation with proportional delay?

This paper is organized as follows. In Section 2, we introduce hybrid functions and its properties. In Section 3, we apply these sets of hybrid functions for approximating the solution of nonlinear Volterra-Fredholm integral equations. Numerical results are reported in Section 4. Finally, Section 5 concludes the paper.

2. Hybrid functions and some of their properties

The orthogonal set of hybrid functions , where is the order for Block-Pulse functions, is the order for Bessel polynomials and is the normalized time, is defined on the interval as

Here, the Bessel polynomials defined in the interval are given by

A set of Block-Pulse functions on the interval is defined as follows

The Block-Pulse functions on are disjoint, so for , we have , also these functions have the property of orthogonality on .

Since is the combination of Bessel polynomials and Block-Pulse functions which are both complete and orthogonal, then the set of hybrid functions is a complete orthogonal system in .

2.1. Function approximation

Any function can be expanded in a hybrid function

Where the hybrid coefficients are given by for , such that denotes the inner product.

3. Numerical solution

The series expansion Eq. (2) contains an infinite number of terms for a smooth . If is piecewise constant or may be approximated as piecewise constant, then the sum in Eq. (2). May be terminated after nm terms, that is

Where

Using we can consider that

Substituting Eqs. Into Eq. Yields:

Suppose that:

Than Eq. Can be rewritten as:

Collocating Eq. in roots of the Hybrid Bessel polynomials and Block-Pulse function method ,the Hybrid Bessel polynomials nodes, we obtain:

Which can be written in the following matrix form:

Where

Finally, the unknown vector can be computed by:

Therefore, the approximate solution of Eq. is given by

4. Illustrative examples

In this section we implemented our method on 3 different examples. Our results achieved by a proper value for m (this feather is experimental) and different values for n and m. The results are tabulated in 3 tables, in these tables the exact solutions are compared with hybrid function solutions and also in examples we compared hybrid functions results by Hybrid Chebyshev Polynomials results [20] for nonlinear Volterra-Fredholm integral equations. It is noticed that our method has quite acceptable results but it is clear for lower values of n, m we have less accuracy in some end points of the interval that by increasing n, m, the results become better.

In order to analyze the error of the method, let and be the exact and approximate solution of Eq. (1), respectively. Then we define the error

Where and

Example 4.1. Consider the following Volterra'Fredholm integral equation

Where

This equation has exact solution

The comparison among the hybrid solution with solved for different values of , , . Besides the solutions of Hybrid Chebyshev Polynomials [20] and exact solutions are shown in Table 1. In Fig. 1 solved for different values with and. In Fig. 2, Fig. 3 solved for different values with ,

Example 4.2. Consider the Volterra'Fredholm integral equation

Where

We have no difficulty finding that this equation has exact solution . We set we see that the approximation solution obtained by the present method has good agreement with the exact solution. Besides the solutions of Hybrid Chebyshev Polynomials [20], In Table 2 and Fig. 4,5 and 6 taking cand error .

Example 4.3. Consider the following Volterra'Fredholm integral equation

Where This equation has exact solution . This example is solved for different values of . In Table 3, respectively and compare the error of the present method Hybrid Chebyshev Polynomials [20] and Fig. 7,8 and 9 taking cand error

m, n Proposed Method

Proposed Method

Proposed Method

Method in [20]

Exact

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0 0.99910

1.10513

1.22149

1.34981

1.49168

1.64874

1.82259

2.01484

2.22711

2.46102

2.71817 0.99996

1.10510

1.22137

1.34987

1.49186

1.64876

1.82217

2.01383

2.22566

2.45973

2.71830 0.999996

1.10517

1.22140

1.34986

1.49183

1.64873

1.82212

2.01376

2.22555

2.45961

2.71828 0.999108

1.10513

1.22149

1.34981

1.49168

1.64874

1.82259

2.01484

2.22711

2.46102

2.71817 1

1.10517

1.22140

1.34986

1.49182

1.64872

1.82212

2.01375

2.22554

2.45960

2.71828

Table 1

Approximate and exact solutions for Example 4.1.

m, n Proposed Method

Proposed Method

Proposed Method

Method in [20]

Exact

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0 0.989538

0.991955

0.981512

0.959068

0.925484

0.881626

0.828357

0.766548

0.697069

0.620793

0.538595 1.000140

0.994926

0.979937

0.955239

0.921024

0.877604

0.825397

0.764922

0.696783

0.621669

0.540337

1

0.995011

0.980074

0.955342

0.921064

0.877584

0.825336

0.764843

0.696707

0.621610

0.540302 0.989538

0.991955

0.981512

0.959068

0.925484

0.881626

0.828357

0.766548

0.697069

0.620793

0.538595 1

0.995004

0.980067

0.955336

0.921061

0.877583

0.825336

0.764842

0.696707

0.621610

0.540302

Table 2

Approximate and exact solutions for Example 4.2. with

m, n Proposed Method

Proposed Method

Proposed Method

Method in [20]

Exact

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0 1.62943''10-14

0.001

0.008

0.027

0.064

0.125

0.216

0.343

0.512

0.729

1 1.40846''10-13

0.001

0.008

0.027

0.064

0.125

0.216

0.343

0.512

0.729

1 3.85463''10-10

0.001

0.008

0.027

0.064

0.125

0.216

0.343

0.512

0.729

1 -4.51878''10-14

0.001

0.008

0.027

0.064

0.125

0.216

0.343

0.512

0.729

1 0

0.001

0.008

0.027

0.064

0.125

0.216

0.343

0.512

0.729

1

Table 3

Approximate and exact solutions for Example 4.3.

Fig.1. with in Example 4.1 Fig.2. with in Example 4.1

Fig.3. with in Example 4.1 Fig.4. with and in a Example 4.2

Fig.5. with and in in Fig.6. with and in

Example 4.2 Example 4.2

Fig.7. with in Example 4.3 Fig.8. with in Example 4.3

Fig.9. with in Example 4.3

5. Conclusion

In this paper, we proposed a numerical method for solving nonlinear Volterra'Fredholm integral equations based on Hybrid Bessel polynomials and Block-Pulse function. This method converts the nonlinear Volterra'Fredholm integral equations into a nonlinear system of algebraic equations. In addition, the comparison of the results obtained by the present method with the exact solution and the other methods reveals that the method is very effective. It is worth mentioning that the presented method can be extended to two- (and higher-) dimensional problems, the method also can be developed and applied to nonlinear integro-differential equations, partial differential equations and functional integral equations with proportional delay. One of the considerable advantages of the method is that the approximate solutions are found very easily by using the computer code written in Mathematica. The method can be to the nonlinear systems Volterra'Fredholm integral equations, but some modi'cations are required.

### References

[1]. F. Bloom, Asymptotic bounds for solutions to a system of damped integro-differential equations of electromagnetic theory, J. Math. Anal. Appl. 73(1980) 524'542.

[2]. M.A. Abdou, Fredholm'Volterra integral equation of the first kind and contact problem, Appl. Math. Comput. 125 (2002) 177'193.

[3]. S.A. Isaacson, R.M. Kirby, Numerical solution of linear Volterra integral equations of the second kind with sharp gradients, J. Comput. Appl. Math. 235(2011) 4283'4301.

[4]. K. Maleknejad', B. Basirat, E. Hashemizadeh, Hybrid Legendre polynomials and Block-Pulse functions approach for nonlinear Volterra'Fredholm integro-differential equations, Computers and Mathematics with Applications 61 (2011) 2821'2828

[5]. F. Mirzaee, S. Fatemeh Hoseini, Application of Fibonacci collocation method for solving Volterra'Fredholm integral equations, Applied Mathematics and Computation 273 (2016) 637'644

[6]. S. Nemati, Numerical solution of Volterra'Fredholm integral equations using Legendre collocation method, Journal of Computational and Applied Mathematics 278 (2015) 29'36

[7]. J. Frankel, A Galerkin solution to regularized Cauchy singular integro-differential equations, Quart. Appl. Math. 24 (1995) 145'258.

[8]. S. Rahbar, E. Hashemizadeh, A Computational Approach to the Fredholm Integral Equation of the Second Kind, Proceedings of the World Congress on Engineering 2008 Vol II WCE 2008, July 2 - 4, 2008, London, U.K

[9]. M. Ghasemi, M.T. Kajani, E. Babolian, Numerical solutions of the nonlinear Volterra'Fredholm integral equations by using homotopy perturbation method, Appl. Math. Comput. 188 (2007) 446'449.

[10]. S. Youse', M. Razzaghi, Legendre wavelets method for the nonlinear Volterra'Fredholm integral equations, Math. Comput. Simul. 70 (2005) 1'8.

[11]. E. Yusufoglu, E. Erbas, Numerical expansion methods for solving Fredholm'Volterra type linear integral equations by interpolation and quadrature rules, Kybernetes 37 (6) (2008) 768'785.

[12]. H. Cerdik-Yaslan, A. Akyuz-Dascioglu, Chebyshev polynomial solution of nonlinear Fredholm'Volterra integro-differential equations, J. Arts. Sci. 5 (2006) 89'101.

[13]. M. Asgari, E. Hashemizadeh, M. Khodabin, K, Maleknejad, Numerical solution of nonlinear stochastic integral equation by stochastic operational matrix based on Bernstein polynomials, Bull. Math. Soc. Sci. Math. Roumanie Tome 57(105) No. 1, 2014, 3-12

[14]. Y. Ordokhani, Solution of nonlinear Volterra'Fredholm'Hammerstein integral equations via rationalized Haar functions, Appl. Math. Comput. 180 (2006) 436'443.

[15]. Y.P. Chen, T. Tang, Spectral methods for weakly singular Volterra integral equations with smooth solutions, J. Comput. Appl. Math. 233 (2009) 938'950.

[16]. C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, 2006.

[17]. Y. Maday, A. Quarteroni, Spectral and pseudo-spectral approximations of Navier'Stokes equations, SIAM J. Numer. Anal. 19 (1982) 761'780.

[18]. J.S.Westhaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-Dependent Problems, Cambridge University Press, Cambridge, 2007.

[19]. S. Yalcinbas, M. Sezer, A Taylor collocation method for the approximate solution of general linear Fredholm'Volterra integro-difference equations with mixed argument, Appl. Math. Comput. 175 (2006) 675'690.

[20]. M. Razzaghi, HR. Marzban, Direct Method for Variational Problems via Hybrid of Block-Pulse and Chebyshev Functions, Mathematical Problems in Engineering Volume 6, pp, 85-97

**...(download the rest of the essay above)**