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

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 , Legendre wavelets method , interpolation method , Chebyshev polynomials method , rationalized Haar functions method  and spectral method .

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  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  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 , 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  and Fig. 7,8 and 9 taking cand error

m, n Proposed Method

Proposed Method

Proposed Method

Method in 

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 

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 

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.

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