ϑ ) are called homotopies.

According to the HPM, we can use the embedding parameter as a “small parameter”, and assume that the solution of Equation (3) can be written as a power series in :

ϑ = φ 0 + φ 1 + 2 φ 2 + (5)

when 1 , the approximate solution of Equation (2) is obtained with

φ = lim 1 ϑ = φ 0 + φ 1 + φ 2 + (6)

The series Equation (6) is convergent for most cases; however, the rate of convergence depends upon the nonlinear operator L [11] .

3. The HPM Applied to Nonlinear Mixed V-FIE

To illustrate the HPM, for nonlinear mixed V-FIE let us consider the Equation (1)

H ( ϑ , ) = ϑ ( , t ) g ( , t ) λ 0 t Ω F ( t , ζ ) k ( | η | ) γ ( η , ζ , ϑ ( η , ζ ) ) d η d ζ = 0 (7)

By the HPM, we can expand ϑ ( , t ) into the form

ϑ ( , t ) = φ 0 ( , t ) + φ 1 ( , t ) + 2 φ 2 ( , t ) + (8)

and the approximate solution is

φ ( , t ) = lim 1 ϑ ( , t ) = φ 0 ( , t ) + φ 1 ( , t ) + φ 2 ( , t ) + (9)

and in sum, according to [15] , He’s HPM considers the nonlinear term γ ( φ ) as

γ ( φ ) = i = 0 i H i = H 0 + H 1 + 2 H 2 + , (10)

where H n s are the so-called He’s polynomials [15] , which can be calculated by using the formula

H n = 1 n n n [ γ ( η , ζ , i = 0 i φ i ) ] = 0 , n = 0 , 1 , 2 , (11)

Substituting (8) and (10) into (7) and equating the terms with identical powers of , we have

0 : φ 0 ( , t ) = g ( , t ) , i + 1 : φ i + 1 ( , t ) = λ 0 t Ω F ( t , ζ ) k ( | η | ) H i d η d ζ , i 0 (12)

The components φ i ( , t ) , i 0 can be computing by using the recursive relations (12).

4. The Combined LHPM Applied to Nonlinear Mixed V-FIE [16]

We assume that the kernel k ( | η | ) of Equation (7) takes the form

k ( | η | ) = 1 ( 2 η 2 )

Applying the Laplace transform to both sides of Equation (7), we represent the linear term ϑ ( , t ) from Equation (8) and the nonlinear term γ ( η , ζ , ϑ ( η , ζ ) ) will be represented by the He’s polynomials from Equation (10) and equating the terms with identical powers of , we have:

0 : l { φ 0 ( , t ) } = l { g ( , t ) } , i + 1 : l { φ i + 1 ( , t ) } = λ l [ { F ( t , ζ ) } { k ( | η | ) } { H i } ] , i 0 (13)

Applying the inverse Laplace transform to the first part of Equation (13) gives φ 0 ( , t ) , that will define H 0 . Utilizing H 0 will enable us to evaluate φ 1 ( , t ) . The determination of φ 0 ( , t ) and φ 1 ( , t ) leads to the determination of H 1 that will allows us to determine φ 2 ( , t ) , and so on. This in turn will lead to the complete determination of the components of φ i , i 0 , upon utilizing the second part of Equation (13). The series solution follows immediately after using Equation (9). The obtained series solution may converge to an exact solution if such a solution exists.

5. Numerical Examples

Example 5.1 [7] :

Consider the linear mixed V-FIE with a generalized Cauchy kernel

φ ( , t ) = g ( , t ) + λ 0 t 1 1 ζ 2 1 ( 2 η 2 ) φ ( η , ζ ) d η d ζ (14)

λ = 1.5 , N = 20 , theexactsolution φ ( , t ) = 5 t 6

we obtain Table 1.

Example 5.2 [8] :

Consider the nonlinear mixed V-FIE with a generalized Cauchy kernel

φ ( , t ) = g ( , t ) + λ 0 t 1 1 ζ 2 1 ( 2 η 2 ) φ 3 ( η , ζ ) d η d ζ (15)

λ = 1.5 , N = 20 , theexactsolution φ ( , t ) = 5 t 6

we obtain Table 2.

The results for this examples using the LHPM obtained in Table 1 and Table 2 are best from the results in [7] [8] where the solution was obtained using TMM.

Table 1. Results obtained for example 1 and error.

Table 2. Results obtained for example 2 and error.

6. Conclusion

In this article, we proposed LHPM and used it for solving nonlinear mixed V- FIE with a generalized singular kernel. As examples show, the displayed technique diminishes the computational difficulties of other methods. An interesting feature of this method is that the error is too small and all the calculations can be done straightforward. It can be concluded that LHPM is a very simple, powerful and effective method.

Acknowledgements

The authors would like to thank the king Abdulaziz city for science and technology.

Cite this paper
Hendi, F. and Al-Qarni, M. (2017) Numerical Solution of Nonlinear Mixed Integral Equation with a Generalized Cauchy Kernel. Applied Mathematics, 8, 209-214. doi: 10.4236/am.2017.82017.
References

[1]   Pachpatta, B.G. (1986) On Mixed Volterra-Fredholm Type Integral Equations. Indian Journal of Pure and Applied Mathematics, 17, 488-496.

[2]   Hacia, L. (1996) On Approximate Solution for Integral Equations of Mixed Type. ZAMM—Journal of Applied Mathematics and Mechanics, 76, 415-416.

[3]   Brunner, H. (1990) On the Numerical Solution of Nonlinear Volterra-Fredholm Integral Equation by Collocation Methods. SIAM Journal on Numerical Analysis, 27, 987-1000.
https://doi.org/10.1137/0727057

[4]   Maleknejad, K. and Hadizadeh, M. (1999) A New Computational Method for Volterra-Fredholm Integral Equations. Journal of Computational and Applied Mathematics, 37, 1-8.
https://doi.org/10.1016/S0898-1221(99)00107-8

[5]   Adomian, G. (1991) A Review of the Decomposition Method and Some Recent Results for Nonlinear Equation. Computers & Mathematics with Applications, 21, 101-127.
https://doi.org/10.1016/0898-1221(91)90220-X

[6]   Wazwaz, A.M. (1999) A Reliable Modification of Adomian’s Decomposition Method. Applied Mathematics and Computation, 102, 77-86.
https://doi.org/10.1016/S0096-3003(98)10024-3

[7]   Abdou, M.A., El-Kalla, I.L. and Al-Bugami, A.M. (2011) Numerical Solution for Volterra-Fredholm Integral Equation with a Generalized Singular Kernel. Journal of Modern Methods in Numerical Mathematics, 2, 1-15.
https://doi.org/10.20454/jmmnm.2011.60

[8]   Al-Bugami, A.M. (2013) Toeplitz Matrix Method and Volterra-Hammerstin Integral Equation with a Generalized Singular Kernel. Progress in Applied Mathematics, 6, 16-42.

[9]   Abdou, M.A., El-Kalla, I.L. and Al-Bugami, A.M. (2011) New Approach for Convergence of the Series Solution to a Class of Hammerstein Integral Equations. International Journal of Applied Mathematics and Computation, 3, 261-269.

[10]   El-Kalla, I.L. and Al-Bugami, A.M. (2012) Numerical Solution for Nonlinear Volterra-Fredholm Integral Equation with Applications in Torsion Problems. International Journal of Computational and Applied Mathematics, 7, 403-418.

[11]   He, J.H. (1999) Homotopy Perturbation Technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262.
https://doi.org/10.1016/S0045-7825(99)00018-3

[12]   He, J.H. (2000) A Coupling Method of a Homotopy Technique and a Perturbation Technique for Non-Linear Problems. International Journal of Non-Linear Mechanics, 35, 37-43.
https://doi.org/10.1016/S0020-7462(98)00085-7

[13]   Abdulaziz, O., Hashim, I. and Chowdhury, M.S.H. (2008) Solving Variational Problems by Homotopy Perturbation Method. International Journal for Numerical Methods in Engineering, 75, 709-721.
https://doi.org/10.1002/nme.2279

[14]   Golbabai, A. and Keramati, B. (2009) Solution of Non-Linear Fredholm Integral Equations of the First Kind Using Modified Homotopy Perturbation Method. Chaos, Solitons and Fractals, 39, 2316-2321.
https://doi.org/10.1016/j.chaos.2007.06.120

[15]   Ghorbani, A. (2009) Beyond Adomian Polynomials: He Polynomials. Chaos, Solitons and Fractals, 39, 1486-1492.
https://doi.org/10.1016/j.chaos.2007.06.034

[16]   Hendi, F.A. (2011) Laplace Adomian Decomposition Method for Solving the Nonlinear Volterra Integral Equation with Weakly Kernels. Studies in Nonlinear Sciences, 2, 129-134.

 
 
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