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In this article, we present approximate solution of the two-dimensional singular nonlinear mixed Volterra-Fredholm integral equations (V-FIE), which is deduced by using new strategy (combined Laplace homotopy perturbation method (LHPM)). Here we consider the V-FIE with Cauchy kernel. Solved examples illustrate that the proposed strategy is powerful, effective and very simple.

The V-FIE arises from parabolic boundary value problems. Many authors have interested in solving the linear and nonlinear integral equation. The time collocation method was introduced by Pachpatta [

We consider the nonlinear mixed V-FIE with a generalized singular kernel

The functions

Many authors have studied solutions of linear and nonlinear integral equations by utilizing different techniques, for example Abdou et al. in [

With the quick advancement of nonlinear sciences, many analytical and numerical techniques have been produced and developed by various scientists, for example, the HPM introduced by He [

In this article, we present new strategy which is the combined LHPM to obtain approximate solutions with high degree of accuracy for the nonlinear mixed V-FIE with a generalized Cauchy kernel.

In this section, we will present the HPM. We consider a general integral equation

where

where

and the process of changing

According to the HPM, we can use the embedding parameter

when

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

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

By the HPM, we can expand

and the approximate solution is

and in sum, according to [

where

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

The components

We assume that the kernel

Applying the Laplace transform to both sides of Equation (7), we represent the linear term

Applying the inverse Laplace transform to the first part of Equation (13) gives

Example 5.1 [

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

we obtain

Example 5.2 [

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

we obtain

The results for this examples using the LHPM obtained in

Exact | App. | Error | |
---|---|---|---|

t = 0.03 | |||

−1.00E+00 | −7.29000E−10 | −7.28829E−10 | 1.71000E−13 |

−8.00E−01 | −2.38879E−10 | −2.38850E−10 | 2.90000E−14 |

−6.00E−01 | −5.66870E−11 | −5.66841E−11 | 2.90000E−15 |

6.00E−01 | 5.66870E−11 | 5.66899E−11 | 2.90000E−15 |

8.00E−01 | 2.38879E−10 | 2.38908E−10 | 2.90000E−14 |

1.00E+00 | 7.29000E−10 | 7.29171E−10 | 1.71000E−13 |

t = 0.7 | |||

−1.00E+00 | −1.17649E−01 | −1.04112E−01 | 1.35370E−02 |

−8.00E−01 | −3.855120E−02 | −3.61709E−02 | 2.38030E−03 |

−6.00E−01 | −9.14839E−03 | −8.90295E−03 | 2.45440E−04 |

6.00E−01 | 9.14839E−03 | 9.40544E−03 | 2.57050E−04 |

8.00E−01 | 3.85512E−02 | 4.12071E−02 | 2.65590E−03 |

1.00E+00 | 1.17649E−01 | 1.34400E−01 | 1.67510E−02 |

Exact | App. | Error | |
---|---|---|---|

t = 0.03 | |||

−1.00E+00 | −7.29000E−10 | −7.29000E−10 | 0.0000E+00 |

−8.00E−01 | −2.38879E−10 | −2.38879E−10 | 0.0000E+00 |

−6.00E−01 | −5.66870E−11 | −5.66870E−11 | 0.0000E+00 |

6.00E−01 | 5.66870E−11 | 5.66870E−11 | 0.0000E+00 |

8.00E−01 | 2.38879E−10 | 2.38879E−10 | 0.0000E+00 |

1.00E+00 | 7.29000E−10 | 7.29000E−10 | 0.0000E+00 |

t = 0.7 | |||

−1.00E+00 | −1.17649E−01 | −1.17573E−01 | 7.60000E−05 |

−8.00E−01 | −3.85512E−02 | −3.85499E−02 | 1.30000E−06 |

−6.00E−01 | −9.14839E−03 | −9.14838E−03 | 1.10000E−08 |

6.00E−01 | 9.14839E−03 | 9.14840E−03 | 1.80000E−08 |

8.00E−01 | 3.85512E−02 | 3.85525E−02 | 1.30000E−06 |

1.00E+00 | 1.17649E−01 | 1.17725E−01 | 7.60000E−05 |

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.

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

Hendi, F.A. and Al-Qarni, M.M. (2017) Numerical Solution of Nonlinear Mixed Integral Equation with a Generalized Cauchy Kernel. Applied Ma- thematics, 8, 209-214. https://doi.org/10.4236/am.2017.82017