^{1}

^{2}

In the paper, the approximate solution for the two-dimensional linear and nonlinear Volterra-Fredholm integral equation (V-FIE) with singular kernel by utilizing the combined Laplace-Adomian decomposition method (LADM) was studied. This technique is a convergent series from easily computable components. Four examples are exhibited, when the kernel takes Carleman and logarithmic forms. Numerical results uncover that the method is efficient and high accurate.

The V-FIE arises from parabolic boundary value problems. In practical applications one frequently encounters the V-FIE with singular kernel of the form

The functions

There are several techniques that have been utilized to handle the integral Equation (1), in [

Many authors have studied solutions of two-dimensional linear and nonlinear integral equations by utilizing different techniques, such as Abdou et al. in [

In this paper, we will discuss the combined (LADM) to approximate solutions with high degree of accuracy for V-FIE with a generalized singular kernel.

Consider the integral equation

The (ADM) introduces the following expression

for the solution

where

Substituting Equation (3) and Equation (4) into Equation (2) yields

The components

Relations (6,7) will enable us to determine the components

We assume that the kernel

Applying the Laplace transform to both sides of Equation (8) gives:

The ADM can be used to handle Equation (9). We represent the linear term

Substituting Equation (3) and Equation (4) into Equation (9) leads to

The ADM introduces the recursive relation

Applying the inverse Laplace transform to the first part of Equation (11) gives

We assume that the kernel

Applying the Laplace transform to both sides of Equation (12) gives:

Using the same method we shall find at the end the required solution by the inverse of Laplace transform.

We consider two examples for the integral equation

We consider the linear and nonlinear cases:

Example 1 [(see 1)]: Consider the V-FIE with Carleman kernel

Using Maple 17, we obtain

Example 2 [(see 1)]: Consider the V-FIE with logarithmic kernel

Using Maple 17, we obtain

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

0.000000000E+00 | 1.00000000E−06 | 0.000000000E+00 | 1.00000000E−06 | 1.000000000E−06 | −1.00E+00 | |

0.000000000E+00 | 2.500000000E−07 | 0.000000000E+00 | 2.500000000E−07 | 2.50000000E−07 | −5.00E−01 | |

0.000000000E+00 | 2.500000000E−07 | 0.000000000E+00 | 2.500000000E−07 | 2.500000000E−07 | 5.00E−01 | |

3.000000000E−16 | 9.999999997E−07 | 3.000000000E−16 | 9.999999997E−07 | 1.000000000E−06 | 1.00E+00 | |

8.402300000E−05 | 4.900840230E−01 | 8.148970000E−05 | 4.900814897E−01 | 4.900000000E−01 | −1.00E+00 | |

1.31121000E−05 | 1.225131121E−01 | 1.025640000E−05 | 1.225102564E−01 | 1.225000000E−01 | −5.00E−01 | |

2.447000000E−05 | 1.224755300E−01 | 1.026020000E−05 | 1.224897398E−01 | 1.225000000E−01 | 5.00E−01 | |

1.568011000E−04 | 4.898431989E−01 | 8.150950000E−05 | 4.899184905E−01 | 4.900000000E−01 | 1.00E+00 |

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

4.690000000E−13 | 9.999995310E−07 | 5.073000000E−13 | 9.999994927E−07 | 1.000000000E−06 | −1.00E+00 | |

1.464000000E−13 | 2.499998536E−07 | 1.277000000E−13 | 2.499998723E−07 | 2.50000000E−07 | −5.00E−01 | |

2.732000000E−13 | 2.499997268E−07 | 1.278000000E−13 | 2.499998722E−07 | 2.500000000E−07 | 5.00E−01 | |

8.754000000E−13 | 9.999991246E−07 | 5.076000000E−13 | 9.999994924E−07 | 1.000000000E−06 | 1.00E+00 | |

1.609239000E−04 | 4.898390761E−01 | 1.739859000E−04 | 4.898260141E−01 | 4.900000000E−01 | −1.00E+00 | |

5.022500000E−05 | 1.224497750E−01 | 4.379890000E−05 | 1.224562011E−01 | 1.225000000E−01 | −5.00E−01 | |

9.366060000E−05 | 1.224063394E−01 | 4.38206000E−05 | 1.224561794E−01 | 1.225000000E−01 | 5.00E−01 | |

3.001406000E−04 | 4.896998594E−01 | 1.740717000E−04 | 4.898259283E−01 | 4.900000000E−01 | 1.00E+00 |

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

6.000000000E−16 | 9.999999994E−07 | 1.000000000E−06 | −1.00E+00 | |

1.000000000E−16 | 2.499999999E−07 | 2.50000000E−07 | −5.00E−01 | |

1.000000000E−16 | 2.500000001E−07 | 2.500000000E−07 | 5.00E−01 | |

1.000000000E−15 | 1.000000001E−06 | 1.000000000E−06 | 1.00E+00 | |

1.467841000E−04 | 4.898532159E−01 | 4.900000000E−01 | −1.00E+00 | |

2.527690000E−05 | 1.224747231E−01 | 1.225000000E−01 | −5.00E−01 | |

2.527990000E−05 | 1.225252799E−01 | 1.225000000E−01 | 5.00E−01 | |

1.467682000E−04 | 4.901467682E−01 | 4.900000000E−01 | 1.00E+00 |

Example 3 [(see 11, 13)]: Consider the V-FIE with generalized Carleman kernel

Using Maple 17, we obtain

Example 4 [(see 11,13)]: Consider the V-FIE with generalized logarithmic kernel

Using Maple 17, we obtain

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

7.500000000E−13 | 1.000000750E−06 | 1.000000000E−06 | −1.00E+00 | |

2.741000000E−13 | 2.500002741E−07 | 2.50000000E−07 | −5.00E−01 | |

2.741000000E−13 | 2.500002741E−07 | 2.500000000E−07 | 5.00E−01 | |

7.500000000E−13 | 1.000000750E−06 | 1.000000000E−06 | 1.00E+00 | |

2.2570318000E−04 | 4.902570318E−01 | 4.900000000E−01 | −1.00E+00 | |

9.399840000E−05 | 1.225939984E−01 | 1.225000000E−01 | −5.00E−01 | |

9.407250000E−05 | 1.225940725E−01 | 1.225000000E−01 | 5.00E−01 | |

2.573282000E−04 | 4.902573282E−01 | 4.900000000E−01 | 1.00E+00 |

Nonlinear | Linear | Exact | ||||
---|---|---|---|---|---|---|

Error | App. | Error | App. | |||

0.000000000E+00 | −4.665600000E−14 | 2.285000000E−20 | −4.665602285E−14 | −4.665600000E−14 | −1.00E+00 | |

0.000000000E+00 | −1.45800000E−15 | 4.980000000E−22 | −1.458000498E−15 | −1.45800000E−15 | −5.00E−01 | |

0.000000000E+00 | 1.45800000E−15 | 7.929000000E−21 | 1.457992071E−15 | 1.45800000E−15 | 5.00E−01 | |

0.000000000E+00 | 4.665600000E−14 | 3.638500000E−19 | 4.665563615E−14 | 4.665600000E−14 | 1.00E+00 | |

2.646100000E−08 | −4.095973539E−03 | 4.223060000E−06 | −4.100223060E−03 | −4.096000000E−03 | −1.00E+00 | |

1.800000000E−11 | −1.279999820E−04 | 1.230657000E−07 | −1.281230657E−04 | −1.280000000E−04 | −5.00E−01 | |

2.862000000E−10 | 1.279997138E−04 | 3.023951300E−06 | 1.249760487E−04 | 1.280000000E−04 | 5.00E−01 | |

4.202460000E−07 | 4.095579754E−03 | 1.374088740E−04 | 3.958591126E−03 | 4.096000000E−03 | 1.00E+00 |

Nonlinear | Linear | Exact | ||||
---|---|---|---|---|---|---|

Error | App. | Error | App. | |||

0.000000000E+00 | −4.665600000E−14 | 1.828950000E−18 | −4.665417105E−14 | −4.665600000E−14 | −1.00E+00 | |

0.000000000E+00 | −1.45800000E−15 | 3.666200000E−20 | −1.457963338E−15 | −1.45800000E−15 | −5.00E−01 | |

0.000000000E+00 | 1.45800000E−15 | 3.666300000E−20 | 1.458036663E−15 | 1.45800000E−15 | 5.00E−01 | |

0.000000000E+00 | 4.665600000E−14 | 1.828990000E−18 | 4.665782899E−14 | 4.665600000E−14 | 1.00E+00 | |

1.965280000E−06 | −4.097965280E−03 | 7.262218110E−04 | −3.369778189E−03 | −4.096000000E−03 | −1.00E+00 | |

1.179900000E−09 | −1.280011799E−04 | 1.400159720E−05 | −1.139984028E−04 | −1.280000000E−04 | −5.00E−01 | |

1.179900000E−09 | 1.280011799E−04 | 1.594017510E−05 | 1.439401751E−04 | 1.280000000E−04 | 5.00E−01 | |

1.966603000E−06 | 4.097966603E−03 | 8.51706301E−04 | 4.947706301E−03 | 4.096000000E−03 | 1.00E+00 |

In this study, we considered linear and nonlinear integral equations of type Volterra-Fredholm with singular kernel. We have proven that the (LADM) is effective and useful technique for solving these kinds of integral equations with singular kernel and many nonlinear problems, efficiency and accuracy of the introduced method are illustrated by four numerical examples which showed simplicity of this method.

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

Fatheah Ahmed Hendi,Manal Mohamed Al-Qarni, (2016) Numerical Treatment of Nonlinear Volterra-Fredholm Integral Equation with a Generalized Singular Kernel. American Journal of Computational Mathematics,06,245-250. doi: 10.4236/ajcm.2016.63025