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R. T. Matoog Assistant Professor. Department of Mathematics, Faculty of Applied Sciences,Umm Al-Qura University, Makkah, Saudi Arabia |

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In this work, the existence of a unique solution of mixed integral equation (MIE) of the second kind is considered, in the space ( ) [0, ] p L C T , where in the domain of integration and t [0,T ] is the time. The kernel of position is considered in a generalized potential form. A numerical method is used to obtain system of Fredholm integral equations (SFIEs). The existence of a unique solution of this system can be proved. Finally, many special cases are considered and established from the work and some numerical results are considered

## Keywords |

Fredholm- Volterra integral equation (F-VIE), generalized potential kernel, degenerate method, linear algebraic system. |

MSC: 45 B05, 45G 10, 65R |

## INTRODUCTION |

Many problems of mathematical physics, engineering and contact problems in the theory of elasticity, fluid mechanics and quantum mechanics lead to one of the form of the integral equations. In [1], Abdou used the separation of variables method to solve the F-VIE of the first kind in the space , where Ã¯ÂÂ in the domain of integration in position, and is the time. The monographs of [2, 3] contain many spectral relationships which are obtained, using the orthogonal polynomial method and potential theory method. In [4], Abdou obtained the spectral relationships for the F-VIE of the first kind in three dimensional. The kernel of FI term is considered in a generalized potential form, while the kernel of VI term is a continuous function in time. Consider the V- FIE |

in the space . Here, the kernel of position takes the form of generalized potential function and the kernel of VI term is a positive continuous function belongs to the classC[0,T ] . The free term f (x , y ,t ) is a known function, is the unknown potential function, Ã¯ÂÂ is the domain of position, Ã¯ÂÂ® is called Poisson ratio,Ã¯ÂÂ is a constant defines the kind of the IE, and Ã¯ÂÂ¬ is a constant, may be complex, and has many physical meaning. In order to guarantee the existence of a unique solution of Eq. (1) we must assume the following conditions: (i) The discontinuous kernel in satisfies |

Theorem (1) (without proof): The IE (1) has an existence and unique solution under the condition . In the reminder part of this work, we represent the generalized potential function in the form of Weber- Sonien integral formula (W-SIf). Then, we represent the W-SIf as a partial differential equation of the first order of Cauchy type. Moreover, the partial second derivatives are represented in the nonhomogeneous wave equation. 2. Weber-Sonien integral formula (W-SIf): |

In this section, we represent the position kernel in the form of a generalized W-SIf. In this aim, after using the polar coordinates in Eqs. (1), (2), we obtain |

Moreover, using the following three formulas, see Bateman and Ergyli [5, 6], |

The position kernel (10) takes a general form of W-SIf. |

3. On the discussion of the W-SIf: |

We derive many special and new cases from the W-SIf of (10) |

(1) Logarithmic kernel: Let, in (10) |

From the previous figures of Carleman function we deduced that as Ã¯ÂÂ® increases the cracks in the material increase. (3) Elliptic kernel: Let, in (10), , we have the elliptic kernel. The importance of the elliptic kernel comes from the work of Kovalenko [7], who developed the FIE of the first kind for the mechanics mixed problem of continuous media and obtained an approximate solution of it. |

General cases: Here, the W-SIf is representing generalized potential form , and as special cases we consider the following: |

Theorem (2): The structure of the kernel W –SIf, represents Cauchy problem for the first order and nonhomogeneous wave equation for the second order. Proof: To prove this we differentiate Eq. (12) with respect to r andÃ¯ÂÂ² respectively, and then adding the result to get |

Using the two famous relations, see Bateman and Ergelyi [6] |

The above formula represents a nonhomogeneous wave equation. So, the second derivative of the generalized potential kernel represents a nonhomogeneous wave equation when m≠0.5. |

## References |

[1] Abdou, M. A. " Associate Professor Fredholm-Volterra integral equation of the first kind and contact problem" J. Appl. Math. Comput. Vol. 125, pp 177-193, 2002. [2] Abdou, M. A. " Spectral relationships for the integral operators in contact problem of impressing stamp" J. Appl. Math. Comput. Vol. 118, pp. 95-111, 2001. [3] Abdou, M. A. "Spectral relationships for the integral equations with Macdonald kernel and contact problem" J. Appl. Math. Comput. Vol. 118, pp. 93-103, 2002. [4] Abdou, M. A. "Fredholm-Volterra integral equation and generalized potential kernel" J. Appl. Math. Comput. Vol. 131, pp. 81-94, 2002. [5] Bateman, G. and Ergelyi, A. "Higher Transcendental Functions" vol. 3 Mc-Graw Hill, London, New York, 1991. [6] Bateman, G. and Ergelyi, A. "Higher Transcendental Functions" vol. 2 Mc-Graw Hill, London, New York, 1989. [7] Kovalenko, E.V. "Some approximate methods of solving integral equations of mixed problems" J. Appl. Math. Mech. Vol. 53, pp. 85-192, 1989. |