Abstract:
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.

Abstract:
A series method is used to separate the variables of position and time for the Fredholm-Volterra integral equation of the first kind and the solution of the system in L_2 [0,1] × C[0,T], 0 ≤ t ≤ T < ∞ is obtained, the Fredholm integral equation is discussed using Krein's method. The kernel is written in a Legendre polynomial form. Some important relations are also, established and discussed.

Abstract:
In this paper, under certain conditions, the solution of mixed type of Fredholm-Volterra integral equation is discussed and obtained in the space L_2 ( 1, 1) × C[0, T ], T < ∞. Here, the singular part of kernel of Fredholm-Volterra integral term is established in a logarithmic form, while the kernel of Fredholm-Volterra integral term is a positive continuous function in time and belongs to the class C[0, T ], T < ∞. The solution, when the mixed type integral, takes a system form of Fredholm integral equation of the first or second kind are discussed.

Abstract:
In the study, we will give a method of solving the solution for the nonlinear Volterra-Fredholm integral equation in the reproducing kernel space W(D). The problem on solving the solution of the nonlinear Volterra-Fredholm integral equation is transformed into the problem of solving system of linear equations. The approximate solution converges to exact solution of the nonlinear Volterra-Fredholm integral equation in the sense of . but also in the sense of . C. In addition, the error of the approximate solution is monotone deceasing and high convergence order in the sense of. Numerical experiments illustrate the method is efficient.

Abstract:
This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differential equations using reproducing kernel Hilbert space method. The solution () is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximate solution () is obtained and it is proved to converge to the exact solution (). Furthermore, the proposed method has an advantage that it is possible to pick any point in the interval of integration and as well the approximate solution and its derivative will be applicable. Numerical examples are included to demonstrate the accuracy and applicability of the presented technique. The results reveal that the method is very effective and simple.

Abstract:
We consider a nonlinear Volterra-Fredholm integral equation (NVFIE) of the second kind. The Volterra kernel is time dependent, and the Fredholm kernel is position dependent. Existence and uniqueness of the solution to this equation, under certain conditions, are discussed. The block-by-block method is introduced to solve such equations numerically. Some numerical examples are given to illustrate our results. 1. Introduction Different methods are used to solve integral equations which are investigated from many physical applications such as the mixed problems in the theory of elasticity. Popov [1] applied the orthogonal polynomials method to solve the mixed problem in the mechanics of continuous media. Badr [2] applied Toeplitz matrix method to solve a NVFIE. Abdou et al. [3] discussed the solution of Harmmerstein-Volterra integral equation of the second kind. In [4], Haci obtained, numerically, the solution of a system of Harmmerstein integral equations in the space . The equivalence between Volterra integral equation with degenerate kernel and a linear system of differential equations is mentioned by Cochran [5]. Although there are some works on Hermite-type collocation method for the second-kind VIEs with smooth kernels, not too many studies have dealt with weakly singular kernel. For example, Papatheodorou and Jesanis [6] used the collocation method and obtained the solution of Volterra integrodifferential equation with weakly singular kernels. More information about different analytical and numerical solutions of Volterra equations can be found in Davis [7], Linz [8], Volterra [9], and Wolkenfelt [10]. In this paper, we consider the following NVFIE: The existence of a unique solution for the above equation, under certain conditions, is granted using fixed point theorem, where is the Fredholm kernel and is the Volterra kernel. is called the free term, and the unknown function, , is called the potential function in the applied mathematics, and it will be determined. Both two functions and are assumed in the same space. The parameter has many physical meanings. A numerical method is applied to this equation, and it is reduced it to a system of Volterra integral equations of the second kind. Finally, the block-by-block method is used to obtain the numerical solution of this system. Some examples are stated to illustrate the results. 2. Existence and Uniqueness of Solution To guarantee the existence and uniqueness of solution to (1.1), we write (1.1) in the integral operator form where Also, we assume the following conditions: (i) and satisfies, in

Abstract:
This study considers an integral equation of Fredholm-Volterra type, where the Fredholm integral term is measured with respect to the position, while Volterra integral term is measured with respect to the time. Also, we obtained the solution of Fredholm-Volterra integral equation in series form.

Abstract:
This
paper proposes the combined Laplace-Adomian decomposition method (LADM) for
solution two dimensional linear mixed integral equations of type Volterra-Fredholm
with Hilbert kernel. Comparison of the obtained results with those obtained by
the Toeplitz matrix method (TMM) demonstrates that the proposed technique is
powerful and simple.