Many researchers studied and discussed (LMVFIE's), Muna M. and Iman N. in [4] using Lagrange polynomials for solving the **linear** **Volterra**-**Fredholm** **integral** **equation**. Hendi F. and Bakodah H. in [5] employed discrete adomain decomposition method to solve **Fredholm**-**Volterra** **integral** **equation** in two dimensional space. Majeed S. and Omran H. in[6] applied the repeated Trapezoidal method and the repeated Simpson's 1/3 method for solving **linear** **Fredholm**- **Volterra** **integral** **equation**, Omran H. in[7] applied the repeated Trapezoidal method and the repeated Simpson's method for solving the first order **linear** **Fredholm**-**Volterra** integro-differential equations. Maleknejad K. and Mahdiani K. in [8] using Piecewise Constant block-pulse functions for solving **linear** two Dimensional **Fredholm**-**Volterra** **Integral** Equations. Hendi F. and Albugami A. in [9] adopt collocation and Galerkin methods for solving **Fredholm**–**Volterra** **integral** **equation** of the second kind.

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In recent years, many researchers have been successfully applying Bernstein polynomials method (BPM) to various **linear** and nonlinear integro-differential **equation**. For example, Bernstein polynomials method applied to find an approximate solution for **Fredholm** integro-Differential **equation** and **integral** **equation** of the second kind in [19]. The propose method is applied to find an approximate solution to initials values problem for high-order nonlinear **Volterra**- **Fredholm** integro differential **equation** of the second kind in [20]. Application of the Bernstein Polynomials for Solving the Nonlinear **Fredholm** Integro-Differential Equations is found in [21]. This method is used to find an approximate Solution of Fractional Integro-Differential Equations in [22].

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In this section, the Aitken's method has been applied successfully on fixed-point method to find the solution of our **integral** **equation**, where the first three approximations are computed by fixed-point method and then substituted in **equation** (6) to get the procedure

Where: "h , g" are given functions , K(x , y) function in two variables are names the kernel of the integration neutralization, λ is a scalar parameter, the given function K(x , y) which depends up on the current variables x as well as the variables y is known as the kernel or nucleus of the integration neutralization. The integration neutralization can be classify for two classes. The first, it is name of "**Volterra** **integral** **equation**" (VIE) where the Volterra’s significant job in this domain was complete in 1884-1896 and the second, name "**Fredholm** **integral** **equation**" (FIE) where the Fredholm’s significant "contribution was made in 1900-1903". **Fredholm** progressing the theory of this integration neutralization such as a limit to the **linear** system of neutralization[1]. Integrat equations play an important role in many branches of sciences such as mathematics, biology, chemistry, physics, mechanics and engineering. Therefore, many different techniques are used to solve these types of equations. Also **Integral** equations diverse evolved directly linked to the number of several branche of mathematics in the differentiation account, integration account ,differential neutralization and rounding issues to addition to the very concepts and physical links issues[2] , [7] . There is equivalence relation between **Integral** equations and ordinary differential equations. There is a close relationship among differentials and integration neutralization, and several issues may be features either way. In example, Green's function, **Fredholm** theory, and Maxwell's neutralization [3]. The of the **integral** equations depends on the type of **integral** **equation** **fredholm** , **volterra** first kind ,or second kind ,**linear** or nonlinear , homogeneous , or non –hom. Also , the solution of **integral** equs. Depend on the kernel of **integral** equs., whether if it is symmetric or difference type , in some case , it is difficulty to solve **integral** equs.[4] . Therefore , there exist approximate and numerical method for solving **integral** equs . **Integral** equations are important in many applications. Problems in which **integral** equations are encountered include radiative transfer, and the oscillation of a string, membrane, or axle. Oscillation problems may also be solved as differential equations . Both **Fredholm** and **Volterra** equations are **linear** **integral** equations, due to the **linear** behavior of y(x) under the **integral** [5] ,[8].

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Because the **equation** in (II.62) combines the differential operator and the **integral** operator ,then it is necessary to deﬁne initial conditions for the determination of the particular solution 𝑢(𝑥) of the nonlinear **Volterra** integro-differential **equation**. The nonlinear **Volterra** integro-differential **equation** appeared after its establishment by **Volterra**. It appears in many physical applications such as glass-forming process, heat transfer, diffusion process in general, neutron diffusion and biological species coexisting together with increasing and decreasing rates of generating. More details about the sources where these equations arise can be found in physics, biology and books of engineering applications. We applied many methods to handle the **linear** **Volterra** integro- differential equations of the second kind. In this section we will use only some of these methods. However, the other methods presented i can be used as well. In what follows we will apply the combined Laplace transform-Adomian decomposition method,

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In this section, we review the basic properties of the SBM that are essential for this work. For more details see [2, 12]. Let the functions α(t, X ), β(t, X ) and γ(t, X) hold in lipschitz conditions and **linear** growth, i.e. there are constants k 1 , k 2 , k 3 , k 4 , k 5 > 0 and k 6 > 0 such that:

In this paper, we applied (LADM) for solution two dimensional **linear** mixed **integral** equations of type **Volterra**- **Fredholm** with Hilbert kernel. Additionally, comparison was made with Toeplitz matrix method (TMM). It could be concluded that (LADM) was an effective technique and simple in finding very good solutions for these sorts of equations.

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-Ado- mian decomposition method (LADM) was studied. This technique is a convergent series from eas- ily 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.

In this paper, we focus on designing feed forward neural network (FFNN) for solving Mixed **Volterra** – **Fredholm** **Integral** Equations (MVFIEs) of second kind in 2–dimensions. in our method, we present a multi – layers model consisting of a hidden layer which has five hidden units (neurons) and one **linear** output unit. Transfer function (Log – sigmoid) and training algorithm (Levenberg – Marquardt) are used as a sigmoid activation of each unit. A comparison between the results of numerical experiment and the analytic solution of some examples has been carried out in order to justify the efficiency and the accuracy of our method. Key words: Feed Forward neural network, Levenberg – Marquardt (trainlm) training algorithm, Mixed **Volterra** - **Fredholm** **integral** equations.

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T T he nonlinear and **linear** **Volterra**-**Fredholm** ordinary **integral** equations arise from vari- ous physical and biological models. The essential features of these models are of wide applicable. These models provide an important tool for mod- eling a numerous problems in engineering and sci- ence [6, 7]. Modelling of certain physical phenom- ena and engineering problems [8, 9, 10, 11, 12] leads to two-dimensional nonlinear and **linear** **Volterra**-**Fredholm** ordinary **integral** equations of the second kind. Some numerical schemes have been inspected for resolvent of two-dimensional ordinary **integral** equations by several probers. Computational complexity of mathematical op- erations is the most important obstacle for solv-

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The Legendre wavelet operational matrix P, together with the integration of the product of two Legendre wavelet vectors functions, are utilized to solve the **integral** **equation**. The present method reduces an **integral** **equation** into a set of algebraic equations. In this paper, we use the 6-base Legendre wavelets, the result for the product with quadrature solution is good. For better results, using the greater N is recommended.

Keywords : F uzzy numbers; F uzzy Volterra-Fredholm integral equation; Fuzzy bivariate Chebyshev.. method; Dual fuzzy linear system; Nonnegative matrix.[r]

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A fully discrete version of a piecewise polynomial collocation method based on new collocation points, is constructed to solve nonlinear **Volterra**-**Fredholm** **integral** equations. In this paper, we obtain existence and uniqueness results and analyze the convergence properties of the collocation method when used to approximate smooth solutions of **Volterra**- **Fredholm** **integral** equations.

x 2 +y 2 ≤ a}, z = 0, and T < ∞. The kernel of the **Fredholm** **integral** term considered in the generalized potential form belongs to the class C([Ω]× [Ω]), while the kernel of **Volterra** **integral** term is a positive and continuous function that belongs to the class C[0,T ]. Also in this work the solution of **Fredholm** **integral** **equation** of the second and ﬁrst kind with a potential kernel is discussed. Many interesting cases are derived and established in the paper.

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The recent available method for solving RHC are Nystrӧm method and numerical formulas by Picard Iteration Method has been done in uniquely and non- uniquely solvable **integral** **equation** for both interior and exterior RHC by (Ismail, 2007) and (Zamzamiar, 2011). While the formula for non-uniquely solvable **integral** **equation** for interior RHC has not been done. Hence, this research aims to construct this formula based Nystrӧm method as well as perform some numerical example, and will not consider the Picard iteration method.

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In this paper, the complex variable function method is used to obtain the hypersingular **integral** equations for the interaction between straight and curved cracks problem in plane elasticity. The curved length coordinate method and suitable quadrature rule are used to solve the integrals for the unknown function, which are later used to evaluate the stress intensity factor, SIF. Three types of stress modes are presented for the numerical results.

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We turn now to the construction of the numerical algorithm for solving **integral** **equation** (1.1) numerically. Obviously the **integral** **equation** (1.2) could either be solved using the same method. The main tool at our disposal is the ability to minimize the error in the nodal polynomial. We begin by constructing statistical spline model, on which the solution to (1.1) and (1.2) are sought, eligible.

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In the section, we have solved two problems about **Fredholm** **integral** **equation** of second kind. For the numerical problem, the analytical solution y1has been known in advance, therefore we test the accuracy of the obtained solutions by computing the deviation: error = absolute(y1-y2), where y2 is the numerical solution.

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This method is also one of the approximate methods that may be used to solve **Volterra** random **integral** **equation** and same time it can be used to solve **linear** and non **linear** random **integral** equations. We will take the trapezoidal rule and consider the non **linear** second kind **Volterra** random **integral** **equation** (1) and by dividing the interval of integration (0,t) in to N-equal subintervals, we have:

[2] H. Brunner, On the numerical solution of nonlinear **Volterra**-**Fredholm** **integral** **equation** by collocation methods, SIAM J. Numer. Anal., 27(4) (1990) 978-1000. [3] A. Cardone, E. Messina and E. Russo, A fast iterative method for discretized **Volterra**-**Fredholm** **integral** equations, J. Comput. Appl. Math., 189 (2006) 568- 579.

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