Linear Volterra-Fredholm Integral Equation

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Vol 6, No 11 (2015)

Vol 6, No 11 (2015)

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 FredholmVolterra integral equation of the second kind.
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Vol 6, No 10 (2015)

Vol 6, No 10 (2015)

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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Existence and Uniqueness of Solution for Linear Mixed Volterra-Fredholm Integral Equations in Banach Space

Existence and Uniqueness of Solution for Linear Mixed Volterra-Fredholm Integral Equations in Banach Space

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

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Reduces Solution of Fredholm Integral Equation to a System of Linear Algebraic Equation

Reduces Solution of Fredholm Integral Equation to a System of Linear Algebraic Equation

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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Reducing Initial Value Problem and Boundary Value Problem to Volterra and Fredholm Integral Equation and Solution of Initial Value Problem
Habeeb Khudhur Kadhim

Reducing Initial Value Problem and Boundary Value Problem to Volterra and Fredholm Integral Equation and Solution of Initial Value Problem Habeeb Khudhur Kadhim

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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Numerical Implementation of Triangular Functions for Solving a Stochastic Nonlinear Volterra-Fredholm Integral Equation

Numerical Implementation of Triangular Functions for Solving a Stochastic Nonlinear Volterra-Fredholm Integral Equation

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:

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Comparison between Adomian’s Decomposition Method and Toeplitz Matrix Method for Solving Linear Mixed Integral Equation with Hilbert Kernel

Comparison between Adomian’s Decomposition Method and Toeplitz Matrix Method for Solving Linear Mixed Integral Equation with Hilbert Kernel

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.

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Numerical Treatment of Nonlinear Volterra Fredholm Integral Equation with a Generalized Singular Kernel

Numerical Treatment of Nonlinear Volterra Fredholm Integral Equation with a Generalized Singular Kernel

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.

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Solving Mixed Volterra - Fredholm Integral Equation (MVFIE) by Designing Neural Network

Solving Mixed Volterra - Fredholm Integral Equation (MVFIE) by Designing Neural Network

In this paper, we focus on designing feed forward neural network (FFNN) for solving Mixed VolterraFredholm 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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Approximation solution of two-dimensional linear stochastic Volterra-Fredholm integral equation via two-dimensional Block-pulse ‎functions

Approximation solution of two-dimensional linear stochastic Volterra-Fredholm integral equation via two-dimensional Block-pulse ‎functions

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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NUMERICAL SOLUTION OF LINEAR FREDHOLM AND VOLTERRA INTEGRAL EQUATION OF THE SECOND KIND BY USING LEGENDRE WAVELETS

NUMERICAL SOLUTION OF LINEAR FREDHOLM AND VOLTERRA INTEGRAL EQUATION OF THE SECOND KIND BY USING LEGENDRE WAVELETS

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.

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Fuzzy bivariate Chebyshev method for solving fuzzy Volterra-Fredholm integral equations

Fuzzy bivariate Chebyshev method for solving fuzzy Volterra-Fredholm integral equations

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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Collocation Method for Nonlinear Volterra Fredholm Integral Equations

Collocation Method for Nonlinear Volterra Fredholm Integral Equations

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.

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Fredholm Volterra integral equation with potential kernel

Fredholm Volterra integral equation with potential kernel

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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Nystrom method for solving non-uniquely solvable interior Riemann-Hilbert problem on region with corners via integral equation

Nystrom method for solving non-uniquely solvable interior Riemann-Hilbert problem on region with corners via integral equation

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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Mode Stresses for the Interaction between Straight and Curved Cracks Problem in Plane Elasticity

Mode Stresses for the Interaction between Straight and Curved Cracks Problem in Plane Elasticity

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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An approach based on statistical spline model for Volterra-Fredholm integral equations

An approach based on statistical spline model for Volterra-Fredholm integral equations

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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Modified Numerical Method for Solving Fredholm Integral Equations

Modified Numerical Method for Solving Fredholm Integral Equations

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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Approximate Solution of the Volterra Random Integral Equations

Approximate Solution of the Volterra Random Integral Equations

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:

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A computational method for nonlinear mixed Volterra-Fredholm integral equations

A computational method for nonlinear mixed Volterra-Fredholm integral equations

[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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