Block-by-Block Method for Solving Nonlinear Volterra-Fredholm Integral Equation

In this paper, the existence and uniqueness of solution of the linear two dimensional Volterra integral equation of the second kind with Continuous Kernel are discussed and proved.RungeKutta method(R. KM)and Block by block method (B by BM) are used to solve this type of two dimensional Volterra integral equation of the second kind. Numerical examples are considered to illustrate the effectiveness of the proposed methods and the error is estimated.

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Runge-Kutta and Block by Block Methods to Solve Linear Two-Dimensional Volterra Integral Equation with Continuous Kernel

JOURNAL OF ADVANCES IN MATHEMATICS ◽

10.24297/jam.v11i5.1247 ◽

2015 ◽

Vol 11 (5) ◽

pp. 5220-5229

Author(s):

Abeer Majed AL-Bugami

Keyword(s):

Integral Equation ◽

Volterra Integral Equation ◽

Existence And Uniqueness ◽

Uniqueness Of Solution ◽

Two Dimensional ◽

Block Method ◽

Numerical Examples ◽

Block Methods ◽

Runge Kutta ◽

Method R

In this paper, the existence and uniqueness of solution of the linear two dimensional Volterra integral equation of the second kind with Continuous Kernel are discussed and proved.RungeKutta method(R. KM)and Block by block method (B by BM) are used to solve this type of two dimensional Volterra integral equation of the second kind. Numerical examples are considered to illustrate the effectiveness of the proposed methods and the error is estimated.

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Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

Mediterranean Journal of Mathematics ◽

10.1007/s00009-021-01822-5 ◽

2021 ◽

Vol 18 (5) ◽

Author(s):

Francesco Aldo Costabile ◽

Maria Italia Gualtieri ◽

Anna Napoli

Keyword(s):

Boundary Value Problem ◽

Boundary Conditions ◽

Existence And Uniqueness ◽

Interpolation Problem ◽

Boundary Value ◽

Computational Tools ◽

Numerical Examples ◽

Odd Order ◽

Uniqueness Of The Solution ◽

Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

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Numerical solution of two-dimensional first kind Fredholm integral equations by using linear Legendre wavelet

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691316500041 ◽

2016 ◽

Vol 14 (01) ◽

pp. 1650004 ◽

Cited By ~ 3

Author(s):

M. Tahami ◽

A. Askari Hemmat ◽

S. A. Yousefi

Keyword(s):

Integral Equation ◽

Tensor Product ◽

Numerical Solution ◽

Fredholm Integral Equation ◽

Fredholm Integral Equations ◽

Wavelet Basis ◽

Two Dimensional ◽

Legendre Wavelets ◽

Numerical Examples ◽

One Dimensional

In one-dimensional problems, the Legendre wavelets are good candidates for approximation. In this paper, we present a numerical method for solving two-dimensional first kind Fredholm integral equation. The method is based upon two-dimensional linear Legendre wavelet basis approximation. By applying tensor product of one-dimensional linear Legendre wavelet we construct a two-dimensional wavelet. Finally, we give some numerical examples.

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Existence and Uniqueness of the Solution for an Integral Equation with Supremum, via w-Distances

Symmetry ◽

10.3390/sym12091554 ◽

2020 ◽

Vol 12 (9) ◽

pp. 1554 ◽

Cited By ~ 1

Author(s):

Veronica Ilea ◽

Diana Otrocol

Keyword(s):

Integral Equation ◽

Fixed Point ◽

Existence And Uniqueness ◽

Stability Result ◽

Fixed Point Problem ◽

Comparison Theorems ◽

Point Problem ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Uniqueness Of The Solution

Following the idea of T. Wongyat and W. Sintunavarat, we obtain some existence and uniqueness results for the solution of an integral equation with supremum. The paper ends with the study of Gronwall-type theorems, comparison theorems and a result regarding a Ulam–Hyers stability result for the corresponding fixed point problem.

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On the Harmonic Problem with Nonlinear Boundary Integral Conditions

International Journal of Analysis ◽

10.1155/2014/976520 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5

Author(s):

Saker Hacene

Keyword(s):

Integral Equation ◽

Existence And Uniqueness ◽

Integral Method ◽

Nonlinear Boundary ◽

Monotone Operators ◽

Boundary Integral ◽

Boundary Integral Method ◽

Integral Conditions ◽

Uniqueness Of The Solution ◽

Harmonic Problem

In the present work, we deal with the harmonic problems in a bounded domain of ℝ2 with the nonlinear boundary integral conditions. After applying the Boundary integral method, a nonlinear boundary integral equation is obtained; the existence and uniqueness of the solution will be a consequence of applying theory of monotone operators.

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Existence, Uniqueness, and Approximation for Solutions of a Functional-integral Equation in Lp Spaces

TEMA (São Carlos) ◽

10.5540/tema.2019.020.03.403 ◽

2019 ◽

Vol 20 (3) ◽

pp. 403

Author(s):

Suzete M Afonso ◽

Juarez S Azevedo ◽

Mariana P. G. Da Silva ◽

Adson M Rocha

Keyword(s):

Integral Equation ◽

Fixed Point ◽

Fixed Point Theorem ◽

Existence And Uniqueness ◽

Functional Integral ◽

Successive Approximation Method ◽

Banach Fixed Point Theorem ◽

Numerical Examples ◽

Lp Spaces ◽

Chebyshev Quadrature

In this work we consider the general functional-integral equation: \begin{equation*}y(t) = f\left(t, \int_{a}^{b} k(t,s)g(s,y(s))ds\right), \qquad t\in [a,b],\end{equation*}and give conditions that guarantee existence and uniqueness of solution in $L^p([a,b])$, with {$1<p<\infty$}.We use Banach Fixed Point Theorem and employ the successive approximation method and Chebyshev quadrature for approximating the values of integrals. Finally, to illustrate the results of this work, we provide some numerical examples.

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A Triangle Algorithm of Padé-Type Approximant for Two-Dimensional Fredholm Integral Equations of the Second Kind

Mathematical Problems in Engineering ◽

10.1155/2018/3194907 ◽

2018 ◽

Vol 2018 ◽

pp. 1-9

Author(s):

Meilan Sun ◽

Chuanqing Gu

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Fredholm Integral Equation ◽

Recursive Algorithm ◽

High Order ◽

Fredholm Integral Equations ◽

Two Dimensional ◽

Initial Value ◽

Numerical Examples ◽

Type Approximation

The function-valued Padé-type approximation (2DFPTA) is used to solve two-dimensional Fredholm integral equation of the second kind. In order to compute 2DFPTA, a triangle recursive algorithm based on Sylvester identity is proposed. The advantage of this algorithm is that, in the process of calculating 2DFPTA to avoid the calculation of the determinant, it can start from the initial value, from low to high order, and gradually proceeds. Compared with the original two methods, the numerical examples show that the algorithm is effective.

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Modified Adomian Decomposition Method for Solving Fuzzy Volterra-Fredholm Integral Equation

Journal of the Indian Mathematical Society ◽

10.18311/jims/2018/16260 ◽

2018 ◽

Vol 85 (1-2) ◽

pp. 53 ◽

Cited By ~ 4

Author(s):

Ahmed A. Hamoud ◽

Kirtiwant P. Ghadle

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Fredholm Integral Equation ◽

Decomposition Method ◽

Adomian Decomposition Method ◽

Fredholm Integral Equations ◽

Modified Adomian Decomposition Method ◽

Adomian Decomposition ◽

Uniqueness Of The Solution ◽

Fuzzy Solution

In this paper, a modied Adomian decomposition method has been applied to approximate the solution of the fuzzy Volterra-Fredholm integral equations of the first and second Kind. That, a fuzzy Volterra-Fredholm integral equation has been converted to a system of Volterra-Fredholm integral equations in crisp case. We use MADM to find the approximate solution of this system and hence obtain an approximation for the fuzzy solution of the Fuzzy Volterra-Fredholm integral equation. A nonlinear evolution model is investigated. Moreover, we will prove the existence, uniqueness of the solution and convergence of the proposed method. Also, some numerical examples are included to demonstrate the validity and applicability of the proposed technique.

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Existence and Uniqueness of the Solution for a Time-Fractional Diffusion Equation with Robin Boundary Condition

Abstract and Applied Analysis ◽

10.1155/2011/321903 ◽

2011 ◽

Vol 2011 ◽

pp. 1-11 ◽

Cited By ~ 9

Author(s):

Jukka Kemppainen

Keyword(s):

Boundary Condition ◽

Integral Equation ◽

Diffusion Equation ◽

Existence And Uniqueness ◽

Original Problem ◽

Robin Boundary Condition ◽

Fractional Diffusion ◽

Uniqueness Of The Solution ◽

Time Fractional Diffusion Equation ◽

Robin Boundary

Existence and uniqueness of the solution for a time-fractional diffusion equation with Robin boundary condition on a bounded domain with Lyapunov boundary is proved in the space of continuous functions up to boundary. Since a Green matrix of the problem is known, we may seek the solution as the linear combination of the single-layer potential, the volume potential, and the Poisson integral. Then the original problem may be reduced to a Volterra integral equation of the second kind associated with a compact operator. Classical analysis may be employed to show that the corresponding integral equation has a unique solution if the boundary data is continuous, the initial data is continuously differentiable, and the source term is Hölder continuous in the spatial variable. This in turn proves that the original problem has a unique solution.

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