Mixed Volterra Fredholm Integral Equations Research Articles

Complex-Valued Suprametric Spaces, Related Fixed Point Results, and Their Applications to Barnsley Fern Fractal Generation and Mixed Volterra–Fredholm Integral Equations

Complex-Valued Suprametric Spaces, Related Fixed Point Results, and Their Applications to Barnsley Fern Fractal Generation and Mixed Volterra–Fredholm Integral Equations

Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space

This paper is dedicated to study the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredohlm integral equations in Banach space, the recent researches considered the study of differential equations of mixed Volterra-Fredholm integral equations with classical order and the study of existence and uniqueness of solutions using approched numerical methodes, the objective of this paper is the study of the existence and uniqueness of fractional order of differential equations with mixed Volterra-Fredholm integral equations using fixed point theory. This work have two important results, the first result was the discussion of the existence of solutions using the Krasnoselskii fixed point theorem after transforming the problem into integral equation firstly then into operator problem suitable for the fixed point theory. The second result will be the existence and uniqueness of solution, this result was obtained by the use of Banach fixed point theorem after the same transformation used in the first one. This work give as conclusion that the boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredholm integral equation has a unique solution in Banach space. Finally, this work was ended with an example to illustrate the results obtained.

Numerical Solution of Stochastic Mixed Volterra–Fredholm Integral Equations Driven by Space-Time Brownian Motion via Two-Dimensional Triangular Functions

Numerical Solution of Stochastic Mixed Volterra–Fredholm Integral Equations Driven by Space-Time Brownian Motion via Two-Dimensional Triangular Functions

Spectral technique with convergence analysis for solving one and two-dimensional mixed Volterra-Fredholm integral equation.

A numerical approach based on shifted Jacobi-Gauss collocation method for solving mixed Volterra-Fredholm integral equations is introduced. The novel technique with shifted Jacobi-Gauss nodes is applied to reduce the mixed Volterra-Fredholm integral equations to a system of algebraic equations that has an easy solved. The present algorithm is extended to solve the one and two-dimensional mixed Volterra-Fredholm integral equations. Convergence analysis for the present method is discussed and confirmed the exponential convergence of the spectral algorithm. Various numerical examples are approached to demonstrate the powerful and accuracy of the technique.

A Computational Technique for Solving Three-Dimensional Mixed Volterra–Fredholm Integral Equations

In this article, a novel and efficient approach based on Lucas polynomials is introduced for solving three-dimensional mixed Volterra–Fredholm integral equations for the two types (3D-MVFIEK2). This method transforms the 3D-MVFIEK2 into a system of linear algebraic equations. The error evaluation for the suggested scheme is discussed. This technique is implemented in four examples to illustrate the efficiency and fulfillment of the approach. Examples of numerical solutions to both linear and nonlinear integral equations were used. The Lucas polynomial method and other approaches were contrasted. A collection of tables and figures is used to present the numerical results. We observe that the exact solution differs from the numerical solution if the exact solution is an exponential or trigonometric function, while the numerical solution is the same when the exact solution is a polynomial. The Maple 18 program produced all of the results.

Legendre-Gauss-Lobatto collocation method for solving multi-dimensional systems of mixed Volterra-Fredholm integral equations

<abstract><p>Integral equations play a crucial role in many scientific and engineering problems, though solving them is often challenging. This paper addresses the solution of multi-dimensional systems of mixed Volterra-Fredholm integral equations (SMVF-IEs) by means of a Legendre-Gauss-Lobatto collocation method. The one-dimensional case is addressed first. Afterwards, the method is extended to two-dimensional linear and nonlinear SMVF-IEs. Several numerical examples reveal the effectiveness of the approach and show its superiority in comparison to other alternative techniques for treating SMVF-IEs.</p></abstract>

Generalized Banach fixed-point theorem and numerical discretization for nonlinear Volterra–Fredholm equations

An investigation of two nonlinear mixed Volterra–Fredholm integral equations of the second kind is undertaken. The equations are constructed as fixed-point problems, under the condition of Lipschitz continuity, the Generalized Banach fixed-point theorem is used to prove the existence of a unique solution without any contraction assumption. A double numerical quadrature algorithm is constructed for the occurring two-dimensional integral. It is theoretically proved, under the existence of relevant partial derivatives, that the proposed numerical scheme has (i) first order of convergence if the nonlinear kernel is multi-variate, and (ii) second order of convergence if the kernel is uni-variate. A Newton–Raphson scheme is then designed and implemented for the resulting nonlinear algebraic system. Some numerical examples are used to demonstrate the accuracy and convergence of the method. The numerical results verify the theoretical results and show that the method performs well.

SOLVING VOLTERRA-FREDHOLM INTEGRAL EQUATIONS BY QUADRATIC SPLINE FUNCTION

Using the quadratic spline function, this paper finds the numerical solution of mixed Volterra-Fredholm integral equations of the second kind. The proposed method is based on employing the quadratic spline function of the unknown function at an arbitrary point and using the integration method to turn the Volterra-Fredholm integral equation into a system of linear equations with respect to the unknown function. An approximate solution can be easily established by solving the given system. This is accomplished with the help of a computer program that runs on Python 3.9..

A Collocation Method for Mixed Volterra–Fredholm Integral Equations of the Hammerstein Type

This paper presents a collocation method for the approximate solution of two-dimensional mixed Volterra–Fredholm integral equations of the Hammerstein type. For a reformulation of the equation, we consider the domain of integration as a planar triangle and use a special type of linear interpolation on triangles. The resulting quadrature formula has a higher degree of precision than expected, leading to a collocation method that is superconvergent at the collocation nodes. The convergence of the method is established, as well as the rate of convergence. Numerical examples are considered, showing the applicability of the proposed scheme and the agreement with the theoretical results.

Analysis of nonlinear fractional optimal control systems described by delay Volterra–Fredholm integral equations via a new spectral collocation method

We aim to introduce a new spectral collocation method for investigating and analyzing nonlinear delay control systems governed by the fractional mixed Volterra–Fredholm integral equations (MVFIEs). The generalized fractional Legendre basis (GFLB) is used as a complete orthogonal basis, and the fractional Legendre–Gaussian nodes are introduced and employed as the fractional collocation points. These nodes correspond to the zeros of the fractional-order Legendre function of degree M. The convergence of the generalized orthogonal basis is discussed in detail based on the Sobolev and L2 norms. Additionally, new fractional integral and delay operators are implemented to reduce the primary control system to a static optimization system. Two benchmark fractional control problems, including delay, are considered to demonstrate the powerfulness and superiority of the new methodology. The introduced collocation approach can be successfully performed for solving even those fractional control problems having any irregularities in the control function, including jump discontinuities and bang–bang behavior.

Usage of numerical methods to solve nonlinear mixed Volterra-Fredholm integral equations and their system

In this paper, we apply the homotopy perturbation method (HPM), modified homotopy perturbation method(MHPM), variational iteration method (VIM), Adomian decomposition method (ADM), and modified Adomian decomposition method (MADM) to solve nonlinear mixed Volterra-Fredholm integral equations and itssystem. We investigate the approximate solution of this equation and its system via proposed methods. Thevalidity and efficiency of these methods are demonstrated through various numerical examples that illustratethe efficiency, accuracy, and simplicity of the proposed methods. Moreover, the convergence and uniquenessof the solution of the suggested methods are confirmed and compared with the exact solutions.

Numerical solution for three-dimensional nonlinear mixed Volterra–Fredholm integral equations via modified moving least-square method

The major purpose of this article is to expand the three-dimensional (3D) modified moving least-square method for the numerical solution of 3D linear and nonlinear Volterra–Fredholm integral equations of the second kind. This approach is very convenient for solving integral equations in high dimensions and it does not require any need for mesh connectivity. The size of the support used is the only factor that has a significant effect on the maximum errors of the MLS method. To overcome this problem, the MMLS approach with a non-singular moment matrix is applied to obtain better results than MLS approximation on using the best support, then applying the method in three dimensions can be easily achieved. The numerical experiments of the MMLS and classical MLS techniques are presented to show the difference between both methods for multi-dimensional problems. The convergence analysis is provided and some numerical tests are given to prove the applicability of this technique.

Numerical solutions of two-dimensional nonlinear integral equations via Laguerre Wavelet method with convergence analysis

In this paper, the approximate solutions for two different type of two-dimensional nonlinear integral equations: two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre wavelet method. To do this, these two-dimensional nonlinear integral equations are transformed into a system of nonlinear algebraic equations in matrix form. By solving these systems, unknown coefficients are obtained. Also, some theorems are proved for convergence analysis. Some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method.

Multivalued generalized graphic θ-contraction on directed graphs and application to mixed Volterra-Fredholm integral inclusion equations

The purpose of the present work is to introduce a generalized graphic θ-contraction conditions on a family of mappings defined on subsets of a metric space endowed with a set-transitive directed graph, and discuss common fixed point results without considering any kind of commutativity and continuity of the family of mappings. Useful examples illustrate the applicability and effectiveness of the given notions and results. We apply our result to the problem of existence of solutions of a pair of mixed Volterra-Fredholm integral inclusion equations followed by a numerical example.

Convergence Analysis for the Homotopy Perturbation Method for a Linear System of Mixed Volterra-Fredholm Integral Equations

In this paper, the homotopy perturbation method (HPM) is presented for treating a linear system of second-kind mixed Volterra-Fredholm integral equations. The method is based on constructing the series whose summation is the solution of the considered system. Convergence of constructed series is discussed and its proof is given; also, the error estimation is obtained. Algorithm is suggested and applied on several examples and the results are computed by using MATLAB (R2015a). To show the accuracy of the results and the effectiveness of the method, the approximate solutions of some examples are compared with the exact solution by computing the absolute errors.

Numerical Solution of Mixed Volterra – Fredholm Integral Equation Using the Collocation Method

Volterra – Fredholm integral equations (VFIEs) have a massive interest from researchers recently. The current study suggests a collocation method for the mixed Volterra - Fredholm integral equations (MVFIEs)."A point interpolation collocation method is considered by combining the radial and polynomial basis functions using collocation points". The main purpose of the radial and polynomial basis functions is to overcome the singularity that could associate with the collocation methods. The obtained interpolation function passes through all Scattered Point in a domain and therefore, the Delta function property is the shape of the functions. The exact solution of selective solutions was compared with the results obtained from the numerical experiments in order to investigate the accuracy and the efficiency of scheme.

A hybrid Legendre block-pulse method for mixed Volterra–Fredholm integral equations

Mixed Volterra–Fredholm integral equations can arise from mathematical modelling of the spatio-temporal development of an epidemic. In this paper a hybrid Legendre block-pulse method is used to provide solutions to a general integral equation of this type. The main idea of this approach is to reduce a Volterra–Fredholm integral equation to a system of algebraic equations. An order convergence analysis is provided. The accuracy of the proposed method is then confirmed by solving some problems, the first being a nonlinear epidemic problem.

Homotopy Perturbation Method and Convergence Analysis for the Linear Mixed Volterra-Fredholm Integral Equations

In this paper, the homotopy perturbation method is presented for solving the second kind linear mixed Volterra-Fredholm integral equations. Then, Aitken method is used to accelerate the convergence. In this method, a series will be constructed whose sum is the solution of the considered integral equation. Convergence of the constructed series is discussed, and its proof is given; the error estimation is also obtained. For more illustration, the method is applied on several examples and programs, which are written in MATLAB (R2015a) to compute the results. The absolute errors are computed to clarify the efficiency of the method.

The existence and uniqueness of solution for linear system of mixed Volterra-Fredholm integral equations in Banach space

In this paper, a linear system of mixed Volterra-Fredholm integral equations is considered. The problem of existence and uniqueness of its solution is investigated and proved in a complete metric space by using the Banach fixed-point theorem. Also, an iterative method of fixed point type is used to approximate the solution of the system. The algorithm is applied on several examples. To show the accuracy of the results and the efficiency of the method, the approximate solutions are compared with the exact solutions.

Numerical solution of nonlinear mixed Volterra-Fredholm integral equations in complex plane via PQWs

This paper presents an efficient numerical method for solving nonlinear mixed Volterra- integral equations in the complex plane. The method is based on the fixed point iteration procedure. In each iteration of this method, periodic quasi-wavelets are used as basis functions to approximate the solution. Also, using the Banach fixed point theorem, some results concerning the error analysis are obtained. Finally, some numerical examples show the implementation and accuracy of this method.