Fredholm Integro-differential Equations Research Articles

Numerical Treatment of the Coupled Fredholm Integro-Differential Equations by Compact Finite Difference Method

Numerical Treatment of the Coupled Fredholm Integro-Differential Equations by Compact Finite Difference Method

Convergence of a refined iterative method and its application to fractional Volterra–Fredholm integro-differential equations

Convergence of a refined iterative method and its application to fractional Volterra–Fredholm integro-differential equations

Single equation approach for linear Fredholm integrodifferential equations

Our primary objective is to develop an innovative approach for the numerical so lution of the Fredholm linear integrodifferential equation. Our overarching goal is to significantly enhance computational efficiency and minimize memory space utilization which is very important in the case of large integration intervals. To begin this en deavor, we establish sufficient conditions that guarantee the existence and uniqueness of the solution. Our novel method is grounded in the single equation approach, which consists in using the variable transformation represented as v(x) = v(a) + ∫ ︁ x a v(t)dt, completed by the trapezoidal rule. This transformation plays a pivotal role in convert ing our equation into an algebraic system, thereby reducing the number of equations and unknowns in the discrete system. Underlying these developments is a fundamental requirement ensuring the existence and uniqueness of the solution. Leveraging this, we formulate theorems that establish the convergence of the approximate solution, ensuring consistency between analytical and numerical investigations. Ultimately, we conduct a comparative analysis between our newly introduced technique and the older method. This comparison serves to highlight the superior computational efficiency and reduced storage space consumption offered by our innovative digital framework. Наша главная цель — значительно повысить вычислительную эффективность и минимизировать использование памяти, что очень важно в случае больших интервалов интегрирования. Сначала определяем достаточные условия, которые гарантируют существование и единственность решения. Наш новый метод основан на подходе с одним уравнением, он заключается в использовании преобразования переменной, представленного как v(x) = v(a) + ∫ ︁ x a v(t)dt, дополненного правилом трапеции. Оно играет ключевую роль в преобразовании нашего уравнения в алгебраическую систему, тем самым уменьшая количество уравнений и неизвестных в дискретной системе. В основе этих разработок лежит фундаментальное требование, гарантирующее существование и единственность решения. Используя это, мы формулируем теоремы, которые устанавливают сходимость приближенного решения, обеспечивая согласованность между аналитическими и численными исследованиями. В конечном итоге мы проводим сравнительный анализ между нашей новой введенной техникой и старым методом. Это сравнение служит для того, чтобы подчеркнуть существенную вычислительную эффективность и сокращенное использование памяти на диске, предлагаемые нашей инновационной цифровой структурой.

Numerical implementation of solving a boundary value problem with parameter for Fredholm integro-differential equation

The boundary value problem with parameter for Fredholm integro-differential equation with degenerate kernel is investigated in this paper. The aim of the paper is to establish the solvability conditions, to construct analytical and numerical solutions of the considered problem. The basis for achieving the goal is the ideas of Dzhumabayev parameterization method, classical numerical methods of solving Cauchy problems and numerical integration techniques. A problem with parameters is obtained by introducing an additional parameter and a new unknown function. A system of equations with respect to parameters is compiled according to the initial data of the considered equation and boundary conditions. The unknown function is found as a solution of the Cauchy problem for the ordinary differential equation. The equivalence of the original problem and the problem with parameters, the conditions of unique solvability are established and the formula for finding an analytical solution is obtained. Test examples of finding analytical and approximate solutions of the original problem are given.

Operational Matrices of Genocchi Polynomials for Solving High-Order Linear Fredholm Integro-Differential Equations

Operational Matrices of Genocchi Polynomials for Solving High-Order Linear Fredholm Integro-Differential Equations

Efficient numerical solution of linear Fredholm integro-differential equations via backward finite difference and Nyström methods

Efficient numerical solution of linear Fredholm integro-differential equations via backward finite difference and Nyström methods

A comparative study on numerical methods for Fredholm integro-differential equations of convection-diffusion problem with integral boundary conditions

This paper numerically solves Fredholm integro-differential equations with small parameters and integral boundary conditions. The solution of these equations has a boundary layer at the right boundary. A central difference scheme approximates the second-order derivative, a backward difference (upwind scheme) approximates the first-order derivative, and the trapezoidal rule is used for the integral term with a Shishkin mesh. It is shown that theoretically, the proposed scheme is uniformly convergent with almost first-order convergence. Further to improve the order of convergence from first order to second order, we use the post-processing and the hybrid scheme. Two numerical examples are computed to support the theoretical results.

Approximate Solution of System of Multi-term Fractional Mixed Volterra–Fredholm Integro-Differential Equations by BPFs

Approximate Solution of System of Multi-term Fractional Mixed Volterra–Fredholm Integro-Differential Equations by BPFs

Uniqueness and Convergence of Solution of Multi-Term Fractional Order Fredholm Integro-Differential Equation

Uniqueness and Convergence of Solution of Multi-Term Fractional Order Fredholm Integro-Differential Equation

Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method

Abstract In this study, a multipoint boundary value problem for Volterra-Fredholm integro-differential equations is considered. The addition of a new function converts the system of Volterra-Fredholm integro-differential equations to a system of Fredholm integro-differential equations. In contrast to the original problem, the dimension of a Fredholm integro-differential equation is determined by the number of matrices in the degenerate kernel of the Volterra integral. A numerical algorithm of Dzhumabaev parameterization method for addressing a multipoint boundary value problem for Volterra-Fredholm integro-differential equations is proposed. The main advantage of the proposed method is splitting the problem into auxiliary Cauchy problems for ordinary differential equations and a system of algebraic equations with respect to the parameters. The conditions for the unique solvability of the multipoint boundary value problem for Fredholm integro-differential equations are established. Finally, various numerical examples are provided to demonstrate the efficiency and correctness of the suggested technique.

Properties of a Partial Fredholm Integro-differential Equations with Nonlocal Condition and Algorithms

Properties of a Partial Fredholm Integro-differential Equations with Nonlocal Condition and Algorithms

Survey of the Layer Behaviour of the Singularly Perturbed Fredholm Integro-Differential Equation

Survey of the Layer Behaviour of the Singularly Perturbed Fredholm Integro-Differential Equation

A Fitted Approximate Method for Solving Singularly Perturbed Volterra–Fredholm Integrodifferential Equations with Integral Boundary Condition

A Fitted Approximate Method for Solving Singularly Perturbed Volterra–Fredholm Integrodifferential Equations with Integral Boundary Condition

Approximating Higher Order Linear Fredholm Integro-Differential Equations by an Efficient Adomian Decomposition Method

Approximating Higher Order Linear Fredholm Integro-Differential Equations by an Efficient Adomian Decomposition Method

Application of Local Integrated Radial Basis Function Method for Solving System of Fredholm Integro-Differential Equations

Application of Local Integrated Radial Basis Function Method for Solving System of Fredholm Integro-Differential Equations

SPLINE QUASI-INTERPOLATION NUMERICAL METHODS FOR INTEGRO-DIFFERENTIAL EQUATIONS WITH WEAKLY SINGULAR KERNELS

In this work, we introduce a numerical approach that utilizes spline quasi-interpolation operators over a bounded interval. This method is designed to provide a numerical solution for a class of Fredholm integro-differential equations with weakly singular kernels. We outline the computational components involved in determining the approximate solution and provide theoretical findings regarding the convergence rate. This convergence rate is analyzed in relation to both the degree of the quasi-interpolant and the grading exponent of the graded grid partition. Finally, we present numerical experiments that validate the theoretical findings.

Physical Models for Sustainability using Fredholm Integro-Differential Equations: Applicability and Analysis of Chebyshev Polynomial Method

Numerical analysis is concerned with the mathematical derivation, explanation and evaluation/analysis of algorithms, models and methods used to obtain numerical solutions for mathematical problems. This paper explores the reliability of the Chebyshev Polynomial Method (CPM) for solving a specific class of equations known as the second-order Fredholm Integro-Differential Equations (FIDEs). A series expansion of the Chebyshev polynomial is derived, used in solving these integral equations, and later on examined in terms of accuracy and convergence of solutions. The evaluation process involves a hybrid approach, combining manual methods and mathematical programs like MAPLE and MATLAB. In addition, three numerical examples were solved in which two truncation points are considered per each example. Furthermore, the performance of the CPM is reported in terms of accuracy, convergence, suitability, reliability and effectiveness in the context of the exact solution.

Solving Fredholm integro-differential equations involving integral condition: A new numerical method

Abstract In this work we investigate a nonlocal problem for the Fredholm integro-differential equation involving integral condition. The main tool used in our considerations is Dzhumabaev parametrization method. We make use of the numerical implementation of the Dzhumabaev parametrization method to obtain the desired result, which is well-supported with numerical examples.

Factorization of abstract operators into two second degree operators and its applications to integro-differential equations

Boundary value problem B1x = f with an abstract linear operator B1, corresponding to an Fredholm integro-differential equation with ordinary or partial differential operator is researched. An exact solution to B1x = f in the case when a bijective operator B1 has a factorization of the form B1 =BB0 where B and B0 are two linear more simple than B1 second degree abstract operators, received. Conditions for factorization of the operator B1 and a criterion for bijectivity of B1 are found.

Numerical scheme for singularly perturbed Fredholm integro-differential equations with non-local boundary conditions

Numerical scheme for singularly perturbed Fredholm integro-differential equations with non-local boundary conditions