Numerical solution for a class of parabolic integro-differential equations subject to integral boundary conditions

In this paper, we consider the solution of nonlinear Volterra–Fredholm integro-differential equation, which contains the first derivative of the function. Our method transforms the nonlinear Volterra-Fredholm integro-differential equations into a system of nonlinear algebraic equations. The method based on the application of the local polynomial splines of the fifth order of approximation is proposed. Theorems about the errors of the approximation of a function and its first derivative by these splines are given. With the help of the proposed splines, the function and the derivative are replaced by the corresponding approximation. Note that at the beginning, in the middle and at the end of the interval of the definition of the integro-differential equation, the corresponding types of splines are used: the left, the right or the middle splines of the fifth order of approximation. When using the spline approximations, we also obtain the corresponding formulas for numerical differentiation. which we also apply for the solution of integro-differential equations. The formulas for approximation of the function and its derivative are presented. The results of the numerical solution of several integro-differential equations are presented. The proposed method is shown that it can be applied to solve integro-differential equations containing the second derivative of the solution.

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Validation and Verification of Numerical Models

Mohammed,

Abbas,

Gudivaka

et al. 2024

Advances in Systems Analysis, Software Engineering, and High Performance Computing

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The conceptual differences between the two expressions are extremely important for the development of this work, since the code used, although analysed through example cases, does not have validation like the proposal of the present work. For a more in-depth discussion on the definitions of validation and verification suggested. This work's main objective is to validate a code based on the finite element method used to simulate incompressible fluid flows. Numerical simulations were carried out for the cylindrical specimen. The two conditions for which data were found in the literature were simulated and experimental tests were carried out. From the simulation results, pressure coefficient graphs around the cylinder were obtained. The curves resulting from the simulations did not coincide with those obtained through experimental tests, for the reason previously explained. However, the pressure coefficient curves resulting from the simulations fit reasonably well with those originating from literature data.

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Numerical solution for a class of parabolic integro-differential equations subject to integral boundary conditions

Cited by 3 publications

References 22 publications

Wavelet-based approximation with nonstandard finite difference scheme for singularly perturbed partial integrodifferential equations

Wavelet-based approximation with nonstandard finite difference scheme for singularly perturbed partial integrodifferential equations

Nonlenear Integro-differential Equations and Splines of the Fifth Order of Approximation

Validation and Verification of Numerical Models

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