A Unified Approach for Solving Linear and Nonlinear Odd-Order Two-Point Boundary Value Problems

Abd-Elhameed, W. M.; Napoli, Anna

doi:10.1007/s40840-019-00840-7

Cited by 6 publications

(1 citation statement)

References 28 publications

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“…Only a limited number of this type of problems can be solved analytically. For this reason many researchers have proposed numerical solutions (see, for instance, [1,13,[19][20][21]33,34,43]). Different approaches have been considered, among which collocation, also with spline functions and Lagrange interpolation, Galerkin weighted residual, decomposition, variational techniques.…”

Section: The Numerical Solutionmentioning

confidence: 99%

Lidstone–Euler interpolation and related high even order boundary value problem

Costabile

Gualtieri

Napoli

2021

Calcolo

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We consider the Lidstone–Euler interpolation problem and the associated Lidstone–Euler boundary value problem, in both theoretical and computational aspects. After a theorem of existence and uniqueness of the solution to the Lidstone–Euler boundary value problem, we present a numerical method for solving it. This method uses the extrapolated Bernstein polynomials and produces an approximating convergent polynomial sequence. Particularly, we consider the fourth-order case, arising in various physical models. Finally, we present some numerical examples and we compare the proposed method with a modified decomposition method for a tenth-order problem. The numerical results confirm the theoretical and computational ones.

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Section: The Numerical Solutionmentioning

confidence: 99%

Lidstone–Euler interpolation and related high even order boundary value problem

Costabile

Gualtieri

Napoli

2021

Calcolo

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Numerical Solutions of High-Order Differential Equations with Polynomial Coefficients Using a Bernstein Polynomial Basis

Ahmed

2023

Mediterr. J. Math.

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The paper presents a novel method that allows one to establish numerical solutions of linear and nonlinear ordinary differential equations—with polynomial coefficients—that contain any finite products of the unknown functions and/or their general derivatives. The presented algorithm provides numerical solutions of these differential equations subject to initial or boundary conditions. This algorithm proposes the desired solution in terms of B-polynomials (Bernstein polynomial basis) and then uses the orthonormal relation of B-polynomials with its weighted dual basis with respect to the Jacobi weight function to construct a linear/nonlinear system in the unknown expansion coefficients which can be solved using a suitable solver. The properties of B-polynomials provide greater flexibility in which to impose the initial or boundary conditions at the end points of the interval [0, R] and enable us to obtain exactly and explicitly some of the unknown expansion coefficients in the form of a suggested numerical solution. Consequently, the presented algorithm leads to a linear or nonlinear algebraic system in the unknown expansion coefficients that has a simpler form than that was obtained by the other algorithms. So that, this procedure is a powerful tool that we may utilize to overcome the difficulties associated with boundary and initial value problems with less computational effort than the other techniques. An accepted agreement is obtained between the exact and approximate solutions for the given examples. The error analysis was also studied, and the obtained numerical results clarified the validity of the theoretical results.

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Least Square Homotopy Perturbation Method for Ordinary Differential Equations

Qayyum

Oscar

2021

Journal of Mathematics

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In this study, a new modification of the homotopy perturbation method (HPM) is introduced for various order boundary value problems (BVPs). In this modification, HPM is hybrid with least square optimizer and named as the least square homotopy perturbation method (LSHPM). The proposed scheme is tested against various linear and nonlinear BVPs (second to seventh order DEs). Validity of the obtained solutions is confirmed by finding absolute errors. To analyze the efficiency of the proposed scheme, tested problems have also been solved through HPM and results are compared with LSHPM. Furthermore, obtained results are also compared with other numerical schemes available in literature. Analysis reveals that LSHPM is a consistent and effective scheme which can be used for more complex BVPs in science and engineering.

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A Unified Approach for Solving Linear and Nonlinear Odd-Order Two-Point Boundary Value Problems

Cited by 6 publications

References 28 publications

Lidstone–Euler interpolation and related high even order boundary value problem

Lidstone–Euler interpolation and related high even order boundary value problem

Numerical Solutions of High-Order Differential Equations with Polynomial Coefficients Using a Bernstein Polynomial Basis

Least Square Homotopy Perturbation Method for Ordinary Differential Equations

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