Abstract
AbstractThe 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.
Publisher
Springer Science and Business Media LLC
Reference50 articles.
1. Abdelhakem, M., Alaa-Eldeen, T., Baleanu, D., Alshehri, M.G., El-Kady, Mamdouh: Approximating real-life BVPs via Chebyshev polynomials first derivative Pseudo-Galerkin method. Fractal Fract. 5(4), 165 (2021)
2. Abdelhakem, M., Youssri, Y.H.: Two spectral Legendre’s derivative algorithms for Lane-Emden, Bratu equations, and singular perturbed problems. Appl. Numer. Math. 169, 243–255 (2021)
3. Abdelhakem, M., Fawzy, M., El-Kady, M., Moussa, H.: An efficient technique for approximated BVPs via the second derivative Legendre polynomials pseudo-Galerkin method: Certain types of applications, Results Phys., 43, Article 106067 (2022)
4. Abd-Elhameed, W.M., Ahmed, H.M.: Tau and Galerkin operational matrices of derivatives for treating singular and Emden-Fowler third-order-type equations. Int. J. Mod. Phys. C. 33(5), 2250061–22500617 (2022)
5. Abd-Elhameed, W.M., Alkenedri, A.M.: Spectral Solutions of Linear and Nonlinear BVPs Using Certain Jacobi Polynomials Generalizing Third- and Fourth-Kinds of Chebyshev Polynomials. Comput. Model. Eng. Sci 126(3), 955–989 (2021)