A Reliable Numerical Treatment of Differential Equations via Hybrid Bernstein and Improved Block-Pulse Functions

Document Type : Original Article

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Mathematics and Computer Science Department, Faculty of Science, Menoufia University, Egypt

Abstract

Many branches of practical mathematics rely heavily on numerical solutions to initial value problems, boundary value problems, and eigenvalue problems for ordinary and partial differential equations. Recently, the authors attempted to solve integral equations using hybrid Bernstein functions and improved block pulse functions. However, this is the first study to present a technical coupling between hybrid Bernstein and improved block-pulse functions for solving differential equations. The current method transforms differential equations into an algebraic system that can be solved with conventional methods. To validate the new method, certain numerical examples are supplied. The findings demonstrated that the method is both promising and highly accurate. The numerical findings reveal that the suggested hybrid approach outperforms the use of Bernstein polynomials, multi derivative hybrid block methods, block-pulse function, and other methods indicated in the numerical section in terms of accuracy. The proposed method can be implemented for more kinds of differential equations. The proposed method is applicable to a broader range of differential equations. The numerical results show that the proposed method are highly accurate and effective.

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