In this paper, we present an efficient, new, and simple programmable method for finding approximate solutions to fractional differential equations based on Bernoulli wavelet approximations. Bernoulli Wavelet functions involve advantages such as orthogonality, simplicity, and ease of usage, in addition to the fact that fractional Bernoulli wavelets have exact operational matrices that improve the accuracy of the applied approach. A fractional differential equation was simplified to a system of algebraic equations using the fractional order integration operational matrices of Bernoulli wavelets. Examples are used to demonstrate the technique’s precision.

In this paper, a hybrid method is presented via combination of the Ramadan Group Integral method (RGITM) with method of Projected Differential Transform(PDTM) for the purpose of solving nonlinear partial differential equations systems. The method’s goal is to produce analytical solutions in series form. In comparison to existing methods, the suggested method makes handling such partial differential equations simple. The outcome demonstrated the method’s effectiveness, accuracy, and validity. The technique can be easily applied to a wide variety of nonlinear issues, and it has the potential to both reduce the amount of computation required and deal with the flaw brought about by the nonlinear components that cannot be resolved by employing recognized integral transforms. Examples will be looked at to help illustrate the proposed analysis.

Recently, the authors attempted to solve integral equations using hybrid improved block pulse and Bernstein functions. However, this is the first article to present a technical coupling between improved block-pulse functions and Bernstein functions for solving differential equations. The current method converts differential equations into an algebraic system that can be solved using traditional methods. To demonstrate the validity of this new approach, some numerical tests are provided. The findings demonstrated that the method is both promising and highly accurate.

This paper proposes a novel mixed numerical scheme for approximating linear Fredholm integral equation systems (LFISs) using a combination of Bernoulli polynomials (BPs) and enhanced block-pulse functions $\left(\mathcal{E}\mathcal{B}\mathcal{P}Fs\right)$. This suggested representation makes use of both the local support nature and computation efficiency of the $\left(\mathcal{E}\mathcal{B}\mathcal{P}Fs\right)$ as well as the high-order approximating nature of BPs. Using the operational matrices, the system of coupled integrals can be transformed into a finite-dimensional algebraic system (AS) of expansion coefficients. A theoretical analysis is established to investigate the solvability, convergence, stability, and approximation error of the resulting scheme. In addition, the effects of polynomial degree and partition refinement on the numerical accuracy are examined. Several test problems are considered, and the obtained results demonstrate that the proposed $\mathfrak{B}\mathcal{E}\mathcal{B}\mathcal{P}\mathcal{F}$ approach provides highly accurate approximations while requiring relatively small basis dimensions. Comparisons with previously reported numerical techniques further illustrate their computational effectiveness and accuracy.

In this work, we offer a novel and accurate method in order to find the solution of the linear differential equations over the intervals [0, 1) based on the generalization of Legendre wavelets. The mechanism is still upon workable implementation of the operational matrix of integration and its derivatives. This method reduces the problems into algebraic equations via the properties of generalized Legendre wavelet (GLW) together with the operational matrix of integration. The function approximation has been picked out in such a way so as to enumerate the connection coefficients in an facile manner. The proposed numerical technique, based on the GLW, has been examined on three linear problems as a consequence of this investigation. The outcomes have shown that this method, as opposed to some other existing numerical and analytical methods, is a very useful and advantageous for tackling such problems.