The generalized Kuramoto–Sivashinsky equation arises frequently in engineering, physics, biology, chemistry, and applied mathematics, and because of its extensive applications, this important model has received much attention regarding obtaining numerical solutions. This article introduces a new hybrid technique based on nonpolynomial splines and finite differences for solving the Kuramoto–Sivashinsky equation approximately. Specifically, the truncation error is studied to examine the convergence order of the proposed scheme, some problems are given to show its viability and effectiveness, and the norm errors are determined to compare the current method with the analytic solution and some other methods from the literature.

In this paper, we look into a group of fractional boundary value problem equations involving fractional derivative fractional orders and there are two boundary value criteria in this equation. The existence and uniqueness solutions are obtained using the Banach fixed point theorem (Contraction mapping theorem) and the Schauder fixed point theorem. based on the method of fractional integral and integral operator, our primary findings are illustrated using examples.

One of the most important generalized inverses is the Drazin inverse, which is defined for square matrices having an index. The objective of this work is to investigate and present a computational tool in the form of an iterative method for computing this task. This scheme reaches the seventh rate of convergence as long as a suitable initial matrix is chosen and by employing only five matrix products per cycle. After some analytical discussions, several tests are provided to show the efficiency of the presented formulation.

The asymptotic behavior of eigenvalues and eigenfunctions of T.Regge fractional boundary value problem has been shown, and we state and prove some theorems for many results, also some necessary definitions and results. In this paper, we look into a group of fractional boundary value problem equations involving fractional derivative fractional orders and with two boundary value conditions. We will which are important to state and prove those theorems; our primary findings are illustrated using examples.

This study are interested on solving T.Regge problem for fractional order by new transformation Rishi transform, where examining the Rishi's transformation technique to the solution of fractional differential equations (FDEs), particularly in the fractional order T. Regge problem. In addition, this article involves the formulation of Rishi transformation formulas applied to fractional integrals and fractional derivatives, with discusses the advantages and limitations of the method and provides several examples to illustrate its effectiveness and applicability. We have demonstrated by examples that the Rishi transformation method offers guaranteed in solving FDEs, which contribute significantly to the field of fractional calculation.

To solve a lacunary interpolation problem with a range of (0, 2, 4), we first look at an appropriate class of insufficient splines. Under suitable assumptions, we prove the existence and uniqueness of the solution. We describe the convergence analysis and show that the approach is of order six and that it can be constructed of any order, with any intermediate lacunary requirements.