This paper proposes a numerical method for solving fractional relaxation-oscillation equations. A relaxation oscillator is a type of oscillator that is based on how a physical system returns to equilibrium after being disrupted. The primary equation of relaxation and oscillation processes is the relaxation-oscillation equation. The fractional derivatives in the relaxation-oscillation equations under consideration are defined in the Φ -Caputo sense. The numerical method relies on a novel type of operational matrix method, namely, the Φ -Haar wavelet operational matrix method. The operational matrix approach has a lower computational complexity. The proposed scheme simplifies the main problem to a set of linear algebraic equations. Numerical examples demonstrate the validity and applicability of the proposed technique.

Purpose: The purpose of this study is to solve two unique but difficult partial differential equations: the foam drainage equation and the nonlinear time-fractional fisher’s equation. Through our methods, we aim to provide accurate solutions and gain a deeper understanding of the intricate behaviors exhibited by these systems.Design/methodology/approach: In this study, we use a dual technique that combines the Aboodh residual power series method and the Aboodh transform iteration method, both of which are combined with the Caputo operator.Findings: We develop exact and efficient solutions by merging these unique methodologies. Our results, presented through illustrative figures and data, demonstrate the efficacy and versatility of the Aboodh methods in tackling such complex mathematical models.Originality/value: Owing to their fractional derivatives and nonlinear behavior, these equations are crucial in modeling complex processes and confront analytical complications in various scientific and engineering contexts.

This article introduces a new numerical approach for solving linear and non-linear boundary value problems for Ψ-fractional differential equations (Ψ-FDEs). This approach relies on the Ψ-Haar wavelet operational integration matrices. The Ψ-operational matrices (Ψ-OMs) are used to convert the Ψ-FDE to an algebraic system of equations. The non-linear fractional boundary value problems are first linearized using the quasi-linearization technique, and then the Ψ-Haar wavelet technique is applied to the linearized problem. The solution is updated by the Ψ-Haar wavelet method in each iteration of the quasi-linearization technique. The proposed method is a good and simple mathematical technique for numerically solving non-linear Ψ-FDEs. The operational matrix (OM) method is computationally more efficient. Several linear and non-linear boundary value problems are discussed to demonstrate the applicability, efficiency, and simplicity of the method. Moreover, the error analysis is carried out resulting a rigorous error bound for the proposed method.

This paper proposes a new numerical technique for solving a specific class of fractional differential equations, which includes the ${\psi}$-Caputo fractional derivative. The class under consideration is nonlinear ${\psi}$-fractional Riccati differential equations (${\psi}$-FRDEs). Our approach relies on the ${\psi}$-Haar wavelet (${\psi}$-HW) operational matrix, which is a novel type of operational matrix of fractional integration. We derive an explicit formula for the ${\psi}$-fractional integral of the HW. This operational matrix has been used successfully to solve nonlinear ${\psi}$-FRDEs.The Quasi-linearization technique is employed to linearize the non-linear $\bm\psi$-FRDEs. This technique reduces the problem to an algebraic equation that can be easily solved. The technique is a useful and straightforward mathematical tool for solving nonlinear ${\psi}$-FRDEs. The computational complexity of the operational matrix technique is minimal. The error analysis of the proposed method is thoroughly investigated. To justify the method's accuracy and efficiency, numerical results are given.

In this study, we propose a two-step iterative procedure for solving absolute value equations. The method includes Simpson's Three-Eighths formula with five points as a corrector step and generalized Newton's approach as a predictor step. For solving large systems, this method is very effective because it is very simple. Moreover, we show the convergence analysis under certain conditions using different theorems. We conducted numerical experiments to examine the efficiency of the presented technique.