Φ -Haar Wavelet Operational Matrix Method for Fractional Relaxation-Oscillation Equations Containing Φ -Caputo Fractional Derivative

Abstract

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 $\Phi$ -Caputo sense. The numerical method relies on a novel type of operational matrix method, namely, the $\Phi$ -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.