<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>Φ</a:mi></a:math>-Haar Wavelet Operational Matrix Method for Fractional Relaxation-Oscillation Equations Containing<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"><c:mi>Φ</c:mi></c:math>-Caputo Fractional Derivative
Pongsakorn Sunthrayuth, Noufe H. Aljahdaly, Amjid Ali, Rasool Shah, Ibrahim Mahariq, Ayékotan M. J. Tchalla
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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M3"><a:mi>Φ</a:mi></a:math> -Caputo sense. The numerical method relies on a novel type of operational matrix method, namely, the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M4"><c:mi>Φ</c:mi></c:math> -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.