$\Phi$-Haar Wavelet Operational Matrix Method for Fractional Relaxation-Oscillation Equations Containing$\Phi$-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 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.

A Comparative Study of the Fractional Coupled Burgers and Hirota–Satsuma KdV Equations via Analytical Techniques

“…In all of these fields of study, investigating the exact and analytical results of fractional differential equations is essential, but as we do not have a technique for finding the exact solution of these types of fractional differential equations, we focus on approximation to the actual result [18][19][20][21][22][23]. In mathematics, determining the exact solution of such fractional differential equations and other applied science applications is challenging.…”

A Comparative Analysis of the Fractional‐Order Coupled Korteweg–De Vries Equations with the Mittag–Leffler Law

“…Caputo and Fabrizio modified the existing Caputo derivative to develop the Caputo-Fabrizio fractional derivative [1][2][3][4][5] based on a nonsingular kernel. Because of its advantages, numerous researchers utilized this operator to investigate various types of fractional-order partial differential equations [6][7][8][9]. To address this issue, Atangana and Baleanu proposed a new fractional operator called the Atangana-Baleanu derivative, which combines Caputo and Riemann-Liouville derivatives.…”

Analysis of the Fractional‐Order Delay Differential Equations by the Numerical Method

“…For these complex problems, a new technique has been used by the researchers known as fractional differential equations (FDEs). In the mathematical modeling of realworld physical problems, FDEs have been widespread due to their numerous applications in engineering and real-life sciences problems [6][7][8][9], such as economics [10], solid mechanics [11], continuum and statistical mechanics [12], oscillation of earthquakes [13], dynamics of interfaces between soft-nanoparticles and rough substrates [14], fluiddynamic traffic model [15], colored noise [16], solid mechanics [11], anomalous transport [17], and bioengineering [18][19][20].…”