Relaxation-oscillation Equation Research Articles

Approximate solution of multi-term fractional differential equations via a block-by-block method

In this paper we introduce a block-by-block method for the numerical solution of multi-term fractional differential equations (MFDEs). The main idea is to convert a MFDE to a Volterra integral equation of weakly singular type, to which a well known block-by-block method is applied. We also provide the error analysis and convergence of the method. Finally, numerical examples involving Bagley–Torvik and relaxation-oscillation equations are given to confirm applications and the theoretical results.

Artificial Intelligence Solution for Relaxation Oscillation Equation

Artificial Intelligence Solution for Relaxation Oscillation Equation

Numerical solution of fractional relaxation–oscillation equation by using residual power series method

The relaxation oscillator is a type of oscillator that is based on the nature of physical phenomena that tend to return to equilibrium after being distributed. The relaxation–oscillation equation is the primary equation of the process of relaxation–oscillation. In this work, the relaxation–oscillation equation is solved using the residual power series method, which is a fractional order differential equation with defined initial conditions. The results obtained by this method are more reliable and accurate as compared to those obtained by other methods studied previously to solve this equation. The reliability and efficiency of this method are demonstrated by means of three examples with exact solutions compared with approximate solutions by means of errors. The pseudocode of the applied methodology has also been discussed in brief. The residual power series method can be used to solve well-known fractional order differential equations.

A numerical scheme based on Gegenbauer wavelets for solving a class of relaxation–oscillation equations of fractional order

A numerical scheme based on Gegenbauer wavelets for solving a class of relaxation–oscillation equations of fractional order

Numerical analysis of block-by-block method for a class of fractional relaxation-oscillation equations

Fractional relaxation-oscillation equation (FROE) is an important physical model describing oscillators. Its numerical solution method has profound theoretical significance and application value. In this paper, we use a non-singularity kernel Volterra integral equation of block-by-block method and construct a block-by-block numerical scheme based on Lagrange basis function interpolation. The analysis proves the stability and convergence of the block-by-block method. Error analysis shows that the convergence order is at least 4 which significantly improves calculation accuracy. Through the error comparison, the error of block-by-block method in this paper is significantly lower than that prediction-correction(P-C) method, which overcomes the shortcomings of the existing methods of low accuracy in solving FROE. Both theoretical analysis and numerical experiments show the high accuracy and effectiveness of the block-by-block numerical scheme, which shows that the method in this paper is efficient and feasible to solve the FROE.

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

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 theΦ-Caputo sense. The numerical method relies on a novel type of operational matrix method, namely, theΦ-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 computational algorithm for simulating fractional order relaxation–oscillation equation

A computational algorithm for simulating fractional order relaxation–oscillation equation

Numerical Solution of the Fractional Relaxation-Oscillation Equation by Using Reproducing Kernel Hilbert Space Method

Numerical Solution of the Fractional Relaxation-Oscillation Equation by Using Reproducing Kernel Hilbert Space Method

Numerical solution of fractional differential equations using fractional Chebyshev polynomials

In this paper, fractional order Chebyshev polynomials are presented and some properties are given. Using definition of fractional order Chebyshev polynomials, we give a numerical scheme for solving fractional differential equation by the collocation method. The Collocation method converts the given fractional differential equation into a matrix equation, which yields a linear algebraic system. Bagley–Torvik equation and fractional relaxation-oscillation equation are solved to show the effectiveness of the given method. This method is compared with some known schemes.

S-asymptotically omega-periodic mild solutions to fractional differential equations

This article concerns the existence of mild solutions to the semilinear fractional differential equation $$ D_t^\alpha u(t)=Au(t)+D_t^{\alpha-1} f(t,u(t)),\quad t\geq 0 $$ with nonlocal conditions \(u(0)=u_0 + g(u)\) where \(D_t^\alpha(\cdot)\) (\(1< \alpha < 2\)) is the Riemann-Liouville derivative, \(A: D(A) \subset X \to X\) is a linear densely defined operator of sectorial type on a complex Banach space \(X\), \(f:\mathbb{R}^+\times X\to X\) is S-asymptotically \(\omega\)-periodic with respect to the first variable. We use the Krsnoselskii's theorem to prove our main theorem. The results obtained are new even in the context of asymptotically \(\omega\)-periodic functions. An application to fractional relaxation-oscillation equations is given.For more information see https://ejde.math.txstate.edu/Volumes/2020/30/abstr.html

A numerical study of fractional relaxation–oscillation equations involving $$\psi $$ ψ -Caputo fractional derivative

We provide a numerical method to solve a certain class of fractional differential equations involving $$\psi $$ -Caputo fractional derivative. The considered class includes as particular case fractional relaxation–oscillation equations. Our approach is based on operational matrix of fractional integration of a new type of orthogonal polynomials. More precisely, we introduce $$\psi $$ -shifted Legendre polynomial basis, and we derive an explicit formula for the $$\psi $$ -fractional integral of $$\psi $$ -shifted Legendre polynomials. Next, via an orthogonal projection on this polynomial basis, the problem is reduced to an algebraic equation that can be easily solved. The convergence of the method is justified rigorously and confirmed by some numerical experiments.

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations Dtαxt=Axt+Dtα-1Ft,xt,Bxt, t∈R, where 1<α<2, A is a linear densely defined operator of sectorial type on a complex Banach space X and B is a bounded linear operator defined on X, F is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity F is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.

Stepanov-like almost automorphic mild solutions for semilinear fractional differential equations

This work is concerned with the existence and uniqueness of Stepanov-like almost automorphic mild solutions for a class of semilinear fractional differential equations
 Dtα x(t) = Ax(t) + Dtα-1F(t,x(t)), t ∈ ℝ,
 where 1 < α < 2, A is a linear densely defined operator of sectorial type of ω < 0 on a complex Banach space X and F is an appropriate function defined on phase space. The fractional derivative is understood in the Riemann-Liouville sense. The results obtained are utilized to study the existence and uniqueness of Stepanov-like almost automorphic mild solutions for a fractional relaxation-oscillation equation.

Generalized wavelet collocation method for solving fractional relaxation–oscillation equation arising in fluid mechanics

In this paper, a generalized wavelet collocation operational matrix method based on Haar wavelets is proposed to solve fractional relaxation–oscillation equation arising in fluid mechanics. Contrary to wavelet operational methods accessible in the literature, we derive an explicit form for the Haar wavelet operational matrices of fractional order integration without using the block pulse functions. The properties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. The performance of the numerical scheme is assessed and tested on specific test problems and the comparisons are given with other methods existing in the recent literature. The numerical outcomes indicate that the method yields highly accurate results and is computationally more efficient than the existing ones.

Pseudo almost automorphy of semilinear fractional differential equations in Banach Spaces

Abstract In this paper, we investigate the existence and uniqueness of (μ,ν)-pseudo almost automorphic mild solutions to semilinear fractional differential equation with Riemann-Liouville derivative in Banach space, where the nonlinear perturbation is of (μ,ν)-pseudo almost automorphic type or Stepanov-like (μ,ν)-pseudo almost automorphic type. As application, we explore the same topic for a fractional relaxation-oscillation equation.

Weighted Stepanov-like pseudo-almost automorphic mild solutions for semilinear fractional differential equations

This work is concerned with the existence and uniqueness of weighted Stepanov-like pseudo-almost automorphic mild solutions for a class of semilinear fractional differential equations, $D_{t}^{\alpha}x(t)=Ax(t)+D_{t}^{\alpha-1}F(t,x(t))$ , $t\in \mathbb{R}$ , where $1<\alpha<2$ , A is a linear densely defined operator of sectorial type of $\omega<0$ on a complex Banach space X and F is an appropriate function defined on phase space. The fractional derivative is understood in the Riemann-Liouville sense. The results obtained are utilized to study the existence and uniqueness of weighted Stepanov-like pseudo-almost automorphic mild solutions for a fractional relaxation-oscillation equation.

Optimal homotopy asymptotic method for solving fractional relaxation-oscillation equation

In this paper, an approximate analytical solution of linear fractional relaxation-oscillation equations in which the fractional derivatives are given in the Caputo sense, is obtained by the optimal homotopy asymptotic method (OHAM). The studied OHAM is based on minimizing the residual error. The results given by OHAM are compared with the exact solutions and the solutions obtained by generalized Taylor matrix method. The reliability and efficiency of the proposed approach are demonstrated in three examples with the aid of the symbolic algebra program Maple.

Numerical approach for solving fractional relaxation–oscillation equation

In this study, we will obtain the approximate solutions of relaxation–oscillation equation by developing the Taylor matrix method. A relaxation oscillator is a kind of oscillator based on a behavior of physical system’s return to equilibrium after being disturbed. The relaxation–oscillation equation is the primary equation of relaxation and oscillation processes. The relaxation–oscillation equation is a fractional differential equation with initial conditions. For this propose, generalized Taylor matrix method is introduced. This method is based on first taking the truncated fractional Taylor expansions of the functions in the relaxation–oscillation equation and then substituting their matrix forms into the equation. Hence, the result matrix equation can be solved and the unknown fractional Taylor coefficients can be found approximately. The reliability and efficiency of the proposed approach are demonstrated in the numerical examples with aid of symbolic algebra program, Maple.

Existence of anti-periodic mild solutions for a class of semilinear fractional differential equations

This work is concerned with the existence of anti-periodic mild solutions for a class of semilinear fractional differential equations D t α x ( t ) = Ax ( t ) + D t α - 1 F ( t , x ( t ) ) , t ∈ R , where 1 < α < 2, A is a linear densely defined operator of sectorial type of ω < 0 on a complex Banach space X and F is an appropriate function defined on phase space, the fractional derivative is understood in the Riemann–Liouville sense. The results obtained are utilized to study the existence of anti-periodic mild solutions to a fractional relaxation-oscillation equation.

Almost Automorphic Solutions to Abstract Fractional Differential Equations

A new and general existence and uniqueness theorem of almost automorphic solutions is obtained for the semilinear fractional differential equation , in complex Banach spaces, with Stepanov-like almost automorphic coefficients. Moreover, an application to a fractional relaxation-oscillation equation is given.