Fractional Oscillator Equation Research Articles

Analytical solution of fractional oscillation equation with two Caputo fractional derivatives

Analytical solution of initial value problem for the fractional oscillation equation with two Caputo fractional derivatives , where the coefficients and orders satisfy and , is investigated by using the Laplace transform and complex inverse integral method on the principal Riemann surface. It is proved by using the argument principle that the characteristic equation has a pair of conjugated simple complex roots with a negative real part on the principal Riemann surface under the assumption that and are not both integers. Then three fundamental solutions, the unit impulse response, the unit initial displacement response, and the unit initial rate response, are derived analytically. Each of these solutions is expressed into a superposition of a classical damped oscillation decaying exponentially and a real Laplace integration decaying in a negative power law. Finally, the asymptotic behaviors of these analytical solutions for sufficiently large are determined as monotonous decays in a power of negative exponent.

A numerical comparative analysis of methods for solving fractional differential equations

In this paper, we present two collocation methods utilizing subdivision schemes and Bernstein polynomials as their basis functions to solve Caputo-type fractional differential equations. These methods transform fractional differential equations into systems of linear equations, which can be solved using various suitable techniques. Notably, these methods are versatile, making them applicable to both boundary value problems and initial value problems. To demonstrate their effectiveness, we applied these methods to three test problems of fractional differential equations, including the Bagley-Torvik equation and fractional oscillation equation. Our results reveal that both methods provide a more accurate approximation to the exact solution when compared to various algorithms found in the existing literature, showcasing their efficiency, accuracy, and consistency.

Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets

This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived by means of fractional Bernstein polynomials. The oscillation equation describes electrical circuits and exhibits a wide range of nonlinear dynamical behaviors. The proposed variable order model is of current interest in a lot of application areas in engineering and applied sciences. The purpose of this study is to analyze the behavior of the fractional force-free and forced oscillation equations under the variable-order fractional operator. The basic idea behind using the approximation technique is that it converts the proposed model into non-linear algebraic equations with the help of collocation nodes for easy computation. Different cases of the proposed model are examined under the selected variable order parameters for the first time in order to show the precision and performance of the mentioned scheme. The dynamic behavior and results are presented via tables and graphs to ensure the validity of the mentioned scheme. Further, the behavior of the obtained solutions for the variable order is also depicted. From the calculated results, it is observed that the mentioned scheme is extremely simple and efficient for examining the behavior of nonlinear random (constant or variable) order fractional models occurring in engineering and science.

Multiplicity of Positive Solutions to Hadamard-Type Fractional Relativistic Oscillator Equation with p-Laplacian Operator

The purpose of this paper is to investigate the initial value problem of Hadamard-type fractional relativistic oscillator equation with p-Laplacian operator. By overcoming the perturbation of singularity to fractional relativistic oscillator equation, the multiplicity of positive solutions to the problem were proved via the methods of reducing and topological degree in cone, which extend and enrich some previous results.

A Comparative Study of Responses of Fractional Oscillator to Sinusoidal Excitation in the Weyl and Caputo Senses

The fractional oscillator equation with the sinusoidal excitation mx″(t)+bDtαx(t)+kx(t)=Fsin(ωt), m,b,k,ω>0 and 0<α<2 is comparatively considered for the Weyl fractional derivative and the Caputo fractional derivative. In the sense of Weyl, the fractional oscillator equation is solved to be a steady periodic oscillation xW(t). In the sense of Caputo, the fractional oscillator equation is solved and subjected to initial conditions. For the fractional case α∈(0,1)∪(1,2), the response to excitation, S(t), is a superposition of three parts: the steady periodic oscillation xW(t), an exponentially decaying oscillation and a monotone recovery term in negative power law. For the two responses to initial values, S0(t) and S1(t), either of them is a superposition of an exponentially decaying oscillation and a monotone recovery term in negative power law. The monotone recovery terms come from the Hankel integrals which make the fractional case different from the integer-order case. The asymptotic behaviors of the solutions removing the steady periodic response are given for the four cases of the initial values. The Weyl fractional derivative is suitable for a describing steady-state problem, and can directly lead to a steady periodic solution. The Caputo fractional derivative is applied to an initial value problem and the steady component of the solution is just the solution in the corresponding Weyl sense.

Exact Solutions of Fractional Order Oscillation Equation with Two Fractional Derivative Terms

The fractional oscillation equation with two fractional derivative terms in the sense of Caputo, where the orders alpha and beta satisfy 1<alpha le 2 and 0<beta le 1, is investigated, and the unit step response and two initial value responses are obtained in two forms by using different methods of inverse Laplace transform. The first method yields series solutions with the nonnegative powers of t, which converge fast for small t. The second method is our emphasis, where the complex path integral formula of the inverse Laplace transform is used. In order to determine singularities of integrand we first seek for the roots of the characteristic equation, which is a transcendental equation with four parameters, two coefficients and two noninteger power exponents. The existence conditions and properties of the roots on the principal Riemann surface are given. Based on the results on the characteristic equation, we derive these responses as a sum of a classical exponentially damped oscillation, which vanishes in an indicated case, and an infinite integral of the Laplace type, which converges fast for large t, with a steady component in the unit step response. Asymptotic behaviors of solutions for large t are derived as algebraic decays in negative power laws characterized by the orders alpha and beta. The fractional system exhibits a magical transition from oscillation to monotonic decay in negative power law.

Laplace inverse and MR approach to existence of a unique solution and the Hyers–Ulam–Wright stability analysis of the nonhomogeneous fractional delay oscillation equation by matrix-valued fuzzy controllers

In this paper, we consider the nonhomogeneous fractional delay oscillation equation with order κ and investigate the existence of a unique solution in matrix-valued fuzzy Banach spaces for this equation using the alternative fixed point theorem. In a fuzzy environment, we introduce a class of the matrix-valued fuzzy Wright controller to investigate the Hyers–Ulam–Wright stability for the NH-FD-O equation with order κ. Finally, an illustrative example to demonstrate the application of the main theorem is also considered.

On the equivalence between fractional and classical oscillators

Recently, much research has been conducted on fractional-order differential calculus to generalise existing mathematical models or propose new ones to describe different phenomena. Fractional models are not as well researched as the classical models, and it is recommended that they should be compared to the classical well-established models. However, questions arise about the relationships between fractional-order and classical integer-order models. The damped harmonic oscillator is the basis for many mathematical models in many areas of research. The physical interpretation and ranges of variability of its parameters are well known. Some fractional models lead to fractional-order differential equations similar in structure to the classical harmonic oscillator equation. Such models are called fractional oscillators.This article considers one of the fractional models based on the Scott Blair constitutive equation for viscoelastic materials, which, when applied to describe mechanical vibrations, leads to a fractional-order differential equation, i.e. the fractional oscillator equation. The aim of this analysis is first to find out whether and in what ranges of the parameters of harmonic oscillator there is a fractional oscillator whose behaviour, with good approximation, is the same as that of the classical harmonic oscillator, then whether there is only one such fractional oscillator, and finally what relationships exist between the parameters of the two oscillators.To answer these questions, we introduce the so-called divergence coefficient between fractional and classical oscillators. By minimising this coefficient, we aim to numerically find a fractional oscillator corresponding to the fixed classical oscillator. The relationships between the parameters of the models were determined for some of their ranges. The formulae were then used to calculate the coefficients of the fractional oscillator equation with a sinusoidal driving force; a good agreement with the classical equation was obtained. The attempt to use the formulae in the model resulting from the fractional Kelvin–Voigt relationship turned out to be only partially successful.

Application of the Explicit Euler Method for Numerical Analysis of a Nonlinear Fractional Oscillation Equation

In this paper, a numerical analysis of the oscillation equation with a derivative of a fractional variable Riemann–Liouville order in the dissipative term, which is responsible for viscous friction, is carried out. Using the theory of finite-difference schemes, an explicit finite-difference scheme (Euler’s method) was constructed on a uniform computational grid. For the first time, the issues of approximation, stability and convergence of the proposed explicit finite-difference scheme are considered. To compare the results, the Adams–Bashford–Moulton scheme was constructed as an experimental method. The theoretical results were confirmed using test examples, the computational accuracy of the method was evaluated, which is consistent with the theoretical one, and the simulation results were visualized. Using the example of a fractional Duffing oscillator, waveforms and phase trajectories, as well as its amplitude–frequency characteristics, were constructed using a finite-difference scheme. To identify chaotic regimes, the spectra of maximum Lyapunov exponents and Poincaré points were constructed. It is shown that an explicit finite-difference scheme can be acceptable under the condition of a step of the computational grid.

Hyers-Ulam-Rassias-Wright Stability for Fractional Oscillation Equation

In this paper, we consider the nonhomogeneous fractional delay oscillation equation with orderσand introduce a class of control functions, i.e., Wright functions. Next, we apply the Cădariu-Radu method to prove the existence of a unique solution and Hyers-Ulam-Rassias-Wright stability of the fractional delay oscillation equation. At the end of the article, by an example, we show the application of the obtained results.

Comparison of Two Different Analytical Forms of Response for Fractional Oscillation Equation

The impulse response of the fractional oscillation equation was investigated, where the damping term was characterized by means of the Riemann–Liouville fractional derivative with the order α satisfying 0≤α≤2. Two different analytical forms of the response were obtained by using the two different methods of inverse Laplace transform. The first analytical form is a series composed of positive powers of t, which converges rapidly for a small t. The second form is a sum of a damped harmonic oscillation with negative exponential amplitude and a decayed function in the form of an infinite integral, where the infinite integral converges rapidly for a large t. Furthermore, the Gauss–Laguerre quadrature formula was used for numerical calculation of the infinite integral to generate an analytical approximation to the response. The asymptotic behaviours for a small t and large t were obtained from the two forms of response. The second form provides more details for the response and is applicable for a larger range of t. The results include that of the integer-order cases, α= 0, 1 and 2.

An accurate numerical technique for fractional oscillation equations with oscillatory solutions

The solution of fractional oscillation equations exhibits highly oscillatory behavior. It is a challenging work to find efficient numerical methods for fractional oscillation equations. The objective of this paper is to develop an accurate numerical technique for fractional oscillation equations with oscillatory solutions. The present method combines the variation‐of‐constants formula with the reproducing kernel function approximation. The merit of our approach is that it can preserve the oscillatory structure of the solution to the considered fractional initial value problems. Illustrative examples clearly show the reliability and efficiency of the approach.

Ordinary Differential Equation with Left and Right Fractional Derivatives and Modeling of Oscillatory Systems

We consider the principle of least action in the context of fractional calculus. Namely, we derive the fractional Euler–Lagrange equation and the general equation of motion with the composition of the left and right fractional derivatives defined on infinite intervals. In addition, we construct an explicit representation of solutions to a model fractional oscillator equation containing the left and right Gerasimov–Caputo fractional derivatives with origins at plus and minus infinity. We derive a representation for the composition of the left and right derivatives with origins at plus and minus infinity in terms of the Riesz potential, and introduce special functions with which we give solutions to the model fractional oscillator equation with a complex coefficient. This approach can be useful for describing dissipative dynamical systems with the property of heredity.

Exact solutions and Hyers–Ulam stability for fractional oscillation equations with pure delay

This paper presents the exact solutions of the homogeneous and the non-homogeneous fractional delay oscillation equations with order α∈(1,2). For non-homogeneous fractional delay oscillation equation, the Laplace transform is taken up as solving method in that it efficiently avoids some restrictions to the non-homogeneous term. Thereafter, we investigate the Hyers–Ulam stability of the fractional delay oscillation equation.

Solution of boundary value problem for fuzzy fractional differential equations by using differential transform method (DTM)

In this study, we implement a well-known transformation technique differential transform method (DTM), to the area of fractional differential equations, in this Paper, the solution to fuzzy fractional Initial Value Problem (FFIVP) under copout-type fuzzy fractional derivatives are define based on Hukuhara difference and strongly gene fuzzy differentiability. Also numerical examples are carried out for various types of problems including the Bagley-Torvik, Ricatti and composite fractional oscillation equations for the application of the method.

Sine‐cosine wavelet method for fractional oscillator equations

PurposeIn this article, a novel computational method is introduced for solving the fractional nonlinear oscillator differential equations on the semi‐infinite domain. The purpose of the proposed method is to get better and more accurate results.Design/methodology/approachThe proposed method is the combination of the sine‐cosine wavelets and Picard technique. The operational matrices of fractional‐order integration for sine‐cosine wavelets are derived and constructed. Picard technique is used to convert the fractional nonlinear oscillator equations into a sequence of discrete fractional linear differential equations. Operational matrices of sine‐cosine wavelets are utilized to transformed the obtained sequence of discrete equations into the systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear oscillator equations.FindingsThe convergence and supporting analysis of the method are investigated. The operational matrices contains many zero entries, which lead to the high efficiency of the method, and reasonable accuracy is achieved even with less number of collocation points. Our results are in good agreement with exact solutions and more accurate as compared with homotopy perturbation method, variational iteration method, and Adomian decomposition method.Originality/valueMany engineers can utilize the presented method for solving their nonlinear fractional models.

Solution of fractional oscillator equations using ultraspherical wavelets

Fractional oscillator type equations are well-known model equations to describe several phenomenon in mathematical physics, engineering and biology. In this paper, a new method incorporated by the ultraspherical wavelet operational matrix of general order integration and block-pulse functions are adopted to investigate the solution of fractional oscillator type equations. To facilitate this, the ultraspherical wavelets are first presented and the corresponding operational matrix of fractional-order integration is derived by virtue of block pulse functions. The properties of ultraspherical wavelets and block pulse functions are used to transform the underlying problem to a system of algebraic equations which can be easily solved by any of the usual numerical methods. The efficiency and accuracy of the proposed method is demonstrated by presenting several benchmark test problems. Moreover, special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.

Well‐posedness and iterative formula for fractional oscillator equations with delays

This paper investigates the well‐posedness and iterative formula for fractional oscillator equations with two different fractional orders and time delays. By the mathematical induction, a new generalized Grönwall's inequality in terms of a multivariate Mittag‐Leffler function is derived with an iteration argument. Subsequently, using the inequality, a generalized Mittag‐Leffler estimation of the solution is proposed. Furthermore, the explicit solution and iterative formula, as well as the unique existence, are presented on the basis of the method of steps. Finally, an example is given to illustrate the validity of our main results.

A new operational matrix of fractional order integration for the Chebyshev wavelets and its application for nonlinear fractional Van der Pol oscillator equation

In this paper, an efficient and accurate computational method based on the Chebyshev wavelets (CWs) together with spectral Galerkin method is proposed for solving a class of nonlinear multi-order fractional differential equations (NMFDEs). To do this, a new operational matrix of fractional order integration in the Riemann–Liouville sense for the CWs is derived. Hat functions (HFs) and the collocation method are employed to derive a general procedure for forming this matrix. By using the CWs and their operational matrix of fractional order integration and Galerkin method, the problems under consideration are transformed into corresponding nonlinear systems of algebraic equations, which can be simply solved. Moreover, a new technique for computing nonlinear terms in such problems is presented. Convergence of the CWs expansion in one dimension is investigated. Furthermore, the efficiency and accuracy of the proposed method are shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As a useful application, the proposed method is applied to obtain an approximate solution for the fractional order Van der Pol oscillator (VPO) equation.

Improvements in a method for solving fractional integral equations with some links with fractional differential equations

Abstract In this work, we apply and extend our ideas presented in [4] for solving fractional integral equations with Riemann-Liouville definition. The approach made in [4] turned any linear fractional integral equation with constant coefficients and rational orders into a similar one, but with integer orders. If the right hand side was smooth enough we could differentiate at both sides to arrive to a linear ODE with constant coefficients and some initial conditions, that can be solved via an standard procedure. In this procedure, there were two major obstacles that did not allow to obtain a full result. These were the assumptions over the smoothness of the source term and the assumption about the rationality of the orders. So, one of the main topics of this document is to describe a modification of the procedure presented in [4], when the source term is not smooth enough to differentiate the required amount of times. Furthermore, we will also study the fractional integral equations with non-rational orders by a limit process of fractional integral equations with rational orders. Finally, we will connect the previous material with some fractional differential equations with Caputo derivatives described in [7]. For instance, we will deal with the fractional oscillation equation, the fractional relaxation equation and, specially, its particular case of the Basset problem. We also expose how to compute these solutions for the Riemann-Liouville case.