Multi-order Fractional Differential Equations Research Articles

Numerical solution of the multi-order fractional differential equation using Legendre wavelets and eigenfunction approach

An eigenfunction approach is implemented in this article to solve the multi-order fractional differential equations (FDEs) with boundary conditions. The approximate unknown solution is expressed as a linear combination of eigenfunctions in the present paper. The proposed approach solved the problem via complex and real eigenfunction. By using the eigenvalue problem Dαϕ(x)=λϕ(x) subjected to u(0)=u(1)=0, first we calculate the eigenvalues and their corresponding eigenfunction and then use the output of this problem in the present scheme to simplify. Exact fractional integration has been utilized to solve the eigenvalue problem. Collocation points have also been used to reduce the problem to a system of algebraic equations. The obtained solutions are compared with the exact solution. In addition, the numerical convergence has also been studied of the proposed technique through three examples. These examples also demonstrate the applicatory and simplicity of the proposed scheme.

Existence and uniqueness of solutions for multi-order fractional differential equations with integral boundary conditions

In this paper, we consider the existence and uniqueness of solutions for the following nonlinear multi-order fractional differential equation with integral boundary conditions {(CD0+αu)(t)+∑i=1mλi(t)(CD0+αiu)(t)+∑j=1nμj(t)(CD0+βju)(t)+∑k=1pξk(t)(CD0+γku)(t)+∑l=1qωl(t)(CD0+δlu)(t)+σ(t)u(t)+f(t,u(t))=0,t∈[0,1],u″(0)=u‴(0)=0,u′(0)=η1∫01u(s)ds,u(1)=η2∫01u(s)ds,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} ({}^{C}D_{0+}^{\\alpha}u)(t)+\\sum_{i=1}^{m}\\lambda _{i}(t)({}^{C}D_{0+}^{\\alpha _{i}}u)(t)+ \\sum_{j=1}^{n}\\mu _{j}(t)({}^{C}D_{0+}^{\\beta _{j}}u)(t)\\\\ \\quad{}+\\sum_{k=1}^{p}\\xi _{k}(t)({}^{C}D_{0+}^{\\gamma _{k}}u)(t)+\\sum_{l=1}^{q}\\omega _{l}(t)({}^{C}D_{0+}^{\\delta _{l}}u)(t)\\\\ \\quad{}+\\sigma (t)u(t)+f(t,u(t))=0,\\quad t\\in [0,1],\\\\ u^{\\prime \\prime}(0)=u^{\\prime \\prime \\prime}(0)=0,\\qquad u^{\\prime}(0)=\\eta _{1}\\int _{0}^{1}u(s)\\,ds,\\qquad u(1)=\\eta _{2}\\int _{0}^{1}u(s)\\,ds, \\end{cases} $$\\end{document} where 0<delta _{1}<delta _{2}<cdots <delta _{q}<1<gamma _{1}<gamma _{2}<cdots <gamma _{p}<2<beta _{1}<beta _{2}<cdots <beta _{n}<3<alpha _{1}<alpha _{2}<cdots <alpha _{m}<alpha <4 and eta _{1}+2(1-eta _{2})neq 0. Using a fixed point theorem and Banach contractive mapping principle, we obtain some existence and uniqueness results.

Pontryagin Maximum Principle for Incommensurate Fractional-Orders Optimal Control Problems

We introduce a new optimal control problem where the controlled dynamical system depends on multi-order (incommensurate) fractional differential equations. The cost functional to be maximized is of Bolza type and depends on incommensurate Caputo fractional-orders derivatives. We establish continuity and differentiability of the state solutions with respect to perturbed trajectories. Then, we state and prove a Pontryagin maximum principle for incommensurate Caputo fractional optimal control problems. Finally, we give an example, illustrating the applicability of our Pontryagin maximum principle.

Collocation Method for the Numerical Solution of Multi-Order Fractional Differential Equations

This study presents a collocation approach for the numerical integration of multi-order fractional differential equations with initial conditions in the Caputo sense. The problem was transformed from its integral form into a system of linear algebraic equations. Using matrix inversion, the algebraic equations are solved and their solutions are substituted into the approximate equation to give the numerical results. The effectiveness and precision of the method were illustrated with the use of numerical examples.

Solving Nonlinear Multi-Order Fractional Differential Equations Using Bernstein Polynomials

Solving Nonlinear Multi-Order Fractional Differential Equations Using Bernstein Polynomials

Event-triggered bipartite synchronization of coupled multi-order fractional neural networks

This paper addresses the bipartite synchronization of coupled multi-order fractional neural networks (MFNNs) with time-varying delays. An effective event-triggered controller is proposed, and sufficient criteria for ensuring the bipartite synchronization are derived by using the Lyapunov function in vector form and the comparison principle for multi-order fractional differential equations. In addition, the preclusion of Zeno behavior is discussed. The results obtained in this paper cover the bipartite synchronization of both fractional neural networks with identical order and integer-order neural networks as special cases. A numerical example is given to verify the effectiveness of the proposed results.

A Two-Grid Spectral Deferred Correction Method for the Multi-Order Fractional Differential Equations

A Two-Grid Spectral Deferred Correction Method for the Multi-Order Fractional Differential Equations

Numerical study of multi-order fractional differential equations with constant and variable coefficients

In this manuscript, a numerical method based on the conjunction of Paraskevopoulos's algorithm and operational matrices is developed to solve numerically the multi-order linear and nonlinear fractional differential equations. By means of this conjunction, the multi-order problem is decomposed into a system of differential equations of fractional order which are then solved by employing operational matrices approach. The accuracy and efficiency of the method is examined by taking some examples. In addition, the numerical results presented in Pak et al. [Analytical solutions of linear inhomogeneous fractional differential equation with continuous variable coefficients. Adv Differ Equ. 2019;2019(256):1–22] are improved in our work.

A Multiple Interval Chebyshev-Gauss-Lobatto Collocation Method for Multi-Order Fractional Differential Equations

A Multiple Interval Chebyshev-Gauss-Lobatto Collocation Method for Multi-Order Fractional Differential Equations

Non-linear System of Multi-order Fractional Differential Equations: Theoretical Analysis and a Robust Fractional Galerkin Implementation.

This paper presents a comprehensive study of non-linear systems of multi-order fractional differential equations from aspects of theory and numerical approximation. In this regard, we first establish the well-posedness of the underlying problem by investigations concerning the existence, uniqueness, and influence of perturbed data on the behavior of the solutions as well as smoothness of the solutions under some assumptions on the given data. Next, from the numerical perspective, we develop and analyze a well-conditioned and high-order numerical approach based on an operational spectral Galerkin method. The main advantage of our strategy is that it characterizes the approximate solution via some recurrence formulas, instead of solving a complex non-linear block algebraic system that requires high computational costs. Moreover, the familiar spectral accuracy is justified in a weighted L^2-norm, and some practical test problems are provided to approve efficiency of the proposed method.

Application of Intelligent Paradigm through Neural Networks for Numerical Solution of Multiorder Fractional Differential Equations.

In this study, the intelligent computational strength of neural networks (NNs) based on the backpropagated Levenberg-Marquardt (BLM) algorithm is utilized to investigate the numerical solution of nonlinear multiorder fractional differential equations (FDEs). The reference data set for the design of the BLM-NN algorithm for different examples of FDEs are generated by using the exact solutions. To obtain the numerical solutions, multiple operations based on training, validation, and testing on the reference data set are carried out by the design scheme for various orders of FDEs. The approximate solutions by the BLM-NN algorithm are compared with analytical solutions and performance based on mean square error (MSE), error histogram (EH), regression, and curve fitting. This further validates the accuracy, robustness, and efficiency of the proposed algorithm.

On a novel impulsive boundary value pantograph problem under Caputo proportional fractional derivative operator with respect to another function

&lt;abstract&gt;&lt;p&gt;In this manuscript, we study the existence and Ulam's stability results for impulsive multi-order Caputo proportional fractional pantograph differential equations equipped with boundary and integral conditions with respect to another function. The uniqueness result is proved via Banach's fixed point theorem, and the existence results are based on Schaefer's fixed point theorem. In addition, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the proposed problem are obtained by applying the nonlinear functional analysis technique. Finally, numerical examples are provided to supplement the applicability of the acquired theoretical results.&lt;/p&gt;&lt;/abstract&gt;

Solution of Multi-order Fractional Differential Equation Based on Conformable Derivative by Shifted Legendre Polynomial

The aim of this article is the way for finding approximation solution of multi-order fractional differential equation with conformable sense with use approximated function by shifted Legendre polynomial, the method is easy and powerful for get our results of the linear and non-linear equation, the background idea behind this method is finding system of algebra after achieving messing variable is that mean obtain approximate solution, a few examples illustrates for presented how much our method is capable.

A novel matrix technique for multi-order pantograph differential equations of fractional order

The main purpose of this article is to investigate a novel set of (orthogonal) basis functions for treating a class of multi-order fractional pantograph differential equations (MOFPDEs) computationally. These polynomials, denoted bySn(x)and calledspecial polynomials, were first discovered in a study of a certain family of isotropic turbulence fields. They are expressible in terms of the generalized Laguerre polynomials and are related to the Bessel and Srivastava–Singhal polynomials. Unlike the Laguerre polynomials, all coefficients of the special polynomials are positive. We further introduce the fractional order of the special polynomials and use them along with some suitable collocation points in a special matrix technique to treat fractional-order MOFPDEs. Moreover, the convergence analysis of these polynomials is established. Through five example applications, the utility and efficiency of the present matrix approach are demonstrated and comparisons with some existing numerical schemes have been performed in this class.

A New Scheme for Solving Multiorder Fractional Differential Equations Based on Müntz–Legendre Wavelets

In this study, we apply the pseudospectral method based on Müntz–Legendre wavelets to solve the multiorder fractional differential equations with Caputo fractional derivative. Using the operational matrix for the Caputo derivative operator and applying the Chebyshev and Legendre zeros, the problem is reduced to a system of linear algebraic equations. We illustrate the reliability, efficiency, and accuracy of the method by some numerical examples. We also compare the proposed method with others and show that the proposed method gives better results.

A generalized operational matrix of mixed partial derivative terms with applications to multi-order fractional partial differential equations

In this paper, a computational approach based on the operational matrices in conjunction with orthogonal shifted Legendre polynomials (OSLPs) is designed to solve numerically the multi-order partial differential equations of fractional order consisting of mixed partial derivative terms. Our computational approach has ability to reduce the fractional problems into a system of Sylvester types matrix equations which can be solved by using MATLAB builtin function lyap(.). The solution is approximated as a basis vectors of OSLPs. The efficiency and the numerical stability is examined by taking various test examples.

On the solutions for generalised multiorder fractional partial differential equations arising in physics

In this article, we have studied solutions of a generalised multiorder fractional partial differential equations involving the Caputo time‐fractional derivative and the Riemann–Liouville space fractional derivatives using Laplace–Fourier transform technique. Proposed generalised multiorder fractional partial differential equation is reducible to Schrödinger equation, wave equation and diffusion equation in a more general sense, and hence, solutions of these equations are specifically noted. Not only this, solutions of equation proposed in the stochastic resetting theory in the context of Brownian motion can also be found in a general regime.

An hp-version Legendre spectral collocation method for multi-order fractional differential equations

In this paper, we consider the multi-order fractional differential equation and recast it into an integral equation. Based on the integral equation, we develop an hp-version Legendre spectral collocation method and the integral terms with the weakly singular kernels are calculated precisely according to the properties of Legendre and Jacobi polynomials. The hp-version error bounds under the L2-norm and the $L^{\infty }$ -norm are derived rigorously. Numerical experiments are included to illustrate the efficiency of the proposed method and the rationality of the theoretical results.

A new integral operational matrix with applications to multi-order fractional differential equations

<abstract> In this article, we propose a numerical method that is completely based on the operational matrices of fractional integral and derivative operators of fractional Legendre function vectors (FLFVs). The proposed method is independent of the choice of the suitable collocation points and expansion of the residual function as a series of orthogonal polynomials as required for Spectral collocation and Spectral tau methods. Consequently, the high efficient numerical results are obtained as compared to the other methods in the literature. The other novel aspect of our article is the development of the new integral and derivative operational matrices in Riemann-Liouville and Caputo senses respectively. The proposed method is computer-oriented and has the ability to reduce the fractional differential equations (FDEs) into a system of Sylvester types matrix equations that can be solved using <italic>MATLAB</italic> builtin function <italic>lyap</italic>(.). As an application of the proposed method, we solve multi-order FDEs with initial conditions. The numerical results obtained otherwise in the literature are also improved in our work. </abstract>

Langevin differential equations with general fractional orders and their applications to electric circuit theory

Multi-order fractional differential equations have been studied due to their applications in modeling, and solved using various mathematical methods. We present explicit analytical solutions for several families of Langevin differential equations with general fractional orders, both homogeneous and inhomogeneous cases. The results can be written, in general and special cases, by means of a recently defined bivariate Mittag-Leffler function and the associated operators of fractional calculus. The novelty of this work is to apply an appropriate norm on the proof of existence and uniqueness theorem, and discuss the application of Langevin differential equation with fractional orders in several interesting cases to the electrical circuit. Moreover, we investigate Ulam–Hyers stability of Caputo type fractional Langevin differential equation. At the end, we provide an example to verify our main results.