A Morgan‐Voyce Polynomial Framework for Solving Variable‐Order Atangana–Baleanu Fractional Differential Equations

In this paper, we examined a wide class of the variable order fractional problems such as linear and nonlin-ear fractional variable order differential equations, variable order fractional functional boundary value problems, variable order fractional pantograph differential equations. The proposed method is a collocation method based on the Bessel polynomials and the operational matrix of derivatives, which transformed equations into a system of non-linear algebraic equations to achieve the approximate solution. By using Caputo fractional derivative, the operational matrix of the variable-order fractional derivatives is constructed. The error analysis shows that the method is convergent. Several numerical results confirm the accuracy and efficiency of the proposed method. Keywords: Bessel collocation method, Variable-order fractional operational matrix, Variable order fractional differential equations, Variable order fractional functional boundary value problems, Variable order fractional panto-graph equations.

In the theory of differential equations, the study of existence and the uniqueness of the solutions are important. In the last few decades, many researchers have had a keen interest in finding the existence–uniqueness solution of constant fractional differential equations, but literature focusing on variable order is limited. In this article, we consider a Caputo type variable order fractional differential equation. First, we present the existence–uniqueness of a solution of the considered problem. Secondly, By borrowing the idea from the theory of ordinary differential equations, we extend the continuation theorem for the variable order fractional differential equation. Further, we prove the global existence results. Finally, we present different types of Ulam–Hyers stability results, which have never been studied before for the Caputo type variable order fractional differential equation.

The symmetry features of fractional differential equations allow effective explanation of physical and biological phenomena in nature. The generalized form of the fractional differential equations is the variable-order fractional differential equations that describe the physical and biological applications. This paper discusses the closed-form traveling wave solutions for the nonlinear space–time variable-order fractional modified Kawahara and (2 + 1)-dimensional Burger hierarchy equations. The variable-order fractional differential equation has a derivative operator in the Caputo sense that is converted into the integer-order ordinary differential equation (ODE) by fractional transformation. The obtained ODE is solved by the exponential rational function method, and as a result, new exact solutions are constructed. Two problems are proposed to confirm the solutions of the space-time variable-order fractional differential equations.

Solution existence for non-autonomous variable-order fractional differential equations

Numerical calculation of the fractional integrals and derivatives is the code tosearch fractional calculus and solve fractional differential equations. The exactsolutions to fractional differential equations are compelling to get in real applications, due to the nonlocality and complexity of the fractional differential operators, especially for variable-order fractional differential equations. Therefore, it is significant to enhanced numerical methods for fractional differential equations. In this work, we consider variable-order fractional differential equations by reproducing kernel method. There has been much attention in the use of reproducing kernels for the solutions to many problems in the recent years. We give two examples to demonstrate how efficiently our theory can be implemented in practice.

Numerical solution method for multi-term variable order fractional differential equations by shifted Chebyshev polynomials of the third kind

PurposeMulti-term variable-order fractional differential equations (VO-FDEs) are powerful tools in accurate modeling of transient-regime real-life problems such as diffusion phenomena and nonlinear viscoelasticity. In this paper the Chebyshev polynomials of the fourth kind is employed to obtain a numerical solution for those multi-term VO-FDEs.Design/methodology/approachTo this end, operational matrices for the approximation of the VO-FDEs are obtained using the Fourth kind Chebyshev Wavelets (FKCW). Thus, the VO-FDE is condensed into an algebraic equation system. The solution of the system of those equations yields a coefficient vector, the coefficient vector in turn yields the approximate solution.FindingsSeveral examples that we present at the end of the paper emphasize the efficacy and preciseness of the proposed method.Originality/valueThe value of the paper stems from the exploitation of FKCWs for the numerical solution of multi-term VO-FDEs. The method produces accurate results even for relatively small collocation points. What is more, FKCW method provides a compact mapping between multi-term VO-FDEs and a system of algebraic equations given in vector-matrix form.

The multiterm fractional variable-order differential equation has a massive application in physics and engineering problems. Therefore, a numerical method is presented to solve a class of variable order fractional differential equations (FDEs) based on an operational matrix of shifted Chebyshev polynomials of the fourth kind. Utilizing the constructed operational matrix, the fundamental problem is reduced to an algebraic system of equations which can be solved numerically. The error estimate of the proposed method is studied. Finally, the accuracy, applicability, and validity of the suggested method are illustrated through several examples.

The multiterm fractional differential equation has a wide application in engineering problems. Therefore, we propose a method to solve multiterm variable order fractional differential equation based on the second kind of Chebyshev Polynomial. The main idea of this method is that we derive a kind of operational matrix of variable order fractional derivative for the second kind of Chebyshev Polynomial. With the operational matrices, the equation is transformed into the products of several dependent matrices, which can also be viewed as an algebraic system by making use of the collocation points. By solving the algebraic system, the numerical solution of original equation is acquired. Numerical examples show that only a small number of the second kinds of Chebyshev Polynomials are needed to obtain a satisfactory result, which demonstrates the validity of this method.

In this paper, a new approach based on a generalized fuzzy hyperbolic model is used for the numerical solution of variable-order fractional differential algebraic equations. The fractional derivative is described in the Atangana–Baleanu sense that is a new derivative with fractional order based on the generalized Mittag–Leffler function. First, by using fuzzy solutions with adjustable parameters, the variable-order fractional differential algebraic equations are reduced to a problem consisting of solving a system of algebraic equations. For adjusting the parameters of fuzzy solutions, an unconstrained optimization problem is then considered. A learning algorithm is also presented for solving the unconstrained optimization problem. Finally, some numerical examples are given to verify the efficiency and accuracy of the proposed approach.

Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets

In this manuscript, an algorithm for the computation of numerical solutions to some variable order fractional differential equations (FDEs) subject to the boundary and initial conditions is developed. We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices. Further, operational matrices are constructed using variable order differentiation and integration. We are finding the operational matrices of variable order differentiation and integration by omitting the discretization of data. With the help of aforesaid matrices, considered FDEs are converted to algebraic equations of Sylvester type. Finally, the algebraic equations we get are solved with the help of mathematical software like Matlab or Mathematica to compute numerical solutions. Some examples are given to check the proposed method’s accuracy and graphical representations. Exact and numerical solutions are also compared in the paper for some examples. The efficiency of the method can be enhanced further by increasing the scale level.

We introduce a machine learning framework that uses the differential evolution algorithm in combination with Adam–Bashforth–Moulton method to learn the parameters in a system of variable order fractional differential equations. In this work, we present out developments with regards to taking care of a class of problem: data‐driven discovery of system of variable order fractional differential equations. The main advantage of the proposed framework is that it works even if data corresponding to only one of the variables in the system of equations is given. We illustrate the working of our framework on several Examples including modeling the 2014–15 Ebola outbreak in Africa via fractional SEIR (susceptible, exposed, infected, removed) model.

Nonlinear fractional differential equations provide suitable models to describe real-world phenomena and many fractional derivatives are varying with time and space. The present study considers the advanced and broad spectrum of the nonlinear (NL) variable-order fractional differential equation (VO-FDE) in sense of VO Caputo fractional derivative (CFD) to describe the physical models. The VO-FDE transforms into an ordinary differential equation (ODE) and then solving by the modified (G′/G)-expansion method. For accuracy, the space-time VO fractional Korteweg-de Vries (KdV) equation is solved by the proposed method and obtained some new types of periodic wave, singular, and Kink exact solutions. The newly obtained solutions confirmed that the proposed method is well-ordered and capable implement to find a class of NL-VO equations. The VO non-integer performance of the solutions is studied broadly by using 2D and 3D graphical representation. The results revealed that the NL VO-FDEs are highly active, functional and convenient in explaining the problems in scientific physics.

Strong convergence of a Euler-Maruyama scheme to a variable-order fractional stochastic differential equation driven by a multiplicative white noise