Operational Matrix Of Fractional Derivative Research Articles

A new approach of B-spline wavelets to solve fractional differential equations

This paper presents a groundbreaking method for solving the multi-order fractional differential (M-OFD), both linear and nonlinear, as well as fractional partial differential equations (FPDE)s. This approach involves constructing an operational matrix of fractional derivatives using linear B-spline (LB-S) wavelet functions with perfect subtlety. The new method has two crucial features. Firstly, it simplifies the problem by converting it into a set of algebraic equations, which is a significant advantage. This makes the method highly accurate and reliable. Secondly, it uses thresholding to dramatically reduce the computational workload in linear problems. This leads to lightning-fast and highly efficient problem-solving. The newly developed scheme underwent a thorough examination of its error estimates and convergence, revealing remarkable results in terms of accuracy and efficiency. The analysis provides a comprehensive understanding of the scheme’s performance, highlighting its potential as a dependable and effective method. Based on the findings, it is evident that the proposed method not only delivers exceptional precision but also operates with remarkable efficiency.

Fractional Pantograph Delay Equations Solving by the Meshless Methods

This work describes two efficient and useful methods for solving fractional pantograph delay equations (FPDEs) with initial and boundary conditions. These two methods depend mainly on orthogonal polynomials, which are the method of the operational matrix of fractional derivative that depends on Bernstein polynomials and the operational matrix of the fractional derivative with Shifted Legendre polynomials. The basic procedure of this method is to convert the pantograph delay equation to a system of linear equations and by using, the operational matrices we get rid of the integration and differentiation operations, which makes solving the problem easier. The concept of Caputo has been used to describe fractional derivatives. Finally, some numerical examples are identified to show the utility and capability of the two proposed approaches. Mathematica®12 program has been relied upon in the calculations.

Approximate solutions for a fractional thermostat model boundary value problem via Bernstein's collocation method with Legendre polynomials

Numerous classical models of thermostats have been analyzed in the literature, and while existence and uniqueness results have been established, the investigation of other properties, such as stability, can present significant challenges. Therefore, there is a need for fractional mathematical models that are nonlocal problems and allow for the straightforward study of the asymptotic behavior of their solutions. However, solving these problems is often exceedingly complex or even impossible, necessitating the use of numerical methods to obtain approximate solutions. The focus of this research is a generalized boundary value problem that is based on a classical thermostat model. Our initial step involves determining the associated integral equation for the problem at hand. The properties of existence and uniqueness are proved by exploiting the classical contraction principle of Banach together with another kind called ‐ ‐contraction. Afterwards, thanks to the Bernstein polynomials and fractional matrix of derivative, the approximate solutions of our nonlinear fractional thermostat model problem are obtained. This approach utilizes the Legendre operational matrix of fractional derivatives in combination with a truncated Legendre series for the numerical integration of fractional differential equations. By doing so, the problem can be reduced to solving a system of algebraic equations, resulting in significant simplification. This method has been successfully used to solve both linear and nonlinear fractional differential equations. Finally, an example on this type of problem have been studied and the approximate solution are plotted.

An Efficient Numerical Scheme for Solving Multiorder Tempered Fractional Differential Equations via Operational Matrix

In this paper, we extend the operational matrix method to solve the tempered fractional differential equation, via shifted Legendre polynomial. Although the operational matrix method is widely used in solving various fractional calculus problems, it is yet to apply in solving fractional differential equations defined in the tempered fractional derivatives. We first derive the analytical expression for tempered fractional derivative for x p , hence, using it to derive the new operational matrix of fractional derivative. By using a few terms of shifted Legendre polynomial and via collocation scheme, we were able to obtain a good approximation for the solution of the multiorder tempered fractional differential equation. We illustrate it using some numerical examples.

Galerkin operational approach for multi-dimensions fractional differential equations

The current manuscript introduces a novel numerical treatment for multi-term fractional differential equations with variable coefficients. The spectral Galerkin approach is developed with the operational matrix of fractional derivatives to get a system of algebraic equations that can be solved using a suitable technique. The operational matrix is introduced based on a combination of the shifted Jacobi polynomials. The proposed approach is applied also for 1+1 and 2+1 multi-term time-fractional diffusion and diffusion-wave equations. In addition, the convergence analysis for the presented approaches is investigated. Some specific numerical examples are given to ascertain the wide applicability and the good efficiency of the suggested algorithms, and to confirm that nonlocal numerical methods are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account.

Appell Type Changhee Polynomials Operational Matrix of Fractional Derivatives and its Applications

Appell Type Changhee Polynomials Operational Matrix of Fractional Derivatives and its Applications

A new operational matrix based on Müntz–Legendre polynomials for solving distributed order fractional differential equations

Our main aim in this work is to find the operational matrix of fractional derivative and the operational matrix of distributed order fractional derivative for the Müntz–Legendre polynomials (MLPs). The operational matrix approach with the tau method or collocation method is applied to reduce the solution of the linear/nonlinear distributed order fractional differential equations (DFDEs) to a system of linear/nonlinear algebraic equations. Moreover, seven numerical examples are included to show the validity and applicability of the suggested methods.

Shifted Genocchi Polynomials Operational Matrix for Solving Fractional Order Stiff System

In this paper, we solve the fractional order stiff system using shifted Genocchi polynomials operational matrix. Different than the well known Genocchi polynomials, we shift the interval from [0, 1] to [1, 2] and name it as shifted Genocchi polynomials. Using the nice properties of shifted Genocchi polynomials which inherit from classical Genocchi polynomials, the shifted Genocchi polynomials operational matrix of fractional derivative will be derived. Collocation scheme are used together with the operational matrix to solve some fractional order stiff system. From the numerical examples, it is obvious that only few terms of shifted Genocchi polynomials is sufficient to obtain result in high accuracy.

Two Fibonacci operational matrix pseudo-spectral schemes for nonlinear fractional Klein–Gordon equation

This paper is devoted to developing spectral solutions for the nonlinear fractional Klein–Gordon equation. The typical collocation method and the tau method are employed for obtaining the desired numerical solutions. In order to do this, a new operational matrix of fractional derivatives of Fibonacci polynomials is established. The idea behind the derivation of this matrix is based on utilizing the connection formula between the Fibonacci and Chebyshev polynomials. The introduced operational matrix is used along with the weighted residual quadrature spectral method and the collocation method to convert the nonlinear fractional Klein–Gordon equation into a system of algebraic equations. By solving the resulting system, we obtain a semi-analytic solution. The convergence and error analysis of the method are discussed. Some numerical results and discussions are presented aiming to illustrate the wide applicability and accuracy of the proposed algorithms.

Numerical solutions for nonlinear multi-term fractional differential equations via Dickson operational matrix

The purpose of this paper is to present a novel numerical technique for solving a class of nonlinear multi-term fractional differential equations. The proposed method mainly relies on using the Dickson polynomial to derive the Dickson operational matrix of fractional derivative, which is then used to solve the given problem. The fundamental significance of this strategy is that it converts the fractional problem into a system of nonlinear equations that can be numerically solved. In addition, the error bound of the suggested method is deduced. Numerical experiments are presented to verify and confirm the accuracy of the suggested technique for dealing with such problems.

Non-polynomial B-spline and shifted Jacobi spectral collocation techniques to solve time-fractional nonlinear coupled Burgers\u2019 equations numerically

This paper proposes two numerical approaches for solving the coupled nonlinear time-fractional Burgers’ equations with initial or boundary conditions on the interval [0, L]. The first method is the non-polynomial B-spline method based on L1-approximation and the finite difference approximations for spatial derivatives. The method has been shown to be unconditionally stable by using the Von-Neumann technique. The second method is the shifted Jacobi spectral collocation method based on an operational matrix of fractional derivatives. The proposed algorithms’ main feature is that when solving the original problem it is converted into a nonlinear system of algebraic equations. The efficiency of these methods is demonstrated by applying several examples in time-fractional coupled Burgers equations. The error norms and figures show the effectiveness and reasonable accuracy of the proposed methods.

Existence, uniqueness, Ulam–Hyers stability and numerical simulation of solutions for variable order fractional differential equations in fluid mechanics

In this paper, a numerical approximation is shown to solve solutions of time fractional linear differential equations of variable order in fluid mechanics where the considered fractional derivatives of variable order are in the Caputo sense. Existence, uniqueness of solutions and Ulam–Hyers stability results are displayed. To solve the considered equations a numerical approximation based on the shifted Legendre polynomials are proposed. To perform the method, an operational matrix of fractional derivative with variable-order is derived for the shifted Legendre polynomials to be applied for developing the unknown function. By substituting the aforesaid operational matrix into the considered equations and using the properties of the shifted Legendre polynomials together with the collocation points, the main equations are reduced to a system of algebraic equations. The approximate solution is calculated by solving the obtained system which is technically easier for checking. We also study the error analysis for the approximate solution yielded by the introduced method. Finally, the accuracy and performance of the proposed method are checked by some illustrative examples. The illustrative examples results establish the applicability and usefulness of the proposed method.

A new operational matrix of fractional derivative based on the generalized Gegenbauer–Humbert polynomials to solve fractional differential equations

In this paper, a new type of wavelet method to solve fractional differential equations (linear or nonlinear) is proposed. The proposed method is based on the generalized Gegenbauer–Humbert polynomial. First, we derived the operational matrices for integer and fractional order derivatives. Then, using these operational matrices with the proposed method, we transformed the given problem into a system of algebraic equations. Then, some linear and nonlinear examples were considered and discussed to confirm the efficiency and accuracy of the proposed method.

The construction of a new operational matrix of the distributed-order fractional derivative using Chebyshev polynomials and its applications

In this paper, the properties of Chebyshev polynomials and the Gauss–Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given. Abbreviations: DFDEs: distributed-order fractional differential equations; DTFDEs: distributed-order time-fractional diffusion equations; DTFWEs: distributed-order time-fractional wave equations; OMFD: operational matrix of fractional derivative; SCP: shifted Chebyshev polynomial

An efficient approximate method for solving two-dimensional fractional optimal control problems using generalized fractional order of Bernstein functions

AbstractWe define a new operational matrix of fractional derivative in the Caputo type and apply a spectral method to solve a two-dimensional fractional optimal control problem (2D-FOCP). To acquire this aim, first we expand the state and control variables based on the fractional order of Bernstein functions. Then we reduce the constraints of 2D-FOCP to a system of algebraic equations through the operational matrix. Now, one can solve straightforward the problem and drive the approximate solution of state and control variables. The convergence of the method in approximating the 2D-FOCP is proved. We demonstrate the efficiency and superiority of the method by comparing the results obtained by the presented method with the results of previous methods in some examples.

Numerical solution of the Bagley\u2013Torvik equation using shifted Chebyshev operational matrix

In this study, an efficient numerical scheme based on shifted Chebyshev polynomials is established to obtain numerical solutions of the Bagley–Torvik equation. We first derive the shifted Chebyshev operational matrix of fractional derivative. Then, by the use of these operational matrices, we reduce the corresponding fractional order differential equation to a system of algebraic equations, which can be solved numerically by Newton’s method. Furthermore, the maximum absolute error is obtained through error analysis. Finally, numerical examples are presented to validate our theoretical analysis.

Fractional-Lucas optimization method for evaluating the approximate solution of the multi-dimensional fractional differential equations

The paper investigates the numerical solution of the multi-dimensional fractional differential equations by applying fractional-Lucas functions (FLFs) and an optimization method. First, the FLFs and their properties are introduced. Then, according to the pseudo-operational matrix of derivative and modified operational matrix of fractional derivative, we present the framework of numerical technique. Also, for computational technique, we evaluate the upper bound of error. As a result, we expound the proposed scheme by solving several kinds of problems. Our computational results demonstrate that the proposed method is powerful and applicable for nonlinear multi-order fractional differential equations, time-fractional convection–diffusion equations with variable coefficients, and time-space fractional diffusion equations with variable coefficients.

Derivation of operational matrix of Rabotnov fractional-exponential kernel and its application to fractional Lienard equation

Our motive in this contribution is to find out the operational matrix of fractional derivative having non singular kernel namely Rabotnov fractional-exponential (RFE) kernel which is recently introduced and seeking numerical solution of non-linear Lienard equation which have Rabotnov fractional-exponential kernel fractional derivative. First we derive an approximation formula of the fractional order derivative of polynomial function zk in term of RFE kernel. Using this formula and some properties of shifted Legendre polynomials, we find out the operational matrix of fractional order differentiation. In the author of knowledge this operational matrix of RFE kernel fractional derivative is derived first time. We solve a new class of fractional partial differential equation (FPDEs) by implementation of this newly derived operational matrix. We show that our newly derived operational matrix is valid by taking an fractional derivative of a polynomial. Also, we study a new model of Lienard equation with RFE kernel fractional derivative and we can easily predict the feasibility of our numerical method to this new model.

Numerical Technique for Solving Fractional Generalized Pantograph-Delay Differential Equations by Using Fractional-Order Hybrid Bessel Functions

This paper introduces a new computational technique for dealing with fractional generalized pantograph-delay differential equations (PDDEs). The main idea of the proposed method is based on the fractional-order hybrid Bessel functions (FHBFs), which are constructed by the combination of fractional Bessel functions with block-pulse functions. The fractional derivative is applied in the Caputo sense. In the numerical technique, the operational matrix of fractional derivative is applied to reduce fractional generalized PDDEs to a system of algebraic equations with unknown fractional-order hybrid Bessel coefficients. In addition, the estimation of the error is analyzed. The error analysis indicates that when the number of FHBFs bases is increased, the obtained results convergent to the exact solution. Illustrative examples will be examined to demonstrate the validity and applicability of the presented technique. Also, the outcomes in comparison with the existing results in other methods illustrate the preference of the presented method.

A new numerical method for fractional order Volterra integro-differential equations

Abstract In this paper, a new numerical method based on the Bernoulli wavelet approximation is presented for the solution of fractional order Volterra integro-differential equations. The operational matrix of fractional derivative of order λ in the Caputo sense derived together with the Gaussian quadrature rule implemented to reduce the Volterra integro-differential equations to system of algebraic equations. Convergence analysis and error estimation of the proposed method have been discussed. The efficiency and accuracy of the proposed method have been depicted through some illustrative examples.