A numerical method for solving boundary value problems for fractional differential equations

A numerical scheme, based on the Haar wavelet operational matrices of integration for solving linear two-point and multi-point boundary value problems for fractional differential equations is presented. The operational matrices are utilized to reduce the fractional differential equation to system of algebraic equations. Numerical examples are provided to demonstrate the accuracy and efficiency and simplicity of the method.     

On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation

In the present analysis, the motion of an immersed plate in a Newtonian fluid described by Torvik and Bagley’s fractional differential equation [1] has been considered. This Bagley Torvik equation has been solved by operational matrix of Haar wavelet method. The obtained result is compared with analytical solution suggested by Podlubny [2]. Haar wavelet method is used because its computation is simple as it converts the problem into algebraic matrix equation.     

Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations

In this paper, a new numerical method for solving fractional differential equations is presented. The fractional derivative is described in the Caputo sense. The method is based upon Bernoulli wavelet approximations. The Bernoulli wavelet is first presented. An operational matrix of fractional order integration is derived and is utilized to reduce the initial and boundary value problems to system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

A wavelet operational method for solving fractional partial differential equations numerically

Fractional calculus is an extension of derivatives and integrals to non-integer orders, and a partial differential equation involving the fractional calculus operators is called the fractional PDE. They have many applications in science and engineering. However not only the analytical solution existed for a limited number of cases, but also the numerical methods are very complicated and difficult. In this paper, we newly establish the simulation method based on the operational matrices of the orthogonal functions. We formulate the operational matrix of integration in a unified framework. By using the operational matrix of integration, we propose a new numerical method for linear fractional partial differential equation solving. In the method, we (1) use the Haar wavelet; (2) establish a Lyapunov-type matrix equation; and (3) obtain the algebraic equations suitable for computer programming. Two examples are given to demonstrate the simplicity, clarity and powerfulness of the new method.     

Wavelet operational matrix method for solving the Riccati differential equation

A Haar wavelet operational matrix method (HWOMM) was derived to solve the Riccati differential equations. As a result, the computation of the nonlinear term was simplified by using the Block pulse function to expand the Haar wavelet one. The proposed method can be used to solve not only the classical Riccati differential equations but also the fractional ones. The capability and the simplicity of the proposed method was demonstrated by some examples and comparison with other methods.     

Discretization algorithm for fractional order integral by Haar wavelet approximation

A discretization algorithm is proposed by Haar wavelet approximation theory for the fractional order integral. In this paper, the integration time is divided into two parts, one presents the effect of the past sampled data, calculated by the iterative method, and the other presents the effect of the recent sampled data at a fixed time interval, calculated by the Haar wavelet. This method can reduce the amount of the stored data effectively and be applied to the design of discrete-time fractional order PID controllers. Finally, several numerical examples and simulation results are given to illustrate the validity of this discretization algorithm.     

A study of fractional Lotka-Volterra population model using Haar wavelet and Adams-Bashforth-Moulton methods

The Lotka-Volterra (LV) system is an interesting mathematical model because of its significant and wide applications in biological sciences and ecology. A fractional LV model in the Caputo sense is investigated in this paper. Namely, we provide a comparative study of the considered model using Haar wavelet and Adams-Bashforth-Moulton methods. For the first method, the Haar wavelet operational matrix of the fractional order integration is derived and used to solve the fractional LV model. The main characteristic of the operational method is to convert the considered model into an algebraic equation which is easy to solve. To demonstrate the efficiency and accuracy of the proposed methods, some numerical tests are provided.     

Numerical solutions to initial and boundary value problems for linear fractional partial differential equations

In this article, Haar wavelets have been employed to obtain solutions of boundary value problems for linear fractional partial differential equations. The differential equations are reduced to Sylvester matrix equations. The algorithm is novel in the sense that it effectively incorporates the aperiodic boundary conditions. Several examples with numerical simulations are provided to illustrate the simplicity and effectiveness of the method.     

A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order

In this paper we present a computational method for solving a class of nonlinear Fredholm integro-differential equations of fractional order which is based on CAS (Cosine And Sine) wavelets. The CAS wavelet operational matrix of fractional integration is derived and used to transform the equation to a system of algebraic equations. Some examples are included to demonstrate the validity and applicability of the technique.     

Generalized wavelet quasilinearization method for solving population growth model of fractional order

The primary aim of this study is to introduce and develop a generalized wavelet method together with the quasilinearization technique to solve the Volterra's population growth model of fractional order. Unlike the existing operational matrix methods based on orthogonal functions, we formulate the wavelet operational matrices of general order integration without using the block pulse functions. Consequently, the governing problem is transformed into an equivalent system of algebraic equations, which can be tackled with any classical method. The applicability of the proposed method is demonstrated via an illustrative comparison of the numerical outcomes with those found by other known methods. The experimental outcomes demonstrate that the proposed method is fast, accurate, simple, and computationally reliable.     

A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations

In this research, a Bernoulli wavelet operational matrix of fractional integration is presented. Bernoulli wavelets and their properties are employed for deriving a general procedure for forming this matrix. The application of the proposed operational matrix for solving the fractional delay differential equations is explained. Also, upper bound for the error of operational matrix of the fractional integration is given. This operational matrix is utilized to transform the problem to a set of algebraic equations with unknown Bernoulli wavelet coefficients. Several numerical examples are solved to demonstrate the validity and applicability of the presented technique.     

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

In this paper, shifted Legendre polynomials will be used for constructing the numerical solution for a class of multiterm variable‐order fractional differential equations. In the proposed method, the shifted Legendre operational matrix of the fractional variable‐order derivatives will be investigated. The fundamental problem is reduced to an algebraic system of equations using the constructed matrix and the collocation technique, which can be solved numerically. The error estimate of the proposed method is investigated. Some numerical examples are presented to prove the applicability, generality, and accuracy of the suggested method.     

Solving a nonlinear fractional differential equation using Chebyshev wavelets

Chebyshev wavelet operational matrix of the fractional integration is derived and used to solve a nonlinear fractional differential equations. Some examples are included to demonstrate the validity and applicability of the technique.     

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

In this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz–Legendre wavelet approximation. We derive a new operational vector for the Riemann–Liouville fractional integral of the Müntz–Legendre wavelets by using the Laplace transform method. Applying this operational vector and collocation method in our approach, the problem can be reduced to a system of linear and nonlinear algebraic equations. The arising system can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, seven test problems are considered to compare our results with other well‐known methods used for solving these problems. The results in the tabulated tables highlighted that the proposed method is an efficient mathematical tool for analyzing distributed order fractional differential equations of the general form.     

The Legendre wavelet method for solving fractional differential equations

Fractional differential equations are solved using the Legendre wavelets. An operational matrix of fractional order integration is derived and is utilized to reduce the fractional differential equations to system of algebraic equations. The illustrative examples are provided to demonstrate the applicability, simplicity of the numerical scheme based on the Legendre.     

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

In this paper, a new two‐dimensional fractional polynomials based on the orthonormal Bernstein polynomials has been introduced to provide an approximate solution of nonlinear fractional partial Volterra integro‐differential equations. For this aim, the fractional‐order orthogonal Bernstein polynomials (FOBPs) are constructed, and its operational matrices of integration, fractional‐order integration, and derivative in the Caputo sense and product operational matrix are derived. These operational matrices are utilized to reduce the under study problem to a nonlinear system of algebraic equations. Using the approximation of FOBPs, the convergence analysis and error estimate associated to the proposed problem have been investigated. Finally, several examples are included to clarify the validity, efficiency, and applicability of the proposed technique via FOBPs approximation.     

Operational solution of fractional differential equations

The main goal of this paper is to solve fractional differential equations by means of an operational calculus. Our calculus is based on a modified shift operator which acts on an abstract space of formal Laurent series. We adopt Weyl’s definition of derivatives of fractional order.     

Wavelet method for a class of fractional convection-diffusion equation with variable coefficients

A wavelet method to the solution for a class of space–time fractional convection-diffusion equation with variable coefficients is proposed, by which combining Haar wavelet and operational matrix together and dispersing the coefficients efficaciously. The original problem is translated into Sylvester equation and computation became convenient. The numerical example shows that the method is effective.     

Numerical methods for nonlinear partial differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order.     

Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet

In this paper, we first construct the second kind Chebyshev wavelet. Then we present a computational method based on the second kind Chebyshev wavelet for solving a class of nonlinear Fredholm integro-differential equations of fractional order. The second kind Chebyshev wavelet operational matrix of fractional integration is derived and used to transform the equation to a system of algebraic equations. The method is illustrated by applications and the results obtained are compared with the existing ones in open literature. Moreover, comparing the methodology with the known technique shows that the present approach is more efficient and more accurate.