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This work is devoted to numerical solution to solve fractional optimal control problems (FOCPs). The fractional-order Boubaker wavelets (FOBWs) are introduced. By applying hypergeometric functions, an exact formula for Riemann–Liouville fractional integral operator for FOBW is obtained. By using this formula, the FOCP is reduced into a corresponding optimization problem which can be solved by applying the Lagrange multiplier method. Also, the convergence for the present method is provided. Seven numerical examples are included to demonstrate the applicability of the proposed technique. In Example 6, it will be shown that we can achieve the exact solutions which were not obtained previously in the literature. In addition, in Example 7, a practical example in a cancer model is provided.

In this study, the orthonormal piecewise Bernoulli functions are generated as a new kind of basis functions. An explicit matrix related to fractional integration of these functions is obtained. An efficient direct method is developed to solve a novel set of optimal control problems defined using a fractional integro-differential equation. The presented technique is based on the expressed basis functions and their fractional integral matrix together with the Gauss–Legendre integration method and the Lagrange multipliers algorithm. This approach converts the original problem into a mathematical programming one. Three examples are investigated numerically to verify the capability and reliability of the approach.

We give a novel method to solve systems of nonlinear fractional differential equations (NFDEs). We first introduce a new class of basis functions called fractional-order generalized Taylor wavelets. The Riemann–Liouville fractional integral operator, of the fractional-order generalized Taylor wavelets, is determined. An exact formula for this operator will be obtained by using the regularized beta function. By applying this exact formula we reduce the given system of NFDEs to a system of algebraic equations. The method is applied to the fractional models in human respiratory syncytial virus infection. We also give numerical examples to show the effectiveness and high accuracy of the present method.

Legendre wavelets and their exact Riemann-Liouville fractional integrals are used to compute numerical solutions to fractional delay differential equations, by reducing the problem to algebraic equations. An error bound of Legendre wavelet approximation is analytically computed, which implies the approximation converges when the degree of the Legendre polynomials or the number of wavelets approaches infinity. Finally, several numerical examples are considered.

We give an effective method for solving fractional Riccati differential equations. We first define the fractional-order Boubaker wavelets (FOBW). Using the hypergeometric functions, we determine the exact values for the Riemann-Liouville fractional integral operator of the FOBW. The properties of FOBW, the exact formula, and the collocation method are used to transform the problem of solving fractional Riccati differential equations to the solution of a set of algebraic equations. These equations are solved via Newton's iterative method. The error estimation for the present method is also determined. The performance of the developed numerical schemes is assessed through several examples. This method yields very accurate results. The given numerical examples support this claim.

In this study, by generalizing the classical clique polynomials, a new set of functions called the hybrid clique functions is defined. These functions retain the useful properties of the original clique polynomials and have an auxiliary parameter that is useful in accurately approximating non-smooth continuous functions. Some useful matrix relations related to these functions are derived. They are successfully utilized in constructing a collocation algorithm for space–time fractional Schrodinger problems with non-smooth continuous solution. This approach converts the original fractional problem solving into solving a system of algebraical equations. Applicability and accuracy of the expressed technique are investigated in three illustrative examples.

Recently, a new family of fractional derivatives called the piecewise fractional derivatives has been introduced, arguing that for some problems, each of the classical fractional derivatives may not be able to provide an accurate statement of the consideration problem alone. In defining this kind of derivatives, several types of fractional derivatives can be used simultaneously.This study introduces a new kind of piecewise fractional derivative by employing the Caputo type distributed-order fractional derivative and ABC fractional derivative. The one- and two-dimensional piecewise fractional Galilei invariant advection-diffusion equations are defined using this piecewise fractional derivative.A new class of basis functions called the orthonormal piecewise Vieta-Lucas (VL) functions are defined. Fractional derivatives of these functions in the Caputo and ABC senses are computed. These functions are utilized to construct two numerical methods for solving the introduced problems under non-local boundary conditions. The proposed methods convert solving the original problems into solving systems of algebraic equations.The accuracy and convergence order of the proposed methods are examined by solving several examples. The obtained results are investigated, numerically.This study introduces a kind of piecewise fractional derivative. This derivative is employed to define the one- and two-dimensional piecewise fractional Galilei invariant advection-diffusion equations. Two numerical methods based on the orthonormal VL polynomials and orthonormal piecewise VL functions are established for these problems. The numerical results obtained from solving several examples confirm the high accuracy of the proposed methods.

The aim of this paper is to introduce a new wavelet method for presenting approximate solutions of multitype variable-order (VO) fractional partial differential equations arising from the modeling of phenomena. In specific, this paper focuses on the numerical solution of the VO-fractional mobile-immobile advection-dispersion equation, Klein Gordon equation and Burgers equation. These equations are converted into a system of algebraic equations with the assistance of the bivariate Genocchi wavelet functions, their operational matrices, and the variable-order fractional Caputo derivative operator. Also, we present a new technique to get the operational matrix of integration and VO-fractional derivative. The modified operational matrices for solving the proposed equations are powerful and effective. So that, the accuracy of these matrices directly affects the implementation process. Finally, we consider numerical examples to confirm the superiority of the scheme, and for each example, exhibit the results through graphs and tables.

This study aims to generate a novel set of basis functions called the orthonormal piecewise Chelyshkov functions to solve a certain category of optimal control problems whose dynamical system is go...

In this article, the non-singular variable-order fractional derivative in the Heydari-Hosseininia concept is used to formulate the variable-order fractional form of the coupled nonlinear Ginzburg-Landau equations. To solve this system, a numerical scheme is constructed based upon the shifted Vieta-Lucas polynomials. In this method, with the help of classical and fractional derivative matrices of the shifted Vieta-Lucas polynomials (which are extracted in this study), solving the studied problem is transformed into solving a system of nonlinear algebraic equations. The convergence analysis and the truncation error of the shifted Vieta-Lucas polynomials in two dimensions are investigated. Numerical problems are demonstrated to confirm the convergence rate of the presented algorithm.

In the present paper, fractional-order generalized Legendre wavelets (FOGLWs) are introduced. We apply the FOGLWs for solving fractional Riccati differential equation. By using the hypergeometric function, we obtain an exact formula for the Riemann–Liouville fractional integral operator (RLFIO) of the FOGLWs. By using this exact formula and the properties of the FOGLWs, we reduce the solution of the fractional Riccati differential equation to the solution of an algebraic system. This algebraic system can be solved effectively. This method gives very accurate results. The given numerical examples support this claim.

In this study, a new fractal-fractional (FF) derivative is defined by coupling the local conformable derivative and non-local Caputo fractional derivative. Using the defined derivative, a space–time FF version of the modified Benjamin–Bona–Mahony type equations is introduced. A collocation technique based on the orthonormal Bernoulli polynomials and their derivative matrices (including the ordinary and FF derivative matrices obtained in this study) is adopted for solving such equations. The presented method converts solving this equation to solve a simple system of algebraic equations. Some numerical problems are provided to show the accuracy of the expressed scheme.

In this paper, we propose a numerical method for solving distributed-order fractional partial differential equations (FPDEs). For this method, we first introduce fractional-order generalized Taylor wavelets (FOGTW). An estimation for the error of the approximation is also studied. In addition, by using the regularized beta function we give a formula for determining the Riemann-Liouville fractional integral operator for the FOGTW. Combining this formula with the Gauss-Legendre quadrature, we obtain a numerical method for solving distributed-order FPDEs. Several illustrative examples are given to show the applicability and the accuracy of the proposed method.

In this article, a class of fractional optimal control problems (FOCPs) are solved using a direct method. We present a new operational matrix of the fractional derivative in the sense of Caputo based on the B-spline functions. Then we reduce the solution of fractional optimal control problem to a nonlinear programming (NLP) one, where some existing well-developed algorithms may be applied. Numerical results demonstrate the efficiency of the presented technique.

In this paper, a numerical approach is proposed based on least squares support vector regression for solving Volterra integral equations of the first and second kind. The proposed method is based on using a hybrid of support vector regression with an orthogonal kernel and Galerkin and collocation spectral methods. An optimization problem is derived and transformed to solving a system of algebraic equations. The resulting system is discussed in terms of the structure of the involving matrices and the error propagation. Numerical results are presented to show the sparsity of resulting system as well as the efficiency of the method.

A new technique based on beta functions is applied to compute the exact formula for the Riemann–Liouville fractional integral of the fractional-order generalized Chelyshkov wavelets. An approximation method based on the wavelets is proposed to effectively solve nonlinear fractional differential equations. Illustrative examples show that the proposed method gives solutions with less errors in comparison with the previous methods.

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.

In this article, a newly modified Bessel wavelet method for solving fractional variational problems is considered. The modified operational matrix of integration based on Bessel wavelet functions is proposed for solving the problems. In the process of computing this matrix, we have tried to provide a high-accuracy operational matrix. We also introduce the pseudo-operational matrix of derivative and the dual operational matrix with the coefficient. Also, we investigate the error analysis of the computational method. In the examples section, the behavior of the approximate solutions with respect to various parameters involved in the construction method is tested to illustrate the efficiency and accuracy of the proposed method.

In the present paper, we apply the fractional-order Bessel wavelets (FBWs) for solving optimal control problems with variable-order (VO) fractional dynamical system. The VO fractional derivative op...

In this article, a pair of wavelets for Hermite cubic spline bases are presented. These wavelets are in C1 and supported on [−1,1]. These spline wavelets are then adapted to the interval [0,1] and ...

In this paper, a numerical method using Münzt–Legendre wavelets for distributed-order fractional optimal control problems (DO-FOCPs) is presented. We first give an exact formula of the Riemann–Liouville fractional integral operator of the Münzt–Legendre wavelets by applying regularized beta function. By using the obtained exact formula, we transform the DO-FOCP to a new optimization problem which can be solved by existing techniques. The error analysis of the method is also given. Several illustrated examples are included to show the advantages of our method.

We present an efficient numerical method for solving fractional pantograph differential equations by applying Taylor wavelets. We give an exact formula for the Riemann-Liouville fractional integral of the Taylor wavelets. We then apply the given formula to reduce our problem to the problem of solving a system of algebraic equations. Several examples are included to show the the effectiveness of our numerical method and in comparison with previous methods.

We introduce fractional-order Bessel functions (FBFs) to obtain an approximate solution for various kinds of differential equations. Our main aim is to consider the new functions based on Bessel polynomials to the fractional calculus. To calculate derivatives and integrals, we use Caputo fractional derivatives and Riemann-Liouville fractional integral definitions. Then, operational matrices of fractional-order derivatives and integration for FBFs are derived. Also, we discuss an error estimate between the computed approximations and the exact solution and apply it in some examples. Applications are given to three model problems to demonstrate the effectiveness of the proposed method.

We present a new numerical method for solving fractional delay differential equations. The method is based on Taylor wavelets. We establish an exact formula to determine the Riemann–Liouville fractional integral of the Taylor wavelets. The exact formula is then applied to reduce the problem of solving a fractional delay differential equation to the problem of solving a system of algebraic equations. Several numerical examples are presented to show the applicability and the effectiveness of this method.

In this study, a new numerical method for the solution of the linear and nonlinear distributed fractional differential equations is introduced. The fractional derivative is described in the Caputo sense. The suggested framework is based upon Legendre wavelets approximations. For the first time, an exact formula for the Riemann–Liouville fractional integral operator for the Legendre wavelets is derived. We then use this formula and the properties of Legendre wavelets to reduce the given problem into a system of algebraic equations. Several illustrative examples are included to observe the validity, effectiveness and accuracy of the present numerical method.

In the current study, a new technique for solving various kinds of fractional partial differential equations based upon Genocchi hybrid functions (GHFs) is projected. Genocchi hybrid collocation technique by utilizing delay operational matrix, integer and the fractional operational matrix of the derivative is introduced. We calculate the delay operational matrix without error, this matrix is a reason to achieve the approximate solution with high accuracy. These operational matrices are applied to convert the proposed equations to a set of algebraic equations with unknown GHFs coefficients. Also, we investigate the error analysis based on Sobolev space. At last, several examples are examined with graphs and tables to demonstrates the efficiency and accuracy of the numerical plan.