Operational Matrix Of Fractional Integration Research Articles

Spectral collocation with generalized Laguerre operational matrix for numerical solutions of fractional electrical circuit models

In this paper, we introduce a pioneering numerical technique that combines generalized Laguerre polynomials with an operational matrix of fractional integration to address fractional models in electrical circuits. Specifically focusing on Resistor-Inductor ($RL$), Resistor-Capacitor ($RC$), Resonant (Inductor-Capacitor) ($LC$), and Resistor-Inductor-Capacitor ($RLC$) circuits within the framework of the Caputo derivative, our approach aims to enhance the accuracy of numerical solutions. We meticulously construct an operational matrix of fractional integration tailored to the generalized Laguerre basis vector, facilitating a transformation of the original fractional differential equations into a system of linear algebraic equations. By solving this system, we obtain a highly accurate approximate solution for the electrical circuit model under consideration. To validate the precision of our proposed method, we conduct a thorough comparative analysis, benchmarking our results against alternative numerical techniques reported in the literature and exact solutions where available. The numerical examples presented in our study substantiate the superior accuracy and reliability of our generalized Laguerre-enhanced operational matrix collocation method in effectively solving fractional electrical circuit models.

A new operational matrix method to solve nonlinear fractional differential equations

Abstract This study aims to propose novel Zernike wavelets and a new method based on the operational matrices for solving nonlinear fractional differential equations. First, non-orthogonal Zernike wavelets are introduced using the Zernike polynomials. Then, a new technique based on combining these wavelets with the block pulse functions is presented to derive the operational matrix of fractional integration and to solve nonlinear fractional differential equations. Moreover, an error analysis is conducted by providing required theorems. Besides, the proposed method is employed to solve a nonlinear fractional competition model of breast cancer. Finally, a parametric study is performed to consider the effect of fractional order on the population of healthy, cancer stem, tumour, and immune cells, as well as the excess estrogen.

The Müntz–Legendre Wavelet Collocation Method for Solving Weakly Singular Integro-Differential Equations with Fractional Derivatives

We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional integration operational matrix is constructed for it. The obtained integral equation is reduced to a system of nonlinear algebraic equations using the collocation method and the operational matrix of fractional integration. The presented method’s error bound is investigated, and some numerical simulations demonstrate the efficiency and accuracy of the method. According to the obtained results, the presented method solves this type of equation well and gives significant results.

A NEW NUMERICAL TECHNIQUE FOR SOLVING <i>ψ</i>-FRACTIONAL RICCATI DIFFERENTIAL EQUATIONS

This paper proposes a new numerical technique for solving a specific class of fractional differential equations, which includes the ${\psi}$-Caputo fractional derivative. The class under consideration is nonlinear ${\psi}$-fractional Riccati differential equations (${\psi}$-FRDEs). Our approach relies on the ${\psi}$-Haar wavelet (${\psi}$-HW) operational matrix, which is a novel type of operational matrix of fractional integration. We derive an explicit formula for the ${\psi}$-fractional integral of the HW. This operational matrix has been used successfully to solve nonlinear ${\psi}$-FRDEs.The Quasi-linearization technique is employed to linearize the non-linear $\bm\psi$-FRDEs. This technique reduces the problem to an algebraic equation that can be easily solved. The technique is a useful and straightforward mathematical tool for solving nonlinear ${\psi}$-FRDEs. The computational complexity of the operational matrix technique is minimal. The error analysis of the proposed method is thoroughly investigated. To justify the method's accuracy and efficiency, numerical results are given.

A Biorthogonal Hermite Cubic Spline Galerkin Method for Solving Fractional Riccati Equation

This paper is devoted to the wavelet Galerkin method to solve the Fractional Riccati equation. To this end, biorthogonal Hermite cubic Spline scaling bases and their properties are introduced, and the fractional integral is represented based on these bases as an operational matrix. Firstly, we obtain the Volterra integral equation with a weakly singular kernel corresponding to the desired equation. Then, using the operational matrix of fractional integration and the Galerkin method, the corresponding integral equation is reduced to a system of algebraic equations. Solving this system via Newton’s iterative method gives the unknown solution. The convergence analysis is investigated and shows that the convergence rate is O(2−s). To demonstrate the efficiency and accuracy of the method, some numerical simulations are provided.

Fractional-Order Chelyshkov Collocation Method for Solving Systems of Fractional Differential Equations

This paper presents an efficient method for solving systems of fractional differential equations by using the fractional-order Chelyshkov functions (FCHFs). The fractional derivative and the fractional integral are considered in the Caputo sense and the Riemann–Liouville sense, respectively. The proposed method is based on using the operational matrix of fractional integration for FCHFs, together with the spectral collocation method to transform the system of fractional differential equations into a system of algebraic equations. The convergence of the presented method is demonstrated. The performance of the presented method is tested through various examples of systems of fractional differential equations. A comparison with some existing methods shows that the presented method can be successfully used to solve systems of fractional differential equations with accuracy and efficiency.

Numerical Simulation of Fractional Delay Differential Equations Using the Operational Matrix of Fractional Integration for Fractional-Order Taylor Basis

In this paper, we consider numerical solutions for a general form of fractional delay differential equations (FDDEs) with fractional derivatives defined in the Caputo sense. A fractional integration operational matrix, created using a fractional Taylor basis, is applied to solve these FDDEs. The main characteristic of this approach is, by utilizing the operational matrix of fractional integration, to reduce the given differential equation to a set of algebraic equations with unknown coefficients. This equation system can be solved efficiently using a computer algorithm. A bound on the error for the best approximation and fractional integration are also given. Several examples are given to illustrate the validity and applicability of the technique. The efficiency of the presented method is revealed by comparing results with some existing solutions, the findings of some other approaches from the literature and by plotting absolute error figures.

Numerical solution for a class of fractional optimal control problems using the fractional-order Bernoulli functions

The main purpose of this paper is to provide an efficient method for solving some types of fractional optimal control problems governed by integro-differential and differential equations, and because finding the analytical solutions to these problems is usually difficult, a numerical method is proposed. In this study, the fractional-order Bernoulli functions (F-BFs) are applied as basis functions and a new operational matrix of fractional integration is constructed for these functions. In the first step, the problem is transformed into an equivalent variational problem. Then the F-BFs, the constructed operational matrix, the Gauss quadrature formula, and necessary conditions for optimization are used to convert the problem into a system of algebraic equations. Finally, with the aid of Newton’s iterative method, the system of algebraic equations is solved and the approximate solution of the problem is obtained. Several numerical examples have been analysed for illustrating the efficiency and accuracy of the proposed method, and the results have been compared with the exact solutions and the results of other methods. The results show that the method provides accurate solutions.

Piecewise Chebyshev cardinal functions: Application for constrained fractional optimal control problems

• A new set of basis functions called the piecewise Chebyshev cardinal functions is introduced. • These basis functions possess many useful properties, such as orthogonality, cardinality and spectral accuracy. • The operational matrix of fractional integration is obtained for these functions. • A direct matrix method is developed to investigate a class of constrained fractional optimal control problems. In this paper, a new set of basis functions called the piecewise Chebyshev cardinal functions is generated to investigate a class of constrained fractional optimal control problems. These basis functions possess many useful properties, such as orthogonality, cardinality and spectral accuracy. The fractional integral matrix of these functions is obtained. A direct scheme based on the these basis functions together with their fractional integral matrix is developed for solving the problem under consideration. The established method transforms solving the original problem into solving a constrained minimization problem by approximating the state and control variables in terms of the piecewise Chebyshev cardinal functions. Some numerical examples are given to show the efficiency of the proposed technique.

Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials

In this paper, the finite integration method and the operational matrix of fractional integration are implemented based on the shifted Chebyshev polynomial. They are utilized to devise two numerical procedures for solving the systems of fractional and classical integro-differential equations. The fractional derivatives are described in the Caputo sense. The devised procedure can be successfully applied to solve the stiff system of ODEs. To demonstrate the efficiency, accuracy and numerical convergence order of these procedures, several experimental examples are given. As a consequence, the numerical computations illustrate that our presented procedures achieve significant improvement in terms of accuracy with less computational cost.

Numerical Investigation of Fractional-Order Differential Equations via φ -Haar-Wavelet Method

In this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with the φ -Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ -Haar-wavelet operational matrix of fractional integration. In this paper, we derived an explicit formula for the φ -fractional integral of the Haar-wavelet by utilizing the φ -fractional integral operator. We also extended our method to nonlinear φ -FDEs. The nonlinear problems are first linearized by applying the technique of quasilinearization, and then, the proposed method is applied to get a numerical solution of the linearized problems. The current technique is an effective and simple mathematical tool for solving nonlinear φ -FDEs. In the context of error analysis, an exact upper bound of the error for the suggested technique is given, which shows convergence of the proposed method. Finally, some numerical examples that demonstrate the efficiency of our technique are discussed.

Discrete Chebyshev Polynomials for Solving Fractional Variational Problems

In ‎the current study, a‎ general formulation of the discrete Chebyshev polynomials is given. ‎The operational matrix of fractional integration for these discrete polynomials is also derived. ‎Then,‎ a numerical scheme based on the discrete Chebyshev polynomials and their operational matrix has been developed to solve fractional variational problems‎. In this method, the need for using Lagrange multiplier during the solution procedure is eliminated.‎ The performance of the proposed scheme is validated through some illustrative examples. ‎Moreover, ‎the obtained numerical results ‎‎‎‎were compared to the previously acquired results by the classical Chebyshev polynomials. Finally, a comparison for the required CPU time is presented, which indicates more efficiency and less complexity of the proposed method.

Alpert wavelet system for solving fractional nonlinear Fredholm integro-differential equations

In this paper, we first construct Alpert wavelet system and propose a computational method for solving a fractional nonlinear Fredholm integro-differential equation. Then we create an operational matrix of fractional integration and use it to simplify the equation to a system of algebraic equations. By using Newtons iterative method, this system is solved and the solution of the fractional nonlinear Fredholm integro-differential equations is achieved. Thresholding parameter is used to increase the sparsity of matrix coefficients and the speed of computations. Finally, the method is demonstrated by examples, and then compared results with CAS wavelet method show that our proposed method is more effective and accurate.

Solving partial fractional differential equations by using the Laguerre wavelet-Adomian method

By using a nonlinear method, we try to solve partial fractional differential equations. In this way, we construct the Laguerre wavelets operational matrix of fractional integration. The method is proposed by utilizing Laguerre wavelets in conjunction with the Adomian decomposition method. We present the procedure of implementation and convergence analysis for the method. This method is tested on fractional Fisher’s equation and the singular fractional Emden–Fowler equation. We compare the results produced by the present method with some well-known results.

An expeditious wavelet-based numerical scheme for solving fractional differential equations

In this article, a numerical scheme based on Euler wavelets for solving fractional differential equations is proposed. An Euler wavelet operational matrix of fractional integration is derived and employed to reduce fractional differential equations into simultaneous algebraic equations. Some examples are given to illustrate the applicability and the high accuracy of the proposed numerical scheme.

Wavelet-Picard iterative method for solving singular fractional nonlinear partial differential equations with initial and boundary conditions

The present study applies the Picard iterative method to nonlinear singular partial fractional differential equations. The Haar and second-kind Chebyshev wavelets operational matrix of fractional integration will be used to solve problems combining linearization technique with the Picard method. The singular problem will be converted to an algebraic system of equations, which can be easily solved. Numerical examples are provided to illustrate the efficiency and accuracy of the technique.

A new approach for solving one-dimensional fractional boundary value problems via Haar wavelet collocation method

In this paper, we propose the numerical method for positive solutions to the m-point boundary value problems of fractional differential equations with p-Laplacian operator and prove the convergence for our scheme. The operational matrix of fractional integration, based on the accurate calculation of fractional integrals of Haar wavelet functions, is combined with the collocation method and the fixed point iterative method to introduce the function approximation and obtain the approximate solution to the given problem. Also, we present some numerical examples to illustrate our main results.

Numerical implementation of nonlinear system of fractional Volterra integral–differential equations by Legendre wavelet method and error estimation

AbstractIn the current study, a numerical scheme for solving the nonlinear system of fractional Volterra integro‐differential equations via Legendre wavelet is proposed. The Legendre wavelet operational matrix of fractional integration is derived and utilized to alter the main system to a system of algebraic equations. In addition, the error estimate of the original system is investigated in detail. Lastly, several numerical examples are set forth to test the proposed method and the obtained numerical results are compared with those from the CAS wavelet method.

Numerical Solutions of Multi-order Fractional Antiperiodic Boundary Value Problems

In this paper, we propose the numerical scheme for solving nonlinear higher multi-order Caputo fractional differential equations with integral and antiperiodic boundary conditions, establish the existence of an approximate solution to our problem, and prove the convergence of our numerical method. The operational matrix of fractional integration based on hat functions is used to solve the given problem. Also, we present a numerical example to illustrate our main results.

Solution of a specific class of nonlinear fractional optimal control problems including multiple delays

SummaryThis research provides a new framework based on a hybrid of block‐pulse functions and Legendre polynomials for the numerical examination of a special class of scalar nonlinear fractional optimal control problems involving delay. The concepts of the fractional derivative and the fractional integral are employed in the Caputo sense and the Riemann‐Liouville sense, respectively. In accordance with the notion of the Riemann‐Liouville integral, we derive a new integral operator related to the proposed basis called the operational matrix of fractional integration. By employing two significant operators, namely, the delay operator and the integral operator connected to the hybrid basis, the system dynamics of the primal optimal control problem converts to another system involving algebraic equations. Consequently, the optimal control problem under study is reduced to a static optimization one that is solved by existing well‐established optimization procedures. Some new theoretical results regarding the new basis are obtained. Various kinds of fractional optimal control problems containing delay are examined to measure the accuracy of the new method. The simulation results justify the merits and superiority of the devised procedure over the existing optimization methods in the literature.