Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel

AbstractThe shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations.

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Numerical Approach Based on Two-Dimensional Fractional-Order Legendre Functions for Solving Fractional Differential Equations

Discrete Dynamics in Nature and Society ◽

10.1155/2017/8630895 ◽

2017 ◽

Vol 2017 ◽

pp. 1-12

Author(s):

Qingxue Huang ◽

Fuqiang Zhao ◽

Jiaquan Xie ◽

Lifeng Ma ◽

Jianmei Wang ◽

...

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Fractional Differential Equations ◽

Operational Matrix ◽

Numerical Approach ◽

Principal Characteristic ◽

Legendre Functions ◽

Algebraic Equations ◽

Linear Algebraic Equations ◽

Fractional Differential

In this paper, a robust, effective, and accurate numerical approach is proposed to obtain the numerical solution of fractional differential equations. The principal characteristic of the approach is the new orthogonal functions based on shifted Legendre polynomials to the fractional calculus. Also the fractional differential operational matrix is driven. Then the matrix with the Tau method is utilized to transform this problem into a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via some examples. It is shown that the FLF yields better results. Finally, error analysis shows that the algorithm is convergent.

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Shifted Fractional-Order Jacobi Collocation Method for Solving Variable-Order Fractional Integro-Differential Equation with Weakly Singular Kernel

Fractal and Fractional ◽

10.3390/fractalfract6010019 ◽

2021 ◽

Vol 6 (1) ◽

pp. 19

Author(s):

Mohamed A. Abdelkawy ◽

Ahmed Z. M. Amin ◽

António M. Lopes ◽

Ishak Hashim ◽

Mohammed M. Babatin

Keyword(s):

Collocation Method ◽

Fractional Order ◽

Jacobi Polynomials ◽

Initial Conditions ◽

Approximate Solutions ◽

Singular Kernel ◽

Variable Order ◽

Algebraic Equations ◽

Weakly Singular Kernel ◽

Weakly Singular

We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fractional integral and derivative and fractional-order shifted Jacobi polynomials, the approximate solutions of VO-FIDE-WSK are derived by solving systems of algebraic equations. The superior accuracy of the method is illustrated through several numerical examples.

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A fast numerical method for fractional partial differential equations

Advances in Difference Equations ◽

10.1186/s13662-019-2390-z ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 2

Author(s):

S. Mockary ◽

E. Babolian ◽

A. R. Vahidi

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Chebyshev Polynomials ◽

Fractional Integration ◽

Operational Matrix ◽

Operational Matrices ◽

Algebraic Equations ◽

Fractional Partial Differential Equations ◽

Partial Differential ◽

Linear Algebraic Equations

Abstract In this paper, we use operational matrices of Chebyshev polynomials to solve fractional partial differential equations (FPDEs). We approximate the second partial derivative of the solution of linear FPDEs by operational matrices of shifted Chebyshev polynomials. We apply the operational matrix of integration and fractional integration to obtain approximations of (fractional) partial derivatives of the solution and the approximation of the solution. Then we substitute the operational matrix approximations in the FPDEs to obtain a system of linear algebraic equations. Finally, solving this system, we obtain the approximate solution. Numerical experiments show an exponential rate of convergence and hence the efficiency and effectiveness of the method.

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NUMERICAL SOLUTION OF VARIABLE-ORDER TIME FRACTIONAL WEAKLY SINGULAR PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS WITH ERROR ESTIMATION

Mathematical Modelling and Analysis ◽

10.3846/mma.2020.11692 ◽

2020 ◽

Vol 25 (4) ◽

pp. 680-701

Author(s):

Haniye Dehestani ◽

Yadollah Ordokhani ◽

Mohsen Razzaghi

Keyword(s):

Approximate Solution ◽

Differential Equations ◽

Collocation Method ◽

Error Estimation ◽

Operational Matrices ◽

Variable Order ◽

Algebraic Equations ◽

Laguerre Functions ◽

Weakly Singular Kernel ◽

Weakly Singular

In this paper, we apply Legendre-Laguerre functions (LLFs) and collocation method to obtain the approximate solution of variable-order time-fractional partial integro-diﬀerential equations (VO-TF-PIDEs) with the weakly singular kernel. For this purpose, we derive the pseudo-operational matrices with the use of the transformation matrix. The collocation method and pseudo-operational matrices transfer the problem to a system of algebraic equations. Also, the error analysis of the proposed method is given. We consider several examples to illustrate the proposed method is accurate.

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Alternative Legendre Polynomials Method for Nonlinear Fractional Integro-Differential Equations with Weakly Singular Kernel

Journal of Mathematics ◽

10.1155/2021/9968237 ◽

2021 ◽

Vol 2021 ◽

pp. 1-13

Author(s):

Guodong Shi ◽

Yanlei Gong ◽

Mingxu Yi

Keyword(s):

Differential Equations ◽

Legendre Polynomials ◽

Numerical Scheme ◽

Operational Matrix ◽

Operational Matrices ◽

Tabular Form ◽

Algebraic Equations ◽

Weakly Singular Kernel ◽

Weakly Singular ◽

Singular Kernels

In this paper, we present a numerical scheme for finding numerical solution of a class of weakly singular nonlinear fractional integro-differential equations. This method exploits the alternative Legendre polynomials. An operational matrix, based on the alternative Legendre polynomials, is derived to be approximated the singular kernels of this class of the equations. The operational matrices of integration and product together with the derived operational matrix are utilized to transform nonlinear fractional integro-differential equations to the nonlinear system of algebraic equations. Furthermore, the proposed method has also been analyzed for convergence, particularly in context of error analysis. Moreover, results of essential numerical applications have also been documented in a graphical as well as tabular form to elaborate the effectiveness and accuracy of the proposed method.

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A New Algorithm Based on Bernstein Polynomials Multiwavelets for the Solution of Differential Equations Governing AC Circuits

Trends in Sciences ◽

10.48048/tis.2021.33 ◽

2021 ◽

Vol 18 (21) ◽

pp. 33

Author(s):

Shweta Pandey ◽

Sandeep Dixit ◽

Sag R Verma

Keyword(s):

Analytical Solution ◽

Differential Equations ◽

Numerical Solution ◽

Bernstein Polynomial ◽

Error Function ◽

Bernstein Polynomials ◽

Operational Matrix ◽

Algebraic Equations ◽

Linear Algebraic Equations ◽

Operational Matrix Of Integration

We extend the application of multiwavelet-based Bernstein polynomials for the numerical solution of differential equations governing AC circuits (LCR and LC). The operational matrix of integration is obtained from the orthonormal Bernstein polynomial wavelet bases, which diminishes differential equations into the system of linear algebraic equations for easy computation. It appeared that fewer wavelet bases gave better results. The convergence and exactness were examined by comparing the calculated numerical solution and the known analytical solution. The error function was calculated and illustrated graphically for the reliability and accuracy of the proposed method. The proposed method examined several physical issues that lead to differential equations. HIGHLIGHTS Differential equations governing AC circuits are converted into the system of linear algebraic equations using Bernstein polynomial multiwavelets operational matrix of integration for easy computation The convergence and exactness were examined by comparing the calculated numerical solution and the known analytical solution The error function is calculated and shown graphically GRAPHICAL ABSTRACT

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Least-Squares Chebyshev Method for Solving Fractional Integro-diffrential Equations

Engineering and Technology Journal ◽

10.47191/etj/v6i7.02 ◽

2021 ◽

Vol 06 (07) ◽

Author(s):

Oyedepo Taiye ◽

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Least Squares ◽

Chebyshev Polynomials ◽

Exact Results ◽

Algebraic Equations ◽

Graphical Solution ◽

Numerical Examples ◽

Chebyshev Method ◽

Linear Algebraic Equations

The main purpose of this study gears towards finding numerical solution to fractional integro-differential equations. The technique involves the application of caputo properties and Chebyshev polynomials to reduce the problem to system of linear algebraic equations and then solved using MAPLE 18. To demonstrate the accuracy and applicability of the presented method some numerical examples are given. Numerical results show that the method is easy to implement and compares favorably with the exact results. The graphical solution of the method is displayed.

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Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

Fractal and Fractional ◽

10.3390/fractalfract5030070 ◽

2021 ◽

Vol 5 (3) ◽

pp. 70

Author(s):

Esmail Bargamadi ◽

Leila Torkzadeh ◽

Kazem Nouri ◽

Amin Jajarmi

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Algebraic System ◽

Operational Matrix ◽

Approximate Solutions ◽

Chebyshev Wavelets ◽

Weakly Singular Kernels ◽

Weakly Singular ◽

Singular Kernels ◽

Chebyshev Wavelet

In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example.

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Solving Some Differential Equations Arising in Electric Engineering Using Hermite Polynomials

Journal of Scientific Research ◽

10.3329/jsr.v12i4.45686 ◽

2020 ◽

Vol 12 (4) ◽

pp. 517-523

Author(s):

G. Singh ◽

I. Singh

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Collocation Method ◽

Hermite Polynomials ◽

Electric Circuit ◽

Algebraic Equations ◽

Numerical Examples ◽

Electric Engineering ◽

Collocation Points ◽

Linear Algebraic Equations

In this paper, a collocation method based on Hermite polynomials is presented for the numerical solution of the electric circuit equations arising in many branches of sciences and engineering. By using collocation points and Hermite polynomials, electric circuit equations are transformed into a system of linear algebraic equations with unknown Hermite coefficients. These unknown Hermite coefficients have been computed by solving such algebraic equations. To illustrate the accuracy of the proposed method some numerical examples are presented.

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Numerical Solution of Nonlinear Fractional Volterra Integro-Differential Equations via Bernoulli Polynomials

Abstract and Applied Analysis ◽

10.1155/2014/162896 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 9

Author(s):

Emran Tohidi ◽

M. M. Ezadkhah ◽

S. Shateyi

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Linear Combination ◽

Fractional Order ◽

Bernoulli Polynomials ◽

Approximate Solutions ◽

Computational Approach ◽

Numerical Results ◽

Algebraic Equations

This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main problems to the solution of algebraic equations systems by expanding the required approximate solutions as the linear combination of the Bernoulli polynomials. Several examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method.

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