scholarly journals THE NUMERICAL TREATMENT OF NONLINEAR FRACTAL–FRACTIONAL 2D EMDEN–FOWLER EQUATION UTILIZING 2D CHELYSHKOV POLYNOMIALS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040042
Author(s):  
M. HOSSEININIA ◽  
M. H. HEYDARI ◽  
Z. AVAZZADEH
Keyword(s):  
Approximate Solution ◽  
Finite Difference ◽  
Test Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Time Step ◽  
Discrete Method ◽  
Difference Formula ◽  
Fowler Equation ◽  
Chelyshkov Polynomials

This paper develops an effective semi-discrete method based on the 2D Chelyshkov polynomials (CPs) to provide an approximate solution of the fractal–fractional nonlinear Emden–Fowler equation. In this model, the fractal–fractional derivative in the concept of Atangana–Riemann–Liouville is considered. The proposed algorithm first discretizes the fractal–fractional differentiation by using the finite difference formula in the time direction. Then, it simplifies the original equation to the recurrent equations by expanding the unknown solution in terms of the 2D CPs and using the [Formula: see text]-weighted finite difference scheme. The differentiation operational matrices and the collocation method play an important role to obtaining a linear system of algebraic equations. Last, solving the obtained system provides an approximate solution in each time step. The validity of the formulated method is investigated through a sufficient number of test problems.

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
Author(s):  
Abdul Ghafoor ◽  
Sirajul Haq ◽  
Manzoor Hussain ◽  
Poom Kumam ◽  
Muhammad Asif Jan
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Quadrature Rule ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Haar Wavelets ◽  
Test Problems ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

scholarly journals Finite Difference and Sinc-Collocation Approximations to a Class of Fractional Diffusion-Wave Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Zhi Mao ◽  
Aiguo Xiao ◽  
Zuguo Yu ◽  
Long Shi
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Fractional Diffusion ◽  
Stability And Convergence ◽  
Algebraic Equations ◽  
Time Step ◽  
Diffusion Wave ◽  
Linear Algebraic Equations ◽  
Sinc Functions ◽  
Exponential Rate Of Convergence

We propose an efficient numerical method for a class of fractional diffusion-wave equations with the Caputo fractional derivative of orderα. This approach is based on the finite difference in time and the global sinc collocation in space. By utilizing the collocation technique and some properties of the sinc functions, the problem is reduced to the solution of a system of linear algebraic equations at each time step. Stability and convergence of the proposed method are rigorously analyzed. The numerical solution is of3-αorder accuracy in time and exponential rate of convergence in space. Numerical experiments demonstrate the validity of the obtained method and support the obtained theoretical results.

scholarly journals A Novel Method for Solution of Fractional Order Two-Dimensional Nonlocal Heat Conduction Phenomena

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Hammad Khalil ◽  
Ishak Hashim ◽  
Waqar Ahmad Khan ◽  
Abuzar Ghaffari
Keyword(s):  
Heat Conduction ◽  
Fractional Order ◽  
Operational Matrix ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Test Problems ◽  
Operational Matrices ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Operational Matrix Method

In this paper, we have extended the operational matrix method for approximating the solution of the fractional-order two-dimensional elliptic partial differential equations (FPDEs) under nonlocal boundary conditions. We use a general Legendre polynomials basis and construct some new operational matrices of fractional order operations. These matrices are used to convert a sample nonlocal heat conduction phenomenon of fractional order to a structure of easily solvable algebraic equations. The solution of the algebraic structure is then used to approximate a solution of the heat conduction phenomena. The proposed method is applied to some test problems. The obtained results are compared with the available data in the literature and are found in good agreement.Dedicated to my father Mr. Sher Mumtaz, (1955-2021), who gave me the basic knowledege of mathematics.

A computational method for time fractional partial integro-differential equations

2020 ◽  
Vol 26 (2) ◽  
pp. 315-323
Author(s):  
M. Mojahedfar ◽  
Abolfazl Tari ◽  
S. Shahmorad
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Error Estimation ◽  
Initial Conditions ◽  
Computational Method ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Numerical Examples

AbstractIn this paper, a class of time fractional partial integro-differential equations (FPIDEs) with initial conditions is studied. Some operational matrices are used to reduce a FPIDE problem to a system of algebraic equations with special properties. The resulted system is solved to give an approximate solution to the problem. Error estimation is also discussed for the approximate solution. Finally, some numerical examples are given to show the accuracy of the proposed method.

A numerical approach for solving fractional optimal control problems with mittag-leffler kernel

2021 ◽  
pp. 107754632110169
Author(s):  
Hossein Jafari ◽  
Roghayeh M Ganji ◽  
Khosro Sayevand ◽  
Dumitru Baleanu
Keyword(s):  
Optimal Control ◽  
Approximate Solution ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Numerical Approach ◽  
Control Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

In this work, we present a numerical approach based on the shifted Legendre polynomials for solving a class of fractional optimal control problems. The derivative is described in the Atangana–Baleanu derivative sense. To solve the problem, operational matrices of AB-fractional integration and multiplication, together with the Lagrange multiplier method for the constrained extremum, are considered. The method reduces the main problem to a system of nonlinear algebraic equations. In this framework by solving the obtained system, the approximate solution is calculated. An error estimate of the numerical solution is also proved for the approximate solution obtained by the proposed method. Finally, some illustrative examples are presented to demonstrate the accuracy and validity of the proposed scheme.

Direct numerical method for isoperimetric fractional variational problems based on operational matrix

2017 ◽  
Vol 24 (14) ◽  
pp. 3063-3076 ◽  
Author(s):  
Samer S Ezz–Eldien ◽  
Ali H Bhrawy ◽  
Ahmed A El–Kalaawy
Keyword(s):  
Direct Method ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Variational Problems ◽  
Test Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Direct Numerical Method ◽  
Fractional Variational Problems ◽  
A Direct Method

In this paper, we applied a direct method for a solution of isoperimetric fractional variational problems. We use shifted Legendre orthonormal polynomials as basis function of operational matrices of fractional differentiation and fractional integration in combination with the Lagrange multipliers technique for converting such isoperimetric fractional variational problems into solving a system of algebraic equations. Also, we show the convergence analysis of the presented technique and introduce some test problems with comparisons between our numerical results with those introduced using different methods.

scholarly journals Toward the Approximate Solution for Fractional Order Nonlinear Mixed Derivative and Nonlocal Boundary Value Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Hammad Khalil ◽  
Mohammed Al-Smadi ◽  
Khaled Moaddy ◽  
Rahmat Ali Khan ◽  
Ishak Hashim
Keyword(s):  
Approximate Solution ◽  
Boundary Conditions ◽  
Operational Matrix ◽  
Nonlocal Boundary ◽  
Coupled System ◽  
Nonlocal Boundary Conditions ◽  
Mixed Derivative ◽  
Test Problems ◽  
Operational Matrices ◽  
Nonlinear Coupled System

The paper is devoted to the study of operational matrix method for approximating solution for nonlinear coupled system fractional differential equations. The main aim of this paper is to approximate solution for the problem under two different types of boundary conditions,m^-point nonlocal boundary conditions and mixed derivative boundary conditions. We develop some new operational matrices. These matrices are used along with some previously derived results to convert the problem under consideration into a system of easily solvable matrix equations. The convergence of the developed scheme is studied analytically and is conformed by solving some test problems.

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

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.

scholarly journals Sinc Collocation Linked with Finite Differences for Korteweg-de Vries Fractional Equation

2020 ◽  
Vol 10 (1) ◽  
pp. 512
Author(s):  
Kamel Al-Khaled
Keyword(s):  
Approximate Solution ◽  
Finite Difference ◽  
Numerical Method ◽  
Finite Differences ◽  
Fractional Derivatives ◽  
Numerical Results ◽  
Algebraic Equations ◽  
De Vries ◽  
Fractional Equation ◽  
Finite Difference Approach

<span>A novel numerical method is proposed for Korteweg-de Vries<br />Fractional Equation. The fractional derivatives are described based<br />on the Caputo sense. We construct the solution using different<br />approach, that is based on using collocation techniques. The method combining a finite difference approach in the time-fractional<br />direction, and the Sinc-Collocation in the space direction, where<br />the derivatives are replaced by the necessary matrices, and a system of algebraic equations is obtained to approximate solution of the problem.  Some numerical results are shown to demonstrate the efficiency of the newly proposed method. The numerical results are shown to demonstrate the efficiency of the newly proposed method. Easy and economical implementation is the strength of this method.</span>

A new finite-difference solution-adaptive method

1992 ◽  
Vol 341 (1662) ◽  
pp. 373-410 ◽  
Keyword(s):  
Finite Difference ◽  
Matrix Equation ◽  
Rapid Change ◽  
Penalty Functions ◽  
Hyperbolic Partial Differential Equations ◽  
Test Problems ◽  
Rarefaction Waves ◽  
Time Step ◽  
Error Measure ◽  
Flow Variables

A new moving finite difference (MFD) method has been developed for solving hyperbolic partial differential equations and is compared with the moving finite element (MFD) method of K. Miller and R. N. Miller. These methods involve the adaptive movement of nodes so as to reduce the number of nodes needed to solve a problem; they are applicable to the solution of non-stationary flow problems that contain moving regions of rapid change in the flow variables, surrounded by regions of relatively smooth variation. Both methods solve simultaneously for the flow variables and the node locations at each time-step, and they move the nodes so as to minimize an ‘error’ measure that contains a function of the time derivatives of the solution. This error measure is manipulated to obtain a matrix equation for node velocities. Both methods make use of penalty functions to prevent node crossing. The penalty functions result in extra terms in the matrix equation that promote node repulsion by becoming large when node separation becomes small. Extensive work applying the MFE and MFD methods to one-dimensional gasdynamic problems has been conducted to evaluate their performance. The test problems include Burgers’ equation, ideal viscid planar flow within a shock-tube, propagation of shock and rarefaction waves through area changes in ducts, and viscous transition through a contact surface and a shock.

Sign in / Sign up

Export Citation Format

Share Document