scholarly journals NUMERICAL SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATION USING RANDOMLY GENERATED FINITE GRIDS AND TWO-DIMENSIONAL FRACTIONAL-ORDER LEGENDRE FUNCTION

2021 ◽  
Author(s):  
Sanaullah Mastoi
Keyword(s):  
Finite Difference ◽  
Numerical Solution ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Polar Coordinates ◽  
Legendre Functions ◽  
Two Dimensional ◽  
Cartesian Coordinates ◽  
Uniform Grids

There are various methods to solve the physical life problem involving engineering, scientific and biological systems. It is found that numerical methods are approximate solutions. In this way, randomly generated finite difference grids achieve an approximation with fewer iterations. The idea of randomly generated grids in cartesian coordinates and polar form are compared with the exact, iterative method, uniform grids, and approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions. The most ideal and benchmarking method is the finite difference method over randomly generated grids on Cartesian coordinates, polar coordinates used for numerical solutions. This concept motivates the investigation of the effects of the randomly generated meshes. The two-dimensional equation is solved over randomly generated meshes to test randomly generated grids and the implementation. The feasibility of the numerical solution is analyzed by comparing simulation profiles.

A new finite difference method for computing approximate solutions of boundary value problems including transition conditions

2021 ◽  
Vol 102 (2) ◽  
pp. 54-61
Author(s):  
S. Çavuşoğlu ◽  
◽  
O.Sh. Mukhtarov ◽  
◽  
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Difference Method ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Approximate Solutions

This article is aimed at computing numerical solutions of new type of boundary value problems (BVPs) for two-linked ordinary differential equations. The problem studied here differs from the classical BVPs such that it contains additional conditions at the point of interaction, so-called transition conditions. Naturally, such type of problems is much more complicated to solve than classical problems. It is not clear how to apply the classical numerical methods to such type of boundary value transition problems (BVTPs). Based on the finite difference method (FDM) we have developed a new numerical algorithm for computing numerical solution of BVTPs for two-linked ordinary differential equations. To demonstrate the reliability and efficiency of the presented algorithm we obtained numerical solution of one BVTP and the results are compared with the corresponding exact solution. The maximum absolute errors (MAEs) are presented in a table.

scholarly journals Numerical Solution of the Fractional Partial Differential Equations by the Two-Dimensional Fractional-Order Legendre Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Fukang Yin ◽  
Junqiang Song ◽  
Yongwen Wu ◽  
Lilun Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Approximate Solutions ◽  
Operational Matrices ◽  
Legendre Functions ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs). The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions (2D-FLFs). The operational matrices of integration and derivative for 2D-FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.

scholarly journals A fractional order epidemic model for the simulation of outbreaks of Ebola

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Weiqiu Pan ◽  
Tianzeng Li ◽  
Safdar Ali
Keyword(s):  
Numerical Solution ◽  
Root Mean Square ◽  
Fractional Order ◽  
Relative Error ◽  
Numerical Solutions ◽  
Real Data ◽  
Mean Square ◽  
Order Model ◽  
The Real ◽  
Fractional Orders

AbstractThe Ebola outbreak in 2014 caused many infections and deaths. Some literature works have proposed some models to study Ebola virus, such as SIR, SIS, SEIR, etc. It is proved that the fractional order model can describe epidemic dynamics better than the integer order model. In this paper, we propose a fractional order Ebola system and analyze the nonnegative solution, the basic reproduction number $R_{0}$ R 0 , and the stabilities of equilibrium points for the system firstly. In many studies, the numerical solutions of some models cannot fit very well with the real data. Thus, to show the dynamics of the Ebola epidemic, the Gorenflo–Mainardi–Moretti–Paradisi scheme (GMMP) is taken to get the numerical solution of the SEIR fractional order Ebola system and the modified grid approximation method (MGAM) is used to acquire the parameters of the SEIR fractional order Ebola system. We consider that the GMMP method may lead to absurd numerical solutions, so its stability and convergence are given. Then, the new fractional orders, parameters, and the root-mean-square relative error $g(U^{*})=0.4146$ g ( U ∗ ) = 0.4146 are obtained. With the new fractional orders and parameters, the numerical solution of the SEIR fractional order Ebola system is closer to the real data than those models in other literature works. Meanwhile, we find that most of the fractional order Ebola systems have the same order. Hence, the fractional order Ebola system with different orders using the Caputo derivatives is also studied. We also adopt the MGAM algorithm to obtain the new orders, parameters, and the root-mean-square relative error which is $g(U^{*})=0.2744$ g ( U ∗ ) = 0.2744 . With the new parameters and orders, the fractional order Ebola systems with different orders fit very well with the real data.

scholarly journals Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
Yongjin Li ◽  
Kamal Shah
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Coupled Systems ◽  
Test Problems ◽  
Operational Matrices ◽  
Partial Differential ◽  
Established Technique

We develop a numerical method by using operational matrices of fractional order integrations and differentiations to obtain approximate solutions to a class of coupled systems of fractional order partial differential equations (FPDEs). We use shifted Legendre polynomials in two variables. With the help of the aforesaid matrices, we convert the system under consideration to a system of easily solvable algebraic equation of Sylvester type. During this process, we need no discretization of the data. We also provide error analysis and some test problems to demonstrate the established technique.

Coherent states for Smorodinsky-Winternitz potentials

Open Physics ◽  
2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Nuri Ünal
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Wave Packets ◽  
Harmonic Oscillators ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Cartesian Coordinates ◽  
Physical Time ◽  
First Case

AbstractIn this study, we construct the coherent states for a particle in the Smorodinsky-Winternitz potentials, which are the generalizations of the two-dimensional harmonic oscillator problem. In the first case, we find the non-spreading wave packets by transforming the system into four oscillators in Cartesian, and also polar, coordinates. In the second case, the coherent states are constructed in Cartesian coordinates by transforming the system into three non-isotropic harmonic oscillators. All of these states evolve in physical-time. We also show that in parametric-time, the second case can be transformed to the first one with vanishing eigenvalues.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
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.

On the Use of Spreadsheets in Heat Conduction Analysis

2000 ◽  
Vol 28 (2) ◽  
pp. 113-139 ◽  
Author(s):  
Esmail M. A. Mokheimer ◽  
Mohamed A. Antar
Keyword(s):  
Heat Conduction ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Single Layer ◽  
Transient Heat Conduction ◽  
New Technique ◽  
Two Dimensional ◽  
A New Technique ◽  
User Friendly

Detailed methodology and different techniques for simply utilizing the widely available and user friendly spreadsheet programs in heat conduction analysis are presented. Evaluation of analytical and numerical solution of heat conduction problems via spreadsheets is investigated. Detailed techniques of obtaining spreadsheet numerical solutions for one- and two-dimensional steady and transient heat conduction problems are introduced. A new technique of marching the transient numerical solution with time, in a single layer spreadsheet, for one- and two-dimensional heat conduction is explained. Creating macros that automate the spreadsheet processes, particularly calculations, is detailed. Utilization of the powerful graphical facility that is built in the spreadsheets to graphically represent the obtained solutions is outlined.

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