scholarly journals The Numerical Method for Solving Differential Equations of Lane-Emden Type by Padé Approximation

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Muhammed Yiğider ◽  
Khatereh Tabatabaei ◽  
Ercan Çelik
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Method ◽  
Padé Approximation ◽  
Pade Approximation ◽  
One Dimensional ◽  
Series Form ◽  
Differential Transformation

Numerical solution differential equation of Lane-Emden type is considered by Padé approximation. We apply these method to two examples. First differential equation of Lane-Emden type has been converted to power series by one-dimensional differential transformation, then the numerical solution of equation was put into Padé series form. Thus, we have obtained numerical solution differential equation of Lane-Emden type.

scholarly journals Numerical Solution Of Partial Integro-Differential Equation Using Legendre Multi Wavelets

2020 ◽  
Vol 55 (2) ◽  
Author(s):  
Amina Kassim Hussain
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Method ◽  
Physical Phenomenon ◽  
Diffusion Method ◽  
Benchmark Problems ◽  
Science And Engineering ◽  
Partial Differential ◽  
Integro Differential Equation

It is very important to state the initial assumptions for the description of physical phenomenon in the case of partial integro-differential equation. The parabolic equations and boundary conditions can be used to define the time-dependent diffusion process. The integro-differential equation is the combination of integration and derivatives. It is part of the technology, which includes science and engineering. Various models that cover the area of science and engineering are available. Moreover, variable techniques are accessible to solve the integro-differential equations. Numerical method is an important way to solve the challenges in the field of science and industry. To improve efficiency, the companies were working on computer simulation. For reliability, flexibility, and inexpensiveness, the numerical methods are preferred. Linear Legendre multi wavelets form a collocated method based on the numerical solution of one-dimensional parabolic partial integro-differential equation of diffusion type. In this study, we aim to study the diffusion method of numerical solution for the integro-partial differential equation. The diffusion method, its basic concept, and other methods used to solve integro-partial differential equations are also studied in detail. The proposed numerical method is useful to different benchmark problems and provides efficient, accurate, and robust results.

Numerical Solution to Volterra-Type Integro-Differential Equations of the Second Kinds by Legendre Collocation Method

2014 ◽  
Vol 687-691 ◽  
pp. 1522-1527
Author(s):  
Ting Jing Zhao
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Method ◽  
Collocation Method ◽  
High Accuracy ◽  
Time Step ◽  
Volterra Type ◽  
Legendre Collocation Method

The purpose of this paper is to propose an efficient numerical method for solving Volterra-type integro-differential equation of the second kinds. This method based on Legendre-Gauss-Radau collocation, which is easy to be implemented especially for nonlinear and possesses high accuracy. Also, the method can be done by proceeding in time step by step. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique, and the results have been compared with the exact solution.

Numerical Solution of Diffusion and Reaction–Diffusion Partial Integro-Differential Equations

2018 ◽  
Vol 15 (06) ◽  
pp. 1850047 ◽  
Author(s):  
Imran Aziz ◽  
Imran Khan
Keyword(s):  
Population Dynamics ◽  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Method ◽  
Reaction Diffusion ◽  
Nonlinear Problems ◽  
Haar Wavelet ◽  
Benchmark Problems ◽  
Two Dimensional ◽  
One Dimensional

In this paper, a collocation method based on Haar wavelet is developed for numerical solution of diffusion and reaction–diffusion partial integro-differential equations. The equations are parabolic partial integro-differential equations and we consider both one-dimensional and two-dimensional cases. Such equations have applications in several practical problems including population dynamics. An important advantage of the proposed method is that it can be applied to both linear as well as nonlinear problems with slide modification. The proposed numerical method is validated by applying it to various benchmark problems from the existing literature. The numerical results confirm the accuracy, efficiency and robustness of the proposed method.

scholarly journals Fractional-Order Logistic Differential Equation with Mittag–Leffler-Type Kernel

2021 ◽  
Vol 5 (4) ◽  
pp. 273
Author(s):  
Iván Area ◽  
Juan J. Nieto
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Differential Equations ◽  
Formal Power Series ◽  
Fractional Order ◽  
Numerical Approximations ◽  
Formal Power

In this paper, we consider the Prabhakar fractional logistic differential equation. By using appropriate limit relations, we recover some other logistic differential equations, giving representations of each solution in terms of a formal power series. Some numerical approximations are implemented by using truncated series.

A Parallel Block Predictor-Corrector Method by Python-Based Distributed Computing

2012 ◽  
Vol 263-266 ◽  
pp. 1315-1318
Author(s):  
Kun Ming Yu ◽  
Ming Gong Lee
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Method ◽  
Initial Value Problem ◽  
Message Passing ◽  
Message Passing Interface ◽  
Parallel Structure ◽  
Initial Value ◽  
Speed Up ◽  
Predictor Corrector

This paper is to discuss how Python can be used in designing a cluster parallel computation environment in numerical solution of some block predictor-corrector method for ordinary differential equations. In the parallel process, MPI-2(message passing interface) is used as a standard of MPICH2 to communicate between CPUs. The operation of data receiving and sending are operated and controlled by mpi4py which is based on Python. Implementation of a block predictor-corrector numerical method with one and two CPUs respectively is used to test the performance of some initial value problem. Minor speed up is obtained due to small size problems and few CPUs used in the scheme, though the establishment of this scheme by Python is valuable due to very few research has been carried in this kind of parallel structure under Python.

scholarly journals Solutions of first level of meromorphic differential equations

1982 ◽  
Vol 25 (2) ◽  
pp. 183-207 ◽  
Author(s):  
W. Balser
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Differential Equations ◽  
Formal Power Series ◽  
Convergent Series ◽  
Constant Matrix ◽  
Image Position ◽  
Formal Power ◽  
Meromorphic Differential Equation ◽  
Meromorphic Differential

Let a meromorphic differential equationbe given, where r is an integer, and the series converges for |z| sufficiently large. Then it is well known that (0.1) is formally satisfied by an expressionwhere F( z) is a formal power series in z–1 times an integer power of z, and F( z) has an inverse of the same kind, L is a constant matrix, andis a diagonal matrix of polynomials qj( z) in a root of z, 1≦ j≦ n. If, for example, all the polynomials in Q( z) are equal, then F( z) can be seen to be a convergent series (see Section 1), whereas if not, then generally the coefficients in F( z) grow so rapidly that F( z) diverges for every (finite) z.

scholarly journals On the Numerical Solution of Fractional Hyperbolic Partial Differential Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Fadime Dal ◽  
Zehra Pinar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Hyperbolic Partial Differential Equations ◽  
Stability Estimates ◽  
Partial Differential ◽  
One Dimensional ◽  
Gauss Elimination ◽  
Gauss Elimination Method

The stable difference scheme for the numerical solution of the mixed problem for the multidimensional fractional hyperbolic equation is presented. Stability estimates for the solution of this difference scheme and for the first and second orders difference derivatives are obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional hyperbolic partial differential equations.

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