scholarly journals Approximate Solutions of Fisher's Type Equations with Variable Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi
Keyword(s):  
Differential Equations ◽  
Legendre Polynomials ◽  
Nonlinear Partial Differential Equations ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Variable Coefficients ◽  
Time Dependent ◽  
First Order ◽  
Spatial Derivatives ◽  
Interpolation Nodes

The spectral collocation approximations based on Legendre polynomials are used to compute the numerical solution of time-dependent Fisher’s type problems. The spatial derivatives are collocated at a Legendre-Gauss-Lobatto interpolation nodes. The proposed method has the advantage of reducing the problem to a system of ordinary differential equations in time. The four-stage A-stable implicit Runge-Kutta scheme is applied to solve the resulted system of first order in time. Numerical results show that the Legendre-Gauss-Lobatto collocation method is of high accuracy and is efficient for solving the Fisher’s type equations. Also the results demonstrate that the proposed method is powerful algorithm for solving the nonlinear partial differential equations.

scholarly journals A Jacobi Collocation Method for Solving Nonlinear Burgers-Type Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
E. H. Doha ◽  
D. Baleanu ◽  
A. H. Bhrawy ◽  
M. A. Abdelkawy
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Type Equation ◽  
High Accuracy ◽  
Time Dependent ◽  
Numerical Techniques ◽  
Spatial Derivatives ◽  
Burgers Type Equations ◽  
Jacobi Collocation Method

We solve three versions of nonlinear time-dependent Burgers-type equations. The Jacobi-Gauss-Lobatto points are used as collocation nodes for spatial derivatives. This approach has the advantage of obtaining the solution in terms of the Jacobi parametersαandβ. In addition, the problem is reduced to the solution of the system of ordinary differential equations (SODEs) in time. This system may be solved by any standard numerical techniques. Numerical solutions obtained by this method when compared with the exact solutions reveal that the obtained solutions produce high-accurate results. Numerical results show that the proposed method is of high accuracy and is efficient to solve the Burgers-type equation. Also the results demonstrate that the proposed method is a powerful algorithm to solve the nonlinear partial differential equations.

scholarly journals Approximate Solutions of Nonlinear Partial Differential Equations Using B-Polynomial Bases

2021 ◽  
Vol 5 (3) ◽  
pp. 106
Author(s):  
Muhammad Bhatti ◽  
Md. Rahman ◽  
Nicholas Dimakis
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Matrix Equation ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Initial Guess ◽  
Partial Differential

A multivariable technique has been incorporated for guesstimating solutions of Nonlinear Partial Differential Equations (NPDE) using bases set of B-Polynomials (B-polys). To approximate the anticipated solution of the NPD equation, a linear product of variable coefficients ai(t) and B-polys Bi(x) has been employed. Additionally, the variable quantities in the anticipated solution are determined using the Galerkin method for minimizing errors. Before the minimization process is to take place, the NPDE is converted into an operational matrix equation which, when inverted, yields values of the undefined coefficients in the expected solution. The nonlinear terms of the NPDE are combined in the operational matrix equation using the initial guess and iterated until converged values of coefficients are obtained. A valid converged solution of NPDE is established when an appropriate degree of B-poly basis is employed, and the initial conditions are imposed on the operational matrix before the inverse is invoked. However, the accuracy of the solution depends on the number of B-polys of a certain degree expressed in multidimensional variables. Four examples of NPDE have been worked out to show the efficacy and accuracy of the two-dimensional B-poly technique. The estimated solutions of the examples are compared with the known exact solutions and an excellent agreement is found between them. In calculating the solutions of the NPD equations, the currently employed technique provides a higher-order precision compared to the finite difference method. The present technique could be readily extended to solving complex partial differential equations in multivariable problems.

scholarly journals On a curve and a system

2020 ◽  
Vol 5 (3) ◽  
pp. 030-052
Author(s):  
Tuba Ağırman Aydın ◽  
Seda Çayan ◽  
Mehmet Sezer ◽  
Abdullah Mağden
Keyword(s):  
Differential Equations ◽  
Constant Width ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
System Of Differential Equations ◽  
Similar System ◽  
First Order ◽  
Cam Design ◽  
Different Types ◽  
Curves Of Constant Width

Curves of constant width, which have a very special place in many fields such as kinematics, engineering, art, cam design and geometry, are specially discussed under this title. In this study, a system of differential equations characterizing the curves of constant width is examined. This is the system of the first order homogenous differential equations with variable coefficients in the normal form. Approximate solutions of the system, by means of two different polynomial approaches, are calculated and error analysis is made. The obtained results are analyzed on a numerical sample and the best method of approach is determined. This system can also constitute a characterization for different types of curves according to different frames in different spaces. Therefore, this study is important not only for this curve type but also for the geometry of all curves that can be expressed in a similar system.

scholarly journals Some Further Results on Oscillations for Neutral Delay Differential Equations with Variable Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fatima N. Ahmed ◽  
Rokiah Rozita Ahmad ◽  
Ummul Khair Salma Din ◽  
Mohd Salmi Md Noorani
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Variable Coefficients ◽  
Oscillatory Behaviour ◽  
Neutral Delay ◽  
Neutral Equations ◽  
First Order ◽  
All Solutions ◽  
Neutral Delay Differential Equations ◽  
Delay Differential

We study the oscillatory behaviour of all solutions of first-order neutral equations with variable coefficients. The obtained results extend and improve some of the well-known results in the literature. Some examples are given to show the evidence of our new results.

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