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 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 Power Series Solution for Solving Nonlinear Burgers-Type Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
E. López-Sandoval ◽  
A. Mello ◽  
J. J. Godina-Nava ◽  
A. R. Samana
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Series Solution ◽  
Solution Method ◽  
Nonlinear Partial Differential Equations ◽  
Time Dependent ◽  
Power Series Solution ◽  
Partial Linear ◽  
Linear Differential ◽  
Burgers Type Equations

Power series solution method has been traditionally used to solve ordinary and partial linear differential equations. However, despite their usefulness the application of this method has been limited to this particular kind of equations. In this work we use the method of power series to solve nonlinear partial differential equations. The method is applied to solve three versions of nonlinear time-dependent Burgers-type differential equations in order to demonstrate its scope and applicability.

A New Numerical Method for Solving Nonlinear Fractional Fokker–Planck Differential Equations

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
BeiBei Guo ◽  
Wei Jiang ◽  
ChiPing Zhang
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Simple Algorithm ◽  
High Accuracy ◽  
Computational Work ◽  
Reproducing Kernel Theory ◽  
Transport Problems ◽  
Fokker Planck

The nonlinear fractional-order Fokker–Planck differential equations have been used in many physical transport problems which take place under the influence of an external force filed. Therefore, high-accuracy numerical solutions are always needed. In this article, reproducing kernel theory is used to solve a class of nonlinear fractional Fokker–Planck differential equations. The main characteristic of this approach is that it induces a simple algorithm to get the approximate solution of the equation. At the same time, an effective method for obtaining the approximate solution is established. In addition, some numerical examples are given to demonstrate that our method has lesser computational work and higher precision.

Wavelet Galerkin method for fourth-order multi-dimensional elliptic partial differential equations

2018 ◽  
Vol 16 (05) ◽  
pp. 1850045 ◽  
Author(s):  
B. V. Rathish Kumar ◽  
Gopal Priyadarshi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Computational Cost ◽  
Stability Estimates ◽  
Partial Differential ◽  
Wavelet Galerkin Method

We describe a wavelet Galerkin method for numerical solutions of fourth-order linear and nonlinear partial differential equations (PDEs) in 2D and 3D based on the use of Daubechies compactly supported wavelets. Two-term connection coefficients have been used to compute higher-order derivatives accurately and economically. Localization and orthogonality properties of wavelets make the global matrix sparse. In particular, these properties reduce the computational cost significantly. Linear system of equations obtained from discretized equations have been solved using GMRES iterative solver. Quasi-linearization technique has been effectively used to handle nonlinear terms arising in nonlinear biharmonic equation. To reduce the computational cost of our method, we have proposed an efficient compression algorithm. Error and stability estimates have been derived. Accuracy of the proposed method is demonstrated through various examples.

scholarly journals A Sixth Order Accuracy Solution to a System of Nonlinear Differential Equations with Coupled Compact Method

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Don Liu ◽  
Qin Chen ◽  
Yifan Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Sixth Order ◽  
Compact Schemes ◽  
Order Convergence ◽  
Order Accuracy ◽  
Partial Differential

A system of coupled nonlinear partial differential equations with convective and dispersive terms was modified from Boussinesq-type equations. Through a special formulation, a system of nonlinear partial differential equations was solved alternately and explicitly in time without linearizing the nonlinearity. Coupled compact schemes of sixth order accuracy in space were developed to obtain numerical solutions. Within couple compact schemes, variables and their first and second derivatives were solved altogether. The sixth order accuracy in space is achieved with a memory-saving arrangement of state variables so that the linear system is banded instead of blocked. This facilitates solving very large systems. The efficiency, simplicity, and accuracy make this coupled compact method viable as variational and weighted residual methods. Results were compared with exact solutions which were obtained via devised forcing terms. Error analyses were carried out, and the sixth order convergence in space and second order convergence in time were demonstrated. Long time integration was also studied to show stability and error convergence rates.

Barycentric Jacobi Spectral Method for Numerical Solutions of the Generalized Burgers-Huxley Equation

2017 ◽  
Vol 18 (1) ◽  
pp. 67-81 ◽  
Author(s):  
Edson Pindza ◽  
M. K. Owolabi ◽  
K.C. Patidar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Point Of View ◽  
Partial Differential ◽  
Computational Point ◽  
Exponential Time Differencing ◽  
Predictor Corrector ◽  
Numerical Approaches

AbstractNumerical solutions of nonlinear partial differential equations, such as the generalized and extended Burgers-Huxley equations which combine effects of advection, diffusion, dispersion and nonlinear transfer are considered in this paper. Such system can be divided into linear and nonlinear parts, which allow the use of two numerical approaches. Barycentric Jacobi spectral (BJS) method is employed for the spatial discretization, the resulting nonlinear system of ordinary differential equation is advanced with a fourth-order exponential time differencing predictor corrector. Comparative numerical results for the values of options are presented. The proposed method is very elegant from the computational point of view. Numerical computations for a wide variety of problems, show that the present method offers better accuracy and efficiency in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.

Thermal Analysis on a Satellite Box during Launch Stage by Analytical Solution

2005 ◽  
Vol 277-279 ◽  
pp. 765-770 ◽  
Author(s):  
Hui Kyung Kim ◽  
Joon Min Choi ◽  
Bum Seok Hyun
Keyword(s):  
Thermal Analysis ◽  
Analytical Solution ◽  
Numerical Solutions ◽  
Thermal Environment ◽  
Time Dependent ◽  
Space Environment ◽  
Numerical Techniques ◽  
Order Ordinary Differential Equation ◽  
Time Line ◽  
Time Period

Simple methods are developed to predict temperatures of a satellite box during the launch stage. The box is mounted on the outer surface of a satellite and directly exposed to a space thermal environment for the time period from fairing jettison to separation. These simple methods involve solving a 1st order ordinary differential equation (ODE), simplified from the full governing equation when several assumptions are made. The existence of an analytical solution for the 1st order ODE is determined depending on the treatment of the time-dependent molecular heating term. Even for the case that the analytical solution is not available due to the time dependent term, the 1st order ODE can be solved by relatively simple numerical techniques. The temperature difference between two different approaches (analytical and numerical solutions) is relatively small (less than 1°C along the time line) when they are applied to the STSAT-I launch scenario. The present methods can be generally used as tools to quickly check whether a satellite box is safe against the space environment during the launch stage for the case that the detailed thermal analysis is not available.

General Deformations of Neo-Hookean Membranes

1973 ◽  
Vol 40 (1) ◽  
pp. 7-12 ◽  
Author(s):  
W. H. Yang ◽  
C. H. Lu
Keyword(s):  
Differential Equations ◽  
Strain Energy ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Strain Energy Function ◽  
Finite Deformations ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Special Cases ◽  
Convergent Algorithm

A set of three nonlinear partial-differential equations is derived for general finite deformations of a thin membrane. The material that composes the membrane is assumed to be hyperelastic. Its mechanical property is represented by the neo-Hookean strain-energy function. The equations reduce to special cases known in the literature. A fast convergent algorithm is developed. The numerical solutions to the finite-difference approximation of the differential equations are computed iteratively with a trivial initial iterant. As an example, the problem of inflating a rectangular membrane with fixed edges by a uniform pressure applied on one side is presented. The solutions and their convergence are displayed and discussed.

scholarly journals Dynamics of water conveying zinc oxide through divergent-convergent channels with the effect of nanoparticles shape when Joule dissipation are significant

PLoS ONE ◽  
2021 ◽  
Vol 16 (1) ◽  
pp. e0245208
Author(s):  
Umair Rashid ◽  
Azhar Iqbal ◽  
Haiyi Liang ◽  
Waris Khan ◽  
Muhammad Waqar Ashraf
Keyword(s):  
Heat Transfer ◽  
Zinc Oxide ◽  
Differential Equations ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Joule Dissipation ◽  
Flow And Heat Transfer ◽  
Shape Effects ◽  
The Magnetic Field

Aim of study The shape effects of nanoparticles are very significant in fluid flow and heat transfer. In this paper, we discuss the effects of nanoparticles shape in nanofluid flow between divergent-convergent channels theoretically. In this present study, various shapes of nanoparticles, namely sphere, column and lamina in zinc oxide-water nanofluid are used. The effect of the magnetic field and joule dissipation are also considered. Research methodology The system of nonlinear partial differential equations (PDEs) is converted into ordinary differential equations (ODES). The analytical solutions are successfully obtained and compared with numerical solutions. The Homotopy perturbation method and NDsolve method are used to compare analytical and numerical results respectively. Conclusion The results show that the lamina shape nanoparticles have higher performance in temperature disturbance and rate of heat transfer as compared to other shapes of nanoparticles.

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