scholarly journals Numerical Method of the Line for Solving One Dimensional Initial- Boundary Singularly Perturbed Burger Equation

2021 ◽  
pp. 4-14
Author(s):  
Kedir Aliyi ◽  
◽  
Hailu Muleta ◽  
Keyword(s):  
Numerical Method ◽  
Burger Equation ◽  
Initial Boundary ◽  
Perturbation Parameter ◽  
Numerical Results ◽  
Singularly Perturbed ◽  
Maximum Point ◽  
Approximation Solution ◽  
Von Neumann ◽  
One Dimensional

In this Research Method of Line is used to find the approximation solution of one dimensional singularly perturbed Burger equation given with initial and boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced into the functional values at each grid points by using the central finite difference method. Then, the resulting first-order linear ordinary differential equation is solved by the fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the perturbation parameter ‘ε’ and mesh sizes in the direction of the temporal variable, t. Numerical results are presented in tables in terms of Maximum point-wise error, EN,Δt and rate of convergence, Pε N,Δt. The stability of this new class of Numerical method is also investigated by using Von Neumann stability analysis techniques. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

scholarly journals Numerical Method of the Line for Solving One Dimensional Initial- Boundary Singularly Perturbed Burger Equation

2021 ◽  
Vol 1 (2) ◽  
pp. 4-14
Author(s):  
Kedir Aliyi ◽  
Hailu Muleta
Keyword(s):  
Numerical Method ◽  
Burger Equation ◽  
Initial Boundary ◽  
Perturbation Parameter ◽  
Numerical Results ◽  
Singularly Perturbed ◽  
Maximum Point ◽  
Approximation Solution ◽  
Von Neumann ◽  
One Dimensional

In this Research Method of Line is used to find the approximation solution of one dimensional singularly perturbed Burger equation given with initial and boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced into the functional values at each grid points by using the central finite difference method. Then, the resulting first-order linear ordinary differential equation is solved by the fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the perturbation parameter ‘  ’ and mesh sizes in the direction of the temporal variable, t. Numerical results are presented in tables in terms of Maximum point-wise error, N t , E  and rate of convergence, N t , P  . The stability of this new class of Numerical method is also investigated by using Von Neumann stability analysis techniques. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

Numerical Method for a Class of Nonlinear Singularly Perturbed Delay Differential Equations Using Parametric Cubic Spline

2018 ◽  
Vol 19 (3-4) ◽  
pp. 357-365 ◽  
Author(s):  
A. S. V. Ravi Kanth ◽  
P. Murali Mohan Kumar
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Delay Differential Equations ◽  
Perturbation Parameter ◽  
Cubic Spline ◽  
Singularly Perturbed ◽  
Delay Term ◽  
Second Order Convergence ◽  
Delay Differential ◽  
Parametric Cubic Spline

AbstractIn this paper, we study the numerical method for a class of nonlinear singularly perturbed delay differential equations using parametric cubic spline. Quasilinearization process is applied to convert the nonlinear singularly perturbed delay differential equations into a sequence of linear singularly perturbed delay differential equations. When the delay is not sufficiently smaller order of the singular perturbation parameter, the approach of expanding the delay term in Taylor’s series may lead to bad approximation. To handle the delay term, we construct a special type of mesh in such a way that the term containing delay lies on nodal points after discretization. The parametric cubic spline is presented for solving sequence of linear singularly perturbed delay differential equations. The error analysis of the method is presented and shows second-order convergence. The effect of delay parameter on the boundary layer behavior of the solution is discussed with two test examples.

scholarly journals Robust numerical method for singularly perturbed differential equations having both large and small delay

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Habtamu Garoma Debela
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Large Delay ◽  
Content Type ◽  
Small Delay ◽  
Numerical Experimentation ◽  
The Stability

PurposeThe purpose of this study is to develop stable, convergent and accurate numerical method for solving singularly perturbed differential equations having both small and large delay.Design/methodology/approachThis study introduces a fitted nonpolynomial spline method for singularly perturbed differential equations having both small and large delay. The numerical scheme is developed on uniform mesh using fitted operator in the given differential equation.FindingsThe stability of the developed numerical method is established and its uniform convergence is proved. To validate the applicability of the method, one model problem is considered for numerical experimentation for different values of the perturbation parameter and mesh points.Originality/valueIn this paper, the authors consider a new governing problem having both small delay on convection term and large delay. As far as the researchers' knowledge is considered numerical solution of singularly perturbed boundary value problem containing both small delay and large delay is first being considered.

scholarly journals Numerical Solution of Partial Integrodifferential Equations of Diffusion Type

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Imran Aziz ◽  
Imran Khan
Keyword(s):  
Applied Sciences ◽  
Dynamical Systems ◽  
Numerical Solution ◽  
Numerical Method ◽  
Collocation Method ◽  
Numerical Results ◽  
Integrodifferential Equations ◽  
Benchmark Problems ◽  
Diffusion Type ◽  
One Dimensional

A collocation method based on linear Legendre multiwavelets is developed for numerical solution of one-dimensional parabolic partial integrodifferential equations of diffusion type. Such equations have numerous applications in many problems in the applied sciences to model dynamical systems. 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 Robust numerical method for a singularly perturbed problem arising in the modelling of enzyme kinetics

BIOMATH ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 2008227
Author(s):  
John J. H. Miller ◽  
Eugene O'Riordan
Keyword(s):  
Enzyme Kinetics ◽  
Numerical Method ◽  
A Priori ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Uniform Mesh ◽  
Initial Value ◽  
Computational Performance ◽  
Explicit Bounds

A system of two coupled nonlinear initial value equations, arising in the mathematical modelling of enzyme kinetics, is examined. The system is singularly perturbed and one of the components will contain steep gradients. A priori parameter explicit bounds on the two components are established. A numerical method incorporating a specially constructed piecewise-uniform mesh is used to generate numerical approximations, which are shown to converge pointwise to the continuous solution irrespective of the size of the singular perturbation parameter. Numerical results are presented to illustrate the computational performance of the numerical method. The numerical method is also remarkably simple to implement. 

FREQUENCY DOMAIN TREATMENT OF ONE-DIMENSIONAL SCALAR WAVES

1993 ◽  
Vol 03 (02) ◽  
pp. 171-194 ◽  
Author(s):  
JIM DOUGLAS ◽  
JUAN E. SANTOS ◽  
DONGWOO SHEEN ◽  
LYNN SCHREYER BENNETHUM
Keyword(s):  
Numerical Method ◽  
Frequency Domain ◽  
Space Variable ◽  
Numerical Results ◽  
One Dimensional ◽  
Scalar Waves

A naturally parallelizable numerical method for approximating scalar waves in a single space variable is developed by going to a frequency domain formulation. General forms of attenuation are permitted. Convergence is established and numerical results are presented.

Sign in / Sign up

Export Citation Format

Share Document