scholarly journals Singularly perturbed Burger-Huxley equation: Analytical solution through iteration

2018 ◽  
Vol 5 (3) ◽  
pp. 45 ◽  
Author(s):  
Deepmala Kamboj ◽  
M.D. Sharma
Keyword(s):  
Analytical Solution ◽  
Singularly Perturbed ◽  
Huxley Equation

THREE-STEP TAYLOR GALERKIN METHOD FOR SINGULARLY PERTURBED GENERALIZED HODGKIN–HUXLEY EQUATION

2010 ◽  
Vol 01 (02) ◽  
pp. 257-276
Author(s):  
VIVEK SANGWAN ◽  
B. V. RATHISH KUMAR ◽  
S. V. S. S. N. V. G. K. MURTHY ◽  
MOHIT NIGAM
Keyword(s):  
Boundary Layer ◽  
Finite Element Method ◽  
Finite Element ◽  
Singular Perturbation ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Galerkin Finite Element Method ◽  
Third Order ◽  
Galerkin Finite Element ◽  
Huxley Equation

A numerical study is carried out for the singularly perturbed generalized Hodgkin–Huxley equation. The equation is nonlinear which mimics the ionic processes at a real nerve membrane. A small parameter called singular perturbation parameter is introduced in the highest order derivative term. Keeping other parameters fixed, as this singular perturbation parameter approaches to zero, a boundary layer occurs in the solution. Three-step Taylor Galerkin finite element method is employed on a piecewise uniform Shishkin mesh to solve the equation. To procure more accurate temporal differencing, the method employs forward-time Taylor series expansion including time derivatives of third order which are evaluated from the governing singularly perturbed generalized Hodgkin–Huxley equation. This yields a generalized time-discretized equation which is successively discretized in space by means of the standard Bubnov–Galerkin finite element method. The method is third-order accurate in time. The code based on the purposed scheme has been validated against the cases for which the exact solution is available. It is also observed that for the Singularly Perturbed Generalized Hodgkin–Huxley equation, the boundary layer in the solution manifests not only by varying the singular perturbation parameter but also by varying the other parameters appearing in the model.

scholarly journals The Analytical Solution Of Singularly Perturbed Boundary Value Problems

2014 ◽  
Vol 10 (01) ◽  
pp. 7-22 ◽  
Author(s):  
S.gh. Hosseini ◽  
S.m. Hosseini ◽  
M. Heydari ◽  
M. Amini
Keyword(s):  
Analytical Solution ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Singularly Perturbed

scholarly journals Solving a Class of Singularly Perturbed Partial Differential Equation by Using the Perturbation Method and Reproducing Kernel Method

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Yu-Lan Wang ◽  
Hao Yu ◽  
Fu-Gui Tan ◽  
Shanshan Qu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Perturbation Method ◽  
Present Method ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Singularly Perturbed ◽  
Partial Differential ◽  
Reproducing Kernel Method

We give the analytical solution and the series expansion solution of a class of singularly perturbed partial differential equation (SPPDE) by combining traditional perturbation method (PM) and reproducing kernel method (RKM). The numerical example is studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.

scholarly journals NUMERICAL SOLUTION TO LINEAR SINGULARLY PERTURBED TWO POINT BOUNDARY VALUE PROBLEMS USING B-SPLINE COLLOCATION METHOD

2016 ◽  
Vol 4 (1) ◽  
pp. 158-164
Author(s):  
Rajashekhar Reddy
Keyword(s):  
Analytical Solution ◽  
Collocation Method ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Point Boundary ◽  
Spline Collocation ◽  
B Spline ◽  
Numerical Examples ◽  
Spline Collocation Method ◽  
Spline Basis

A Recursive form cubic B-spline basis function is used as basis in B-spline collocation method to solve second linear singularly perturbed two point boundary value problem. The performance of the method is tested by considering the numerical examples with different boundary conditions. Results of numerical examples show the robustness of the method when compared with the analytical solution.

Sign in / Sign up

Export Citation Format

Share Document