In this work we generate the numerical solutions of the Burgers’ equation by applying the Crank-Nicolson method directly to the Burgers’ equation, i.e., we do not use Hopf-Cole transformation to reduce Burgers’ equ...In this work we generate the numerical solutions of the Burgers’ equation by applying the Crank-Nicolson method directly to the Burgers’ equation, i.e., we do not use Hopf-Cole transformation to reduce Burgers’ equation into the linear heat equation. Absolute error of the present method is compared to the absolute error of the two existing methods for two test problems. The method is also analyzed for a third test problem, nu-merical solutions as well as exact solutions for different values of viscosity are calculated and we find that the numerical solutions are very close to exact solution.展开更多
In this paper, we construct the group-invariant (exact) solutions for the generalised Fisher type equation using both classical Lie point and the nonclassical symmetry techniques. The generalised Fisher type equation ...In this paper, we construct the group-invariant (exact) solutions for the generalised Fisher type equation using both classical Lie point and the nonclassical symmetry techniques. The generalised Fisher type equation arises in theory of population dynamics. The diffusion term and coefficient of the source term are given as the power law functions of the spatial variable. We introduce the modified Hopf-Cole transformation to simplify a nonlinear second Order Ordinary Equation (ODE) into a solvable linear third order ODE.展开更多
文摘In this work we generate the numerical solutions of the Burgers’ equation by applying the Crank-Nicolson method directly to the Burgers’ equation, i.e., we do not use Hopf-Cole transformation to reduce Burgers’ equation into the linear heat equation. Absolute error of the present method is compared to the absolute error of the two existing methods for two test problems. The method is also analyzed for a third test problem, nu-merical solutions as well as exact solutions for different values of viscosity are calculated and we find that the numerical solutions are very close to exact solution.
文摘In this paper, we construct the group-invariant (exact) solutions for the generalised Fisher type equation using both classical Lie point and the nonclassical symmetry techniques. The generalised Fisher type equation arises in theory of population dynamics. The diffusion term and coefficient of the source term are given as the power law functions of the spatial variable. We introduce the modified Hopf-Cole transformation to simplify a nonlinear second Order Ordinary Equation (ODE) into a solvable linear third order ODE.
