We develop a new sixth-order accurate numerical scheme for the solution of two point boundary value problems.The scheme uses nonpolynomial spline basis and high order finite difference approximations.With the help of ...We develop a new sixth-order accurate numerical scheme for the solution of two point boundary value problems.The scheme uses nonpolynomial spline basis and high order finite difference approximations.With the help of spline functions,we derive consistency conditions and it is used to derive high order discretizations of the first derivative.The resulting difference schemes are solved by the standard Newton’s method and have very small computing time.The new method is analyzed for its convergence and the efficiency of the proposed scheme is illustrated by convection-diffusion problem and nonlinear Lotka–Volterra equation.The order of convergence and maximum absolute errors are computed to present the utility of the new scheme.展开更多
In this article,we present two new novel finite difference approximations of order two and four,respectively,for the three dimensional non-linear triharmonic partial differential equations on a compact stencil where t...In this article,we present two new novel finite difference approximations of order two and four,respectively,for the three dimensional non-linear triharmonic partial differential equations on a compact stencil where the values of u,δ^(2)u/δn^(2)andδ^(4)u/δn^(4)are prescribed on the boundary.We introduce new ideas to handle the boundary conditions and there is no need to discretize the derivative boundary conditions.We require only 7-and 19-grid points on the compact cell for the second and fourth order approximation,respectively.The Laplacian and the biharmonic of the solution are obtained as by-product of the methods.We require only system of three equations to obtain the solution.Numerical results are provided to illustrate the usefulness of the proposed methods.展开更多
In this paper,we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyper-bolic partial differential equation of the form u...In this paper,we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyper-bolic partial differential equation of the form utt=A(x,y,t)uxx+B(x,y,t)uyy+g(x,y,t,u,ux,uy,ut),0<x,y<1,t>0 subject to appropriate initial and Dirichlet boundary conditions.We use only five evaluations of the function g and do not require any fictitious points to discretize the differential equation.The proposed method is directly applicable to wave equation in polar coordinates and when applied to a linear telegraphic hyperbolic equation is shown to be unconditionally stable.Numerical results are provided to illustrate the usefulness of the proposed method.展开更多
文摘We develop a new sixth-order accurate numerical scheme for the solution of two point boundary value problems.The scheme uses nonpolynomial spline basis and high order finite difference approximations.With the help of spline functions,we derive consistency conditions and it is used to derive high order discretizations of the first derivative.The resulting difference schemes are solved by the standard Newton’s method and have very small computing time.The new method is analyzed for its convergence and the efficiency of the proposed scheme is illustrated by convection-diffusion problem and nonlinear Lotka–Volterra equation.The order of convergence and maximum absolute errors are computed to present the utility of the new scheme.
基金This research was supported by’The University of Delhi’under research grant No.Dean(R)/R&D/2010/1311.
文摘In this article,we present two new novel finite difference approximations of order two and four,respectively,for the three dimensional non-linear triharmonic partial differential equations on a compact stencil where the values of u,δ^(2)u/δn^(2)andδ^(4)u/δn^(4)are prescribed on the boundary.We introduce new ideas to handle the boundary conditions and there is no need to discretize the derivative boundary conditions.We require only 7-and 19-grid points on the compact cell for the second and fourth order approximation,respectively.The Laplacian and the biharmonic of the solution are obtained as by-product of the methods.We require only system of three equations to obtain the solution.Numerical results are provided to illustrate the usefulness of the proposed methods.
基金“The University of Delhi”under research grant No.Dean(R)/R&D/2010/1311.
文摘In this paper,we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyper-bolic partial differential equation of the form utt=A(x,y,t)uxx+B(x,y,t)uyy+g(x,y,t,u,ux,uy,ut),0<x,y<1,t>0 subject to appropriate initial and Dirichlet boundary conditions.We use only five evaluations of the function g and do not require any fictitious points to discretize the differential equation.The proposed method is directly applicable to wave equation in polar coordinates and when applied to a linear telegraphic hyperbolic equation is shown to be unconditionally stable.Numerical results are provided to illustrate the usefulness of the proposed method.
