Abstract
When working with mathematical models, to keep the model errors as small as possible, a special system of linear equations is constructed whose solution vector yields accurate discretized values for the exact solution of the second-order linear inhomogeneous ordinary differential equation (ODE). This case involves a 1D spatial variable x with an arbitrary coefficient function kappa (x) and an arbitrary source function f(x) at each grid point under Dirichlet or/and Neumann boundary conditions. This novel exact scheme is developed considering the recurrence relations between the variables. Consequently, this scheme is similar to those obtained using the finite difference, finite element, or finite volume methods; however, the proposed scheme provides the exact solution without any error. In particular, the adequate test functions that provide accurate values for the solution of the ODE at arbitrarily located grid points are determined, thereby eliminating the errors originating from discretization and numerical approximation.
Highlights
Based on the fundamental solutions corresponding to differential operators, several schemes applicable to first-order ordinary differential equation (ODE) have been reported [3]
Consider the following linear second-order inhomogeneous ordinary differential equation (ODE) in self-adjoint form: d d– κ(x) u(x) = f (x) (0 < x < ), (1.1)dx dx where κ(x) ≥ κ0 > 0 is a positive function ensuring the existence of the integrals applied for the solution procedure
On the basis of our research work, this paper proposes a scheme that provides the values for the exact solutiona of Eq (1.1) in arbitrarily chosen grid points
Summary
Based on the fundamental solutions corresponding to differential operators, several schemes applicable to first-order ODEs have been reported [3]. 4 Exact scheme for Dirichlet boundary conditions We use the basic recurrence relation of Eq (3.10) to obtain the exact values of solution (1.1) at all (n + 2) node points: u(x0), u(x1), . Un)T lead to the same values as the solution u(x) of the second-order ODE (1.1) with Dirichlet boundary conditions (4.2) at the interior grid points (2.1) without any error, i.e., u(xi) = ui
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
