Abstract
In this paper, we use the method of lines to convert elliptic and hyperbolic partial differential equations (PDEs) into systems of boundary value problems and initial value problems in ordinary differential equations (ODEs) by replacing the appropriate derivatives with central difference methods. The resulting system of ODEs is then solved using an extended block Numerov-type method (EBNUM) via a block unification technique. The accuracy and speed advantages of the EBNUM over the finite difference method (FDM) are established numerically.
Highlights
The method of lines approach which involves replacing the spatial derivatives with finite difference approximations is commonly used for solving partial differential equations (PDEs); whereby, the PDE is transformed into systems of ordinary differential equations (ODEs) and solved by reliable ODE solvers
The method of lines approach which involves replacing the spatial derivatives with finite difference approximations is commonly used for solving PDEs; whereby, the PDE is transformed into systems of ODEs and solved by reliable ODE solvers
( ) Remark 4 We found that ρ q2 ≤ 1 if q2 ∈[0,12], the stability interval for the extended block Numerov-type method (EBNUM) is [0,12] ; which is twice the stability interval of the standard Numerov method
Summary
The method of lines approach which involves replacing the spatial derivatives with finite difference approximations is commonly used for solving PDEs; whereby, the PDE is transformed into systems of ODEs and solved by reliable ODE solvers (see Lambert [1], Ramos and Vigo-Aguiar [2], Brugnano and Trigiante [3], D’Ambrosio and Paternoster [4], and). Our objective is to convert the elliptic and hyperbolic PDEs into systems of ODEs by replacing the appropriate derivatives with central difference methods. The resulting systems of ODEs are solved using an EBNUM via a block unification technique. (2015) Block Unification Scheme for Elliptic, Telegraph, and Sine-Gordon Partial Differential Equations. S. Jator are spatial variables, g ( x, y) is a distributed source, and when r ( x, y) = 0 , (1) becomes the two-dimensional convection diffusion equation given in Sun and Zhang [5]. We invoke the method of lines approach in which for real numbers a,b, c, d , we seek a solution in the strip [a,b]×[c, d ] by first fixing the grid in the spatial variable x; approximating this spatial derivative using central difference methods, and solving the resulting system of second order time independent ODEs in the spatial variable y.
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