On the numerical solution of partial differential equations defined on domains limited by curves using segmented forms of the tau-lines method

Abstract

This paper discusses solving partial differential equations defined over domains limited by curves using segmented forms of the Tau-Lines Method introduced by El Misiery and Ortiz. The paper consists of two parts. In the first part discussion on how to apply the Method to a linear partial differential equation defined on a rectangular domain, with mixed boundary conditions, is given. In the second part discussion and numerical application are carried out on the Poisson equation defined: (i) on a square domain, (ii) on a domain limited by a parabolic arc. The result obtained for example (i) has been favorably compared with results reported by using two different techniques. Second and fourth order accurate Finite Difference Approximations have been used and their merits have been discussed. Four systems of equally spaced mesh-lines have been experimented and their related results have been compared. The approximate solutions, on the lines, are sought as finite expansions in terms of orthogonal polynomial bases. These solutions have shown a good and rapid convergence since their numerical values have agreed up to at least five decimals for approximation orders ranging from six to ten for both Legendre and Chebyshev polynomials.

Keywords:

• Partial differential equations
• finite difference approximations
• Tau-Method
• method of lines
• segmented lines
• parallel shooting
• ordinary differential equations
• orthogonal expansions

C.R Categories:

• G.1.8

^*Present address: Faculty of Science and Technology, computer Science Dept., Balamand University, PO Box 100 Tripoli, Lebanon

^*Present address: Faculty of Science and Technology, computer Science Dept., Balamand University, PO Box 100 Tripoli, Lebanon

Notes

^*Present address: Faculty of Science and Technology, computer Science Dept., Balamand University, PO Box 100 Tripoli, Lebanon