The investigation of a nonlinear differential equation using numerical methods

Christie, Alan M (1966) The investigation of a nonlinear differential equation using numerical methods. MSc(R) thesis, University of Glasgow.

Abstract

The equation investigated was [equation] the parameters a and c being varied. The boundary conditions inposed upon the equation were [equation] where t[m] was the position of the first maximum after the origin. It was most fully investigated for a 20, this being the region in which the solutions were exponentioally decaying. Although no analytic solution was discovered for the full equation, full solutions were found when a = 0. By suitable transformations the solution for c>0 was [equation] where M, t[0], k and q were constants. For c<0 the solution was [equation]. These, as might be expected were periodic solutions. The four numerical methods used were (1) Finite Difference (2) Step-by-Step (3) Picards (4) Perturbation The first two were purely numeric and the second two, semi-analytic. The Finite Difference technique was used to dind the solution between the boundary values, and the Step-by-Step method then was used to integrate along the curve until the value of y dropped to 0.01. The initial conditions for this latter method were found from the Finite Difference solution. Picard's Method and Perturbation which were used over the whole region both gave solutions in terms of exponential series. This series was of the form [equation] where the A[rs]'s were constant coefficients and ? and ? were the exponents of the linear solution [equation]. In all the methods except the Step-by-Step, the maximum had to be iterated onto by some means or another. In the Finite Difference method the second point was adjusted until this condition had been satisfied. In the two semianalytic approaches, the coefficients were in effect, altered to suit the condition. There was good agreement in results between the boundary conditions for all methods, but as might be expected for large values of c, the accuracy outside this region was not good, when the numerical methods were compared with the semi-analytic. This was due to the fact that the semi-analytic solutions were essentially sollutions expanded about a point. In comparing the two numerical solutions when the Finite Difference methid was used over the whole region, there was good agreement.

• Item Type: Thesis (MSc(R))
• Qualification Level: Masters
• Additional Information: Adviser: Gilles
• Keywords: Mathematics
• Date of Award: 1966
• Depositing User: Enlighten Team
• Unique ID: glathesis:1966-73640
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 14 Jun 2019 08:56
• Last Modified: 14 Jun 2019 08:56
• URI: https://theses.gla.ac.uk/id/eprint/73640