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Computation of Eigenfunctions to Singular Eigenvalue Problem

Petr Girg
Organization: University of West Bohemia

2006 Wolfram Technology Conference
Conference location

Champaign IL

Boundary value problems for quasilinear differential equations appear in many mathematical models nowadays. Solvability and dependence of the number of solutions on parameters are often asked by the people from praxis. A prototype of such a problem is the Dirichlet problem for the so-called p-Laplacian. In this talk we consider p-Laplacian in dimension one. The solvability of

* equation omitted here

can be approached using bifurcation theory (bifurcation from the infinity). In this context we arrive at a linearized problem, solvability of which is determined by linear fredholm alternative. In order to use this approach, we need to know eigenvalues and eigenfunctions to the weighted linear problem

* equation omitted here

The eigenvalue problem is solved by a shooting method, starting with ψ'(0) = 1 and seeking for λ such that ψ(1) = 0. The weight functions | φk' |^(p - 2) and | φk (|)^(p -2) are singular or degenerated at some points. Another problem is that the function φk, which appears in the weight functions, is not known explicitly.

Moreover asymptotic behavior of | φk' |^(p - 2) and | φk (|)^(p -2) close to points x where either φk' (x) = 0 or φk (x) = 0 is known. For that reason we treat Eq.(2) using the event locator - when the values of φk' or φk are sufficiently far from zero, we compute these weights by numerical ODE integration, when they are close to zero we use an asymptotic formula. The initial value problem corresponding to Eq.(2) and the initial value problem Eq.(3) are solved simultaneously, and we require that

* equation omitted here

The talk concentrates on underlying ideas of general interest and their implementation. Theoretical reasons for why to compute these eigenfunctions will be presented on a poster by Benedikt, Čepička, Girg, and Takáč.