Numerical solutions to singular ϕ-Laplacianwith Dirichlet boundary conditions

Abstract

In this note we are concerned with numerical solutions to Dirichlet problem

$$[\phi(u')]' =f(x) \quad \mbox{in} [\alpha, \beta]; \quad u(\alpha)=A, \; u(\beta)=B, $$

where $\phi :(-\eta , \eta ) \to \mathbb {R}$ $(\eta <+ \infty )$ is an increasing diffeomorphism with $\phi '(y)\geq d >0$ for all $y\in (-\eta , \eta )$. The obtained algorithm combines the shooting method with Euler’s method and it is convergent whenever the problem is solvable. We provide numerical experiments confirming the theoretical aspects.