Numerical solutions to singular ϕ-Laplacianwith Dirichlet boundary conditions

AbstractIn this note we are concerned with numerical solutions to Dirichlet problem [ϕ(u′)]′=f(x)in[α,β];u(α)=A,u(β)=B,$$[\phi(u')]' =f(x) \quad \mbox{in} [\alpha, \beta]; \quad u(\alpha)=A, \; u(\beta)=B, $$ where ϕ:(−η,η)→ℝ$\phi :(-\eta , \eta ) \to \mathbb {R}$(η<+∞)$(\eta <+ \infty )$ is an increasing diffeomorphism with ϕ′(y)≥d>0$\phi '(y)\geq d >0$ for all y∈(−η,η)$y\in (-\eta , \eta )$. 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.