Critical lengths for semilinear singular parabolic mixed boundary value problems

VARIATIONAL DISCRETIZATION AND MIXED METHODS FOR SEMILINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS WITH INTEGRAL CONSTRAINT

The aim of this work is to investigate the variational discretization and mixed finite element methods for optimal control problem governed by semi linear parabolic equations with integral constraint. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is not discreted. Optimal error estimates in L2 are established for the state...

Singular discrete and continuous mixed boundary value problems

For each n ∈ N, n ≥ 2 we prove the existence of a positive solution of the singular discrete problem 1 h2 ∆uk−1 + f(tk, uk) = 0, k = 1, . . . , n− 1, ∆u0 = 0, un = 0, where T ∈ (0,∞), h = T n , tk = hk, f : [0, T ] × (0,∞) is continuous and has a singularity at x = 0. We prove that for n → ∞ the sequence of solutions of the above discrete problems converges to a solution y of the corresponding ...

Initial - Boundary Value Problems for Parabolic Equations

Boundary Value Problems with Singular Boundary Conditions

Singular boundary conditions are formulated for non-selfadjoint Sturm-Liouville operators with singularities and turning points. For boundary value problems with singular boundary conditions properties of the spectrum are studied and the completeness of the system of root functions is proved.

Singular boundary value problems for ODEs

This paper is concerned with the numerical solution of a system of ordinary di erential equations ODEs y Sy t f t y p on an inter val b subject to boundary conditions g y y b p The ODEs have a coe cient that is singular at t but it is assumed that the boundary value problem BVP has a smooth solution Some popular methods for BVPs evaluate the ODEs at t This paper deals with the practical issues ...