Analyticity of solutions of semi-linear equations with double characteristics

Least Squares Solutions of Inconsistent Fuzzy Linear Matrix Equations

Analyticity and Nonanalyticity of Solutions of Delay-Differential Equations

We consider the equation ẋ(t) = f(t, x(t), x(η(t))) with a variable time shift η(t). Both the nonlinearity f and the shift function η are given, and are assumed to be analytic (that is, holomorphic) functions of their arguments. Typically the time shift represents a delay, namely, that η(t) = t−r(t) with r(t) ≥ 0. The main problem considered is to determine when solutions (generally C∞ and ofte...

Exact and numerical solutions of linear and non-linear systems of fractional partial differential equations

The present study introduces a new technique of homotopy perturbation method for the solution of systems of fractional partial differential equations. The proposed scheme is based on Laplace transform and new homotopy perturbation methods. The fractional derivatives are considered in Caputo sense. To illustrate the ability and reliability of the method some examples are provided. The results ob...

Positive solutions to second order semi-linear elliptic equations

Here G ⊆ R (N ≥ 2) is an unbounded domain, and L is a second-order elliptic operator. We mainly confine ourselves to the cases F (x, u) = W (x)u with real p and W (x) a real valued function on G, and F (x, u) = g(u) with g : R→ R continuous and g(0) = 0. The operator L = H − V is of Schrödinger type, namely V = V (x) is a real potential and H = −∆ or more generally H = −∇ · a · ∇ is a second or...

Solutions of Linear Equations

For simplicity, we follow the rules: x denotes a set, i, j, k, l, m, n denote natural numbers, K denotes a field, N denotes a without zero finite subset of N, a, b denote elements of K, A, B, B1, B2, X, X1, X2 denote matrices over K, A′ denotes a matrix over K of dimension m × n, B′ denotes a matrix over K of dimension m × k, and M denotes a square matrix over K of dimension n. We now state a n...