Abstract
In this paper solution of mixed complex boundary value problem of first order is considered. The basic term in the problem with respect to space variables, has Cauchy-Riemann operator. We first use Laplace transformation to introduce spectral problem. Then we investigate for corresponding Fredholm's type. The spectral problem here is different from classical boundary value problems. Here boundary conditions are nonlocal and global and in general linear.At the end we find asymptotic expansionfor the solution of spectral problemwhich depends on unknown complex parameter. With the help of this asymptotic expansion we prove existence and uniqueness of mixed problem.
Highlights
Let time t ∈ (0, ∞), space variables x = (x1, x2 ) ∈ IR2 and D ⊂ IR2 be a bounded connected region
Mixed partial differential equations are basically considered for parabolic (Heat equation) and hyperbolic kinds of problems ([4], [5], [6])
If the boundary condition is Dirichlet’s the problem has no solution, i.e. the problem is not well posed. In this problem boundary condition is nonlocal and the number of boundary conditions is equal to highest order of derivative with respect to space variables ([7], [8], [9])
Summary
Mixed partial differential equations are basically considered for parabolic (Heat equation) and hyperbolic (wave equation) kinds of problems ([4], [5], [6]) In these cases number of boundary conditions (for local conditions) is the half of highest order of derivative of unknown function with respect to space variables, (for even orders), ([4], [5], [6]). If the boundary condition is Dirichlet’s (with any unknown equation in all over the boundary Γ ) the problem has no solution, i.e. the problem is not well posed In this problem boundary condition is nonlocal and the number of boundary conditions is equal to highest order of derivative with respect to space variables ([7], [8], [9]).
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