On an application of Laplace transforms

In this study, complex differential equations are solved us ing laplace transform. Firstly we seperate real and imagine r parts of equation. Thus from one unknown equation is obtained two u nknown equation system. Later we obtain laplace transforms of real and imaginer parts of solutions using laplace transform. In the latest we obtain real and imaginer parts of solution usin g inverse laplace transform.


Introduction
In real, general solutions of some equations, especially type of elliptic, are not found. Real partial differential equation systems when number of independent variables are even can be transformed to a complex partial differential equations. The solving a complex equation can more easier with complex methods. For example, Laplace equation hasn't got general solution in R 2 , but it can be written u zz = 0 with the relation ∆ = ∂ 2 ∂ z∂ z and the solution of the equation is given as where f is analytic, g is anti analytic arbitrary functions. A partial differential equation system which has two real dependant and two real independant variables can be transformed to a complex equation. For example, Cauchy Riemann system transforms to complex equation where w = u + iv, z = x + iy. All solutions of this complex equation are analytic functions.
Moreover any order complex differential equation can be transformed to real partial differential equation system which has two unknowns, two independent variables by seperating the real and imaginer parts. The solution of complex equation can be put forward helping solutions of this real system.
In this study, we investigate solutions of first order constant coefficients complex equations with laplace transforms. Laplace transform using several areas of mathematics is a integral transform. We can solve ordinary differential equations, system of ordinary differential equation, integral equations, integro differential equations, difference equations, integro difference equations and also calculate some generalized integrals with laplace transform. Moreover we can use laplace transform in electrical circuits. Therefore we can solve fractional differential equations via laplace transforms [2,3]. Nonlinear differential equations can be solved laplace decomposition method [4].

Basic definitions and theorems
Since integral of (2) is a function of s, then we can write L(F(t)) = f (s).

Theorem 2.
Laplace transforms of partial derivatives of u(x,t) are given as follow.

Complex derivatives
3 Solution of complex differential equations from first order which is constant coeffients Theorem 3. Let A, B,C are real constants, F(z, z) is a polynomial of z, z and w = u + iv is a complex function. Then the solution of is given as
From the above theorem Example 2. Solve the following problem   = 2xy − y.