Partial Differential Equations Possessing New Complex Derivatives

Abstract

The recently proposed new operator for complex differentiation is applied to partial differential equations for the first time. The definition involves complex numbers in the differentiation operators. In this sense, it does not possess any resemblance with the real number differentiation of complex valued functions. First, the operator is defined with its basic properties. Several partial differential equations are considered involving the new complex number derivatives. First, the homogenous constant real coefficient linear partial differential equations are considered. If the real and imaginary components of the derivatives are equal, the imaginary parts can be ignored. For complex valued coefficients, for non-homogenous equations and for derivatives having different real and imaginary parts, the imaginary parts cannot be ignored. Some nonlinear partial differential equations are also considered. In the calculations, the Cauchy-Riemann as well as the Laplace equations appears frequently. Expression of the Hamilton’s equations and the Schrodinger equation in ¨ terms of the new differential operators are outlined. The new operators have the potential to be applied to mathematical modeling of several physical problems in the future