Solutions of several general quadratic partial differential-difference equations in ℂ ²

Abstract In this article, we have introduced general transformation to solving the general quadratic equations. It is of interest to know about the existence and form of the solutions of general quadratic functional equations. By utilizing Nevanlinna theory for several complex variables and the transformation, we study the existence and form of the solutions to the general quadratic partial differential or partial differential-difference equations of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>a</m:mi> <m:msup> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mn>2</m:mn> <m:mi>α</m:mi> <m:mi>f</m:mi> <m:mi>g</m:mi> <m:mo>+</m:mo> <m:mi>b</m:mi> <m:msup> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mn>2</m:mn> <m:mi>β</m:mi> <m:mi>f</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> <m:mi>γ</m:mi> <m:mi>g</m:mi> <m:mo>+</m:mo> <m:mi>C</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:math> a{f}^{2}+2\alpha fg+b{g}^{2}+2\beta f+2\gamma g+C=0 in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> {{\mathbb{C}}}^{2} . Consequently, we obtain certain corollaries of the main results of this article concerning trinomial equations which generalize many results. In addition, some examples relevant to the content of the article has been exhibited.