Taylor polynomial solutions of second order linear partial differential equations
Polynomial approximation for the solutions of second order linear PDE with two variables and variable coefficients of the form \[ A(x,y)\frac{\partial^2u}{\partial x^2}+ B(x,y)\frac{\partial^2u}{\partial x\partial y}+ C(x,y)\frac{\partial^2u}{\partial y^2}+ D(x,y)\frac{\partial u}{\partial x}+ E(x,y)\frac{\partial u}{\partial y}+F(x,y)u=G(x,y), \] where \(A(x,y)\), \(B(x,y)\), \(C(x,y)\), \(D(x,y)\), \(E(x,y)\), \(F(x,y)\), and \(G(x,y)\) are functions having Taylor expressions on the domain \(a\leq x,y\leq b\), under the conditions: \[ \sum_{i=0}^1\left(u^{(0,i)}(x,a)+ u^{(0,i)}(x,b)+ u^{(0,i)}(x,c_1)\right)= f(x), \] \[ \sum_{i=0}^1\left(u^{(i,0)}(a,y)+ u^{(i,0)}(b,y)+ u^{(i,0)}(c_0,y)\right)= g(y), \] \(u(a,a)+ u(a,b)+u(a,c_1)+ u(b,a)+u(b,b)+u(b,c_1)+u(c_0,a)+ u(c_0,b)+ u(c_0,c_1)=\lambda\) is considered. The truncated Taylor series of the functions from the above equation are taken and then substituted into the equation and given conditions. Thereby the considered problem for the PDE is reduced to a matrix equation. Two illustrative examples are considered.
scientific article; zbMATH DE number 2102853