On the solution of partial differential equations with specified boundary conditions
By a certain decomposition, explicit solutions to linear partial differential equations are computed. As an example, the equation \(\partial_{xx}u+\partial_{yy}u=g(x,y)\) is written as \[ u=c_ 1k_ 1(y)+c_ 2k_ 2(y)x+\partial^{-1}_{xx}g-\partial^{- 1}_{xx}\partial_{yy}u, \] \[ u=c_ 3k_ 3(x)+c_ 4k_ 4(x)y+\partial^{-1}_{yy}g-\partial^{-1}_{yy}\partial_{xx}u. \] This is used iteratively with appropriate boundary conditions to compute the solution \(u(x,y)=x^ 2y^ 2/2.\) Several other (more complicated) examples are considered.
scientific article; zbMATH DE number 4111961