A method for solving partial differential equations with homogeneous boundary or initial conditions (Q2583654)

Relationship between homogeneous functions of zero degree in the space \(\mathbb R^3\) and functions on the plane are considered. Let \(u(x,y,z)=\ln[((x^2+y^2+z^2)^{1/2}+z)(x^2+y^2)^{-1/2}]\) and \(v(x,y)=\arg(x+iy)\). Assertion of Theorem 1 is: If \(f(u,v)\) is a twice continuously differentiable function in the trip \(0\leq v\leq2\pi\) such that \(f(u,0)=f(u,2\pi)\) and \(F(x,y,z)= f(u(x,y,z), v(x,y))\) then \((x^2+y^2)\Delta_{xyz}F=\Delta_{uv} f\), where \(\Delta_{xyz}F=F''_{xx}+F''_{yy}+ F''_{zz}\). Let \(a=x((x^2+ y^2+z^2)^{1/2}-z)^{-1}\), \(b=y((x^2+y^2+z^2)^{1/2}-z)^{-1}\) and \(\psi(a,b)=F(a,b,(a^2+b^2-1)/2)\) then \(((x^2+ y^2+z^2)^{1/2}- z)^{2}\Delta_{xyz}F=\Delta_{ab}\psi\). Assertion 1: Each harmonic function \(\psi(x,y)\) induces a homogeneous harmonic function \(F(x,y,z)\) of zero degree in three-dimensional space. Assertion 2: Each homogeneous harmonic function \(F(x,y,z)\) defines a harmonic function of two variables in any plane passing through the origin and that function is specific for each plane. Application the obtained results for finding solutions of second-order partial differential equations is given in two examples.