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Symmetry group methods for fundamental solutions. (English) Zbl 1065.35016

The authors use Lie symmetry group methods to find fundamental solutions for a class of partial differential equations of the form \(u_t=xu_{xx}+f(x)u_x\), \(x\geq 0\), i.e. to find the kernel function \(p(t,x,y)\), such that \(u(x,t)=\int_0^\infty\varphi(y)p(t,x,y)\,dy\) is a solution of the relevant Cauchy problem with \(u(x,0)=\varphi(x)\).

MSC:

35A30 Geometric theory, characteristics, transformations in context of PDEs
35K55 Nonlinear parabolic equations
58J70 Invariance and symmetry properties for PDEs on manifolds
35K65 Degenerate parabolic equations
35A08 Fundamental solutions to PDEs
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References:

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