The Cauchy Problem for Partial Differential Equations of the Second Order and the Method of Ascent
United States Air Force, Office of Scientific Research, 1961 - Differential equations, Partial - 112 pages
A method of ascent is used to solve the Cauchy problem for linear partial differential equations of the second order in p space variables with constant coefficients i.e., the pure wave equation, the damped wave equation, and the heat equation. This method consists of inferring the solution of the problem referred to from the well known solution of the same problem for one space variable. The commutability of repeated pf integral, the solution deduced by the method of singularities for the Cauchy problem for the damped wave equation, and the solution of singular integral equations of the Volterra type are also considered. (Author).
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The wave equation
The damped wave equation
The heat equation
1 other sections not shown
_k+l 2k+2 dx 2k+l 2pu.(p d pda a2k+l abbreviate ap-l Appendix at2k Cauchy problem 14-15 ck+1 Commutability of repeated consider damped wave equation define denote divergent integrals Dk+1 dp d a drp-l dT f dvk d a dx pf dx x-y dy pf f 2k+2J f e kt f g(y f t2 f u(y f w(r,a f X U(y finally obtains finite fX f'(y heat equation hypersphere induction integer Lemma lim _ U(r;t method of ascent Numbers partial differential equations pf f IQ Pf rfc pff U(y PffX U(y polynomial problem 2-3 Proof r w(r r2k+1 dr regular function singular integral equations solution of 9 solve the Cauchy space variables starting spherical mean value Suppose u(x+r a u(xjt vk+1 Volterra type Wk(a