Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: GENERAL AND PARTICULAR SOLUTIONSdescribes the quantum mechanical wavefunction u(r,t) of a non-relativistic particleof mass m; is Planck’s constant divided by 2π. Like the diffusion equation it issecond order in the three spatial variables and first order in time.20.2 General form of solutionBefore turning to the methods by which we may hope to solve PDEs such asthose listed in the previous section, it is instructive, as for ODEs in chapter 14, tostudy how PDEs may be formed from a set of possible solutions. Such a studycan provide an indication of how equations obtained not from possible solutionsbut from physical arguments might be solved.For definiteness let us suppose we have a set of functions involving twoindependent variables x and y. Without further specification this is of course avery wide set of functions, and we could not expect to find a useful equation thatthey all satisfy. However, let us consider a type of function u i (x, y) inwhichx andy appear in a particular way, such that u i can be written as a function (howevercomplicated) of a single variable p, itself a simple function of x and y.Let us illustrate this by considering the three functionsu 1 (x, y) =x 4 +4(x 2 y + y 2 +1),u 2 (x, y) =sinx 2 cos 2y +cosx 2 sin 2y,u 3 (x, y) = x2 +2y +23x 2 +6y +5 .These are all fairly complicated functions of x and y and a single differentialequation of which each one is a solution is not obvious. However, if we observethat in fact each can be expressed as a function of the variable p = x 2 +2y alone(withnootherx or y involved) then a great simplification takes place. Writtenin terms of p the above equations becomeu 1 (x, y) =(x 2 +2y) 2 +4=p 2 +4=f 1 (p),u 2 (x, y) =sin(x 2 +2y) =sinp = f 2 (p),u 3 (x, y) = (x2 +2y)+23(x 2 +2y)+5 = p +23p +5 = f 3(p).Let us now form, for each u i , the partial derivatives ∂u i /∂x and ∂u i /∂y. Ineachcase these are (writing both the form for general p and the one appropriate toour particular case, p = x 2 +2y)∂u i∂x = df i(p) ∂pdp ∂x =2xf′ i,∂u i∂y = df i(p) ∂pdp ∂y =2f′ i,for i = 1, 2, 3. All reference to the form of f i can be eliminated from these680