Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: GENERAL AND PARTICULAR SOLUTIONSWe now tackle the problem of solving some types of second-order PDE withconstant coefficients by seeking solutions that are arbitrary functions of particularcombinations of independent variables, just as we did for first-order equations.Following the discussion of the previous section, we can hope to find suchsolutions only if all the terms of the equation involve the same total numberof differentiations, i.e. all terms are of the same order, although the numberof differentiations with respect to the individual independent variables may bedifferent. This means that in (20.19) we require the constants D, E and F to beidentically zero (we have, of course, already assumed that R(x, y) iszero),sothatwe are now considering only equations of the formA ∂2 u∂x 2 + B ∂2 u∂x∂y + C ∂2 u=0, (20.20)∂y2 where A, B and C are constants. We note that both the one-dimensional waveequation,∂ 2 u∂x 2 − 1 ∂ 2 uc 2 ∂t 2 =0,and the two-dimensional Laplace equation,∂ 2 u∂x 2 + ∂2 u∂y 2 =0,are of this form, but that the diffusion equation,κ ∂2 u∂x 2 − ∂u∂t =0,is not, since it contains a first-order derivative.Since all the terms in (20.20) involve two differentiations, by assuming a solutionof the form u(x, y) =f(p), where p is some unknown function of x and y (or t),we may be able to obtain a common factor d 2 f(p)/dp 2 as the only appearance off on the LHS. Then, because of the zero RHS, all reference to the form of f canbe cancelled out.We can gain some guidance on suitable forms for the combination p = p(x, y)by considering ∂u/∂x when u is given by u(x, y) =f(p), for then∂u∂x = df(p) ∂pdp ∂x .Clearly differentiation of this equation with respect to x (or y) will not lead to asingle term on the RHS, containing f only as d 2 f(p)/dp 2 , unless the factor ∂p/∂xis a constant so that ∂ 2 p/∂x 2 and ∂ 2 p/∂x∂y are necessarily zero. This shows thatp must be a linear function of x. In an exactly similar way p must also be a linearfunction of y, i.e.p = ax + by.If we assume a solution of (20.20) of the form u(x, y) =f(ax+by), and evaluate688