Mathematical Methods for Physics and Engineering - Matematica.NET

21.2 SUPERPOSITION OF SEPARATED SOLUTIONSIn order to satisfy the boundary condition u → 0ast →∞, λ 2 κ must be > 0. Since κis real and > 0, this implies that λ is a real non-zero number and that the solution issinusoidal in x and is not a disguised hyperbolic function; this was our reason for choosingthe separation constant as −λ 2 . ◭As a final example we consider Laplace’s equation in Cartesian coordinates;this may be treated in a similar manner.◮Use the method of separation of variables to obtain a solution for the two-dimensionalLaplace equation,∂ 2 u∂x + ∂2 u=0. (21.13)2 ∂y2 If we assume a solution of the form u(x, y) =X(x)Y (y) then, following the above method,and taking the separation constant as λ 2 , we findX ′′ = λ 2 X, Y ′′ = −λ 2 Y.Taking λ 2 as > 0, the general solution becomesu(x, y) =(A cosh λx + B sinh λx)(C cos λy + D sin λy). (21.14)An alternative form, in which the exponentials are written explicitly, may be useful forother geometries or boundary conditions:u(x, y) =[A exp λx + B exp(−λx)](C cos λy + D sin λy), (21.15)with different constants A and B.If λ 2 < 0 then the roles of x and y interchange. The particular combination of sinusoidaland hyperbolic functions and the values of λ allowed will be determined by the geometricalproperties of any specific problem, together with any prescribed or necessary boundaryconditions. ◭We note here that a particular case of the solution (21.14) links up with the‘combination’ result u(x, y) =f(x + iy) of the previous chapter (equations (20.24)and following), namely that if A = B and D = iC then the solution is the sameas f(p) =AC exp λp with p = x + iy.21.2 Superposition of separated solutionsIt will be noticed in the previous two examples that there is considerable freedomin the values of the separation constant λ, the only essential requirement beingthat λ has the same value in both parts of the solution, i.e. the part dependingon x and the part depending on y (or t). This is a general feature for solutionsin separated form, which, if the original PDE has n independent variables, willcontain n − 1 separation constants. All that is required in general is that weassociate the correct function of one independent variable with the appropriatefunctions of the others, the correct function being the one with the same valuesof the separation constants.If the original PDE is linear (as are the Laplace, Schrödinger, diffusion andwave equations) then mathematically acceptable solutions can be formed by717