Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: GENERAL AND PARTICULAR SOLUTIONSwill be satisfactory solutions of the equation and that the general solution will beu(x, t) =f(x − ct)+g(x + ct), (20.23)where f and g are arbitrary functions. This solution is discussed further in section 20.4. ◭The method used to obtain the general solution of the wave equation may alsobe applied straightforwardly to Laplace’s equation.◮ Find the general solution of the two-dimensional Laplace equation∂ 2 u∂x + ∂2 u=0. (20.24)2 ∂y2 Following the established procedure, we look for a solution that is a function f(p) ofp = x + λy, where from (20.24) λ satisfies1+λ 2 =0.This requires that λ = ±i, and satisfactory variables p are p = x ± iy. The general solutionrequired is therefore, in terms of arbitrary functions f and g,u(x, y) =f(x + iy)+g(x − iy). ◭It will be apparent from the last two examples that the nature of the appropriatelinear combination of x and y depends upon whether B 2 > 4AC or B 2 < 4AC.This is exactly the same criterion as determines whether the PDE is hyperbolicor elliptic. Hence as a general result, hyperbolic and elliptic equations of theform (20.20), given the restriction that the constants A, B and C are real, have assolutions functions whose arguments have the form x+αy and x+iβy respectively,where α and β themselves are real.The one case not covered by this result is that in which B 2 =4AC, i.e.aparabolic equation. In this case λ 1 and λ 2 are not different and only one suitablecombination of x and y results, namelyu(x, y) =f(x − (B/2C)y).To find the second part of the general solution we try, in analogy with thecorresponding situation for ordinary differential equations, a solution of the formu(x, y) =h(x, y)g(x − (B/2C)y).Substituting this into (20.20) and using A = B 2 /4C results in()A ∂2 h∂x 2 + B ∂2 h∂x∂y + C ∂2 h∂y 2 g =0.Therefore we require h(x, y) to be any solution of the original PDE. There areseveral simple solutions of this equation, but as only one is required we take thesimplest non-trivial one, h(x, y) =x, to give the general solution of the parabolicequationu(x, y) =f(x − (B/2C)y)+xg(x − (B/2C)y). (20.25)690