Mathematical Methods for Physics and Engineering : A Comprehensive Guide

20.3 GENERAL AND PARTICULAR SOLUTIONS We could, of course, have takenh(x, y)=y, but this only leads to a solution that is alread ...

PDES: GENERAL AND PARTICULAR SOLUTIONS Then, providing the factor in brackets does not vanish, for which the required condition ...

20.4 THE WAVE EQUATION 20.4 The wave equation We have already found that the general solution of the one-dimensional wave equati ...

PDES: GENERAL AND PARTICULAR SOLUTIONS discussed in the next chapter. Nevertheless, we now considerD’Alembert’s solution u(x, t) ...

20.5 THE DIFFUSION EQUATION term is a little less obvious. It can be viewed as representing the accumulated transverse displacem ...

PDES: GENERAL AND PARTICULAR SOLUTIONS in whichκis a constant with the dimensions length^2 ×time−^1. The physical constants that ...

20.5 THE DIFFUSION EQUATION written entirely in terms ofη, 4 η d^2 f(η) dη^2 +(2+η) df(η) dη =0. This is a straightforward ODE, ...

PDES: GENERAL AND PARTICULAR SOLUTIONS
An infrared laser delivers a pulse of (heat) energyEto a pointPon a large insulated shee ...

20.6 CHARACTERISTICS AND THE EXISTENCE OF SOLUTIONS 20.6 Characteristics and the existence of solutions So far in this chapter w ...

PDES: GENERAL AND PARTICULAR SOLUTIONS were discussed. Comparing (20.41) with (20.12) we see that the characteristics are merely ...

20.6 CHARACTERISTICS AND THE EXISTENCE OF SOLUTIONS 1 1 2 y x − 1 x=1 c=1 y=c/x^2 Figure 20.2 The characteristics of equation (2 ...

PDES: GENERAL AND PARTICULAR SOLUTIONS C y x dx dy nˆds dr Figure 20.4 A boundary curveCand its tangent and unit normal at a giv ...

20.6 CHARACTERISTICS AND THE EXISTENCE OF SOLUTIONS we may differentiate the two first derivatives∂u/∂xand∂u/∂yalong the boundar ...

PDES: GENERAL AND PARTICULAR SOLUTIONS 0 ct x+ct=constant x−ct=constant L x Figure 20.5 The characteristics for theone-dimension ...

20.7 UNIQUENESS OF SOLUTIONS Equation type Boundary Conditions hyperbolic open Cauchy parabolic open Dirichlet or Neumann ellipt ...

PDES: GENERAL AND PARTICULAR SOLUTIONS As an important example let us consider Poisson’s equation in three dimensions, ∇^2 u(r)= ...

20.8 EXERCISES We also note that often the same general method, used in the above example for proving the uniqueness theorem for ...

PDES: GENERAL AND PARTICULAR SOLUTIONS (b)y ∂u ∂x −x ∂u ∂y =3x, u(1,0) = 2; (c) y^2 ∂u ∂x +x^2 ∂u ∂y =x^2 y^2 (x^3 +y^3 ), no bo ...

20.8 EXERCISES 20.14 Solve ∂^2 u ∂x∂y +3 ∂^2 u ∂y^2 =x(2y+3x). 20.15 Find the most general solution of∂^2 u/∂x^2 +∂^2 u/∂y^2 =x^ ...

PDES: GENERAL AND PARTICULAR SOLUTIONS (b) The tube initially has a small transverse displacementu=acoskxand is suddenly release ...