1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

168 Chapter 2 The Heat Equation u( 0 ,t)=T 0 , ∂∂ux(a,t)= 0 , 0 <t, u(x, 0 )=T 0 , 0 <x<a. 6.Solve this problem for the ...

2.5 Example: Different Boundary Conditions 169 construct the functionG(x)with these properties: G(x)=g(x), 0 <x<a, G(x)=g( ...

170 Chapter 2 The Heat Equation In these equations,Dis the diffusion constant,Lis the length of the cylin- der, andC 0 is the sa ...

2.6 Example: Convection 171 w( 0 ,t)= 0 , hw(a,t)+κ∂w∂x(a,t)= 0 , 0 <t, (7) w(x, 0 )=f(x)−v(x)≡g(x), 0 <x<a. (8) The so ...

172 Chapter 2 The Heat Equation A n 0. 2500 0. 5000 1. 0000 2. 0000 4. 0000 12. 5704 2. 2889 2. 0288 1. 8366 1. 7155 25. 3540 5. ...

2.6 Example: Convection 173 that makeswn(x,t)=φn(x)Tn(t)a solution of the partial differential equa- tion (6) and the boundary c ...

174 Chapter 2 The Heat Equation (a) (b) Figure 7 Solution of Eqs. (1)–(4) withT 0 =20,T 1 =100, andf(x)=0. Graphs (a) and (b) co ...

2.7 Sturm–Liouville Problems 175 Find the coefficientsbmcorresponding to g(x)= 1 , 0 <x<a. Using the solution of Exercise ...

176 Chapter 2 The Heat Equation which is proved true if the left-hand side is zero: ∫r l ( φn′′φm−φ′′mφn ) dx= 0. This integral ...

2.7 Sturm–Liouville Problems 177 Let us carry out the procedure used in the preceding with this problem. The eigenfunctions sati ...

178 Chapter 2 The Heat Equation is called aregular Sturm–Liouville problemif the following conditions are ful- filled: a.s(x),s′ ...

2.7 Sturm–Liouville Problems 179 this section. In particular, the problem φ′′+λ^2 φ= 0 , 0 <x<a, φ( 0 )= 0 , hφ(a)+κφ′(a)= ...

180 Chapter 2 The Heat Equation with boundary conditions a. φ( 0 )=0, φ′(a)=0, b. φ′( 0 )=0, φ(a)=0, c. φ( 0 )=0, φ(a)+φ′(a)=0, ...

2.8 Expansion in Series of Eigenfunctions 181 Showthat0isaneigenvalueoftheproblem (sφ′)′+λ^2 pφ= 0 , l<x<r, φ′(l)= 0 ,φ′ ...

182 Chapter 2 The Heat Equation mis a fixed integer) and integrating fromltoryields ∫r l f(x)φm(x)p(x)dx= ∑∞ n= 1 cn ∫r l φn(x)φ ...

2.8 Expansion in Series of Eigenfunctions 183 (xφ′)′+λ^2 ( 1 x ) φ= 0 , 1 <x<b, φ( 1 )= 0 ,φ(b)= 0. Find the expansion of ...

184 Chapter 2 The Heat Equation 6.Show that, for the function in Exercise 5, ∫r l f^2 (x)p(x)dx= ∑∞ n= 1 b^2 n. 7.What are the n ...

2.9 Generalities on the Heat Conduction Problem 185 Assuming thatc 1 andc 2 are constants, we must first find the steady-state s ...

186 Chapter 2 The Heat Equation and, on dividing through bypφT,wefindtheseparatedequation (sφ′)′ pφ =T ′ kT , l<x<r, 0 < ...

2.9 Generalities on the Heat Conduction Problem 187 We now begin to assemble the solution. For eachn= 1 , 2 , 3 ,...,wn(x,t) =φn ...