1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

188 Chapter 2 The Heat Equation EXERCISES 1.Find the explicit form forv(x)in terms of the function in Eq. (8) assuming a. α 1 =β ...

2.10 Semi-Infinite Rod 189 differential equation can be separated into two ordinary equations as usual: φ′′(x) φ(x)= T′(t) kT(t) ...

190 Chapter 2 The Heat Equation we must use an integral — the continuous analogue of a sum or series — to include all possibilit ...

2.10 Semi-Infinite Rod 191 Figure 8 Graphs of the solution of the example,u(x,t)as a function ofxover the interval 0<x< 3 ...

192 Chapter 2 The Heat Equation ∂^2 u ∂x^2 =^1 k ∂u ∂t , 0 <x, 0 <t, ∂u ∂x ( 0 ,t)= 0 , 0 <t, u(x, 0 )=f(x), 0 <x. 5 ...

2.11 Infinite Rod 193 differential equation is ∂C ∂t =D (∂ (^2) C ∂x^2 −a^2 C ) , whereCis the concentration of SO 2 as a functi ...

194 Chapter 2 The Heat Equation Using the same techniques as before, we look for solutions in the form u(x,t)=φ(x)T(t)so that th ...

2.11 Infinite Rod 195 Figure 9 Solution of example problem. Att=0, the temperature isT 0 >0for −a<x<aand is 0 in the re ...

196 Chapter 2 The Heat Equation substitute the formulas forA(λ)andB(λ)into Eq. (4): u(x,t)= 1 π ∫∞ 0 [∫∞ −∞ f(x′)cos(λx′)dx′cos( ...

2.11 Infinite Rod 197 in the integrand of Eq. (4) will be nearly zero, except for smallλ.Thus,Eq.(4) is approximately u(x,t)∼= ∫ ...

198 Chapter 2 The Heat Equation u( 0 ,t)= 0 , 0 <t, u(x, 0 )=f(x), 0 <x, can be expressed as u(x,t)=√^1 4 πkt ∫∞ 0 f(x′) [ ...

2.12 The Error Function 199 Figure 10 Graph of the error function erf(z)for− 3 <z<3. 2.12 The Error Function InSection11we ...

200 Chapter 2 The Heat Equation And finally, the error function supplies the integral ∫b a e−y^2 dy= √ π 2 ( erf(b)−erf(a) ) . ( ...

2.12 The Error Function 201 We are interested in the error function because of its role in solving the heat equation. First we s ...

202 Chapter 2 The Heat Equation (a) (b) Figure 11 (a) Graph of exp(−y^2 )and (b) graph of sgn(x+y √ 4 kt)exp(−y^2 ). The tails b ...

2.12 The Error Function 203 EXERCISES Show that erf(−z)=−erf(z),thatis,thaterfisanoddfunction. Carry out the integration indica ...

204 Chapter 2 The Heat Equation u( 0 ,t)=Ub, 0 <t, u(x, 0 )=Ui, 0 <x. (Hint: What conditions doesu(x,t)−Ubsatisfy?) 9.Assu ...

2.13 Comments and References 205 i m− 3 − 2 − 10123 00 0 0 1 0 0 10 0 0. 500 .50 0 20 0. 25 0 0. 50 0. 25 0 30. 125 0 0. 375 0 0 ...

206 Chapter 2 The Heat Equation the ordinary heat equation has this form if we takeA=1,E=− 1 /kand all other coefficients equal ...

Miscellaneous Exercises 207 ∂u ∂x(^0 ,t)=0, ∂u ∂x(a,t)=0,^0 <t, u(x, 0 )=Ta^1 x,0<x<a. ∂ (^2) u ∂x^2 =^1 k ∂u ∂t ,0&l ...