1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

148 Chapter 2 The Heat Equation Can you think of a physical interpretation of this problem? Note the differ- ence between the pa ...

2.3 Example: Fixed End Temperatures 149 This problem describes the diffusion of a substance in a medium that is moving with spe ...

150 Chapter 2 The Heat Equation anddeterminedthatwsatisfies the boundary value–initial value problem ∂^2 w ∂x^2 = 1 k ∂w ∂t,^0 & ...

2.3 Example: Fixed End Temperatures 151 allxin the interval 0<x0, the common value of the two sides must be a constant, varyi ...

152 Chapter 2 The Heat Equation Ifφhas the form given in the preceding, the boundary conditions require that φ( 0 )=c 1 =0, leav ...

2.3 Example: Fixed End Temperatures 153 Notice that the choice of the coefficientsbndoes not enter into the check- ing of the pa ...

154 Chapter 2 The Heat Equation w(a,t)= 0 , 0 <t, w(x, 0 )=−T 0 −(T 1 −T 0 )x a ≡g(x), 0 <x<a. According to the precedi ...

2.3 Example: Fixed End Temperatures 155 Figure 3 The solution of the example withT 1 =100 andT 0 =20. The function u(x,t)is grap ...

156 Chapter 2 The Heat Equation 5.g(x)=T 0 (constant). 6.g(x)=βx(βis constant). 7.g(x)=β(a−x)(βis constant). 8.g(x)=    2 ...

2.4 Example: Insulated Bar 157 e.Use the formula indto findtexplicitly fora= 5 × 10 −^6 m,D= 10 −^11 cm^2 /s. Be careful to chec ...

158 Chapter 2 The Heat Equation and separate these equalities into two ordinary differential equations linked by the common para ...

2.4 Example: Insulated Bar 159 the valuesπ/a, 2 π/a, 3 π/a,.... We label the eigenvalues with a subscript: λ^2 n= ( nπ a ) 2 ,φn ...

160 Chapter 2 The Heat Equation Figure 4 The solution of the example,u(x,t), as a function ofxfor several times. The initial tem ...

2.4 Example: Insulated Bar 161 EXERCISES Using the initial condition u(x, 0 )=T 1 x a , 0 <x<a, find the solutionu(x,t)o ...

162 Chapter 2 The Heat Equation c. Show that the functionu(x,t)=A(kt+x^2 / 2 )+Bxsatisfies the heat equation for arbitraryAandBa ...

2.5 Example: Different Boundary Conditions 163 12.This table gives values ofu( 0 ,t)for the functionufound in the example and sh ...

164 Chapter 2 The Heat Equation The boundary conditions take the form φ( 0 )T(t)= 0 , 0 <t, (9) φ′(a)T(t)= 0 , 0 <t. (10) ...

2.5 Example: Different Boundary Conditions 165 As in previous cases, we assemble the general solution of the homogeneous problem ...

166 Chapter 2 The Heat Equation Figure 5 Solution of the example, Eq. (20):u(x,t)is shown as a function ofx for various times, w ...

2.5 Example: Different Boundary Conditions 167 Solve Solve the eigenvalue problem forφ. That is, find the values ofλ^2 for which ...