1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

248 Chapter 3 The Wave Equation 6.Find an analytic (integral) solution of this wave problem ∂^2 u ∂x^2 =^1 c^2 ∂^2 u ∂t^2 , −∞&l ...

Miscellaneous Exercises 249 time-varying boundary condition u( 0 ,t)=    sin ( ct a ) , 0 <t<πca, 0 , πa c <t. Ske ...

250 Chapter 3 The Wave Equation 18.Fort<0, water flows steadily through a long pipe connected atx= 0 to a large reservoir and ...

Miscellaneous Exercises 251 Show that ∂u ∂t +V∂u ∂x = 2 V∂v ∂ξ . 23.Assume thatu(x,y,t)has the product form shown in what follow ...

252 Chapter 3 The Wave Equation can be obtained fromφ(x−ct)andφ(x− ̄ct).(Herep= √ ω/ 2 kand ̄cis the complex conjugate ofc. Refe ...

Miscellaneous Exercises 253 u( 0 ,t)= 0 , u(a,t)= 0 , ∂^2 u ∂x^2 (^0 ,t)=^0 , ∂^2 u ∂x^2 (a,t)=^0 ,^0 <t. 32.Show thatφ(x)=si ...

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The Potential Equation CHAPTER 4 4.1 Potential Equation The equation for the steady-state temperature distribution in two dimens ...

256 Chapter 4 The Potential Equation it is clear that the temperature cannot be greater at one point than at all other nearby po ...

Chapter 4 The Potential Equation 257 From these equations we easily find that the Laplacian in polar coordinates is ∇^2 v=∂ (^2) ...

258 Chapter 4 The Potential Equation (a) (b) (c) (d) Figure 1 (a)uis displacement of a membrane; the graph off(x)is an isosceles ...

4.2 Potential in a Rectangle 259 4.2 Potential in a Rectangle One of the simplest and most important problems in mathematical ph ...

260 Chapter 4 The Potential Equation Under the new assumption, Eq. (6) separates into X′′+λ^2 X= 0 , Y′′−λ^2 Y= 0. (8) The first ...

4.2 Potential in a Rectangle 261 must be thenth Fourier sine coefficient off 2 .Sinceanis known,bncan be determined from the fol ...

262 Chapter 4 The Potential Equation (a) (b) Figure 2 (a) Level curves of the solutionu(x,y)of the example problem (see Eq. (12) ...

4.2 Potential in a Rectangle 263 It is evident thatu 1 +u 2 is the solution of the original problem Eqs. (13)– (17). Also, each ...

264 Chapter 4 The Potential Equation 5.Solve the problem ∇^2 u= 0 , 0 <x<a, 0 <y<b, u( 0 ,y)= 0 , u(a,y)= 0 , 0 < ...

4.3 Further Examples for a Rectangle 265 pected) that X′′(x) X(x)=− Y′′(y) Y(y)=constant. The conditions atx=0andx=abecome X′( 0 ...

266 Chapter 4 The Potential Equation We have seen that the success of the separation of variables method depends on having homog ...

4.3 Further Examples for a Rectangle 267 Then at the left and bottom boundaries, we have ∂u ∂x ( 0 ,y)= 0 + 0 , ∂u ∂y (x, 0 )=S+ ...