1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

268 Chapter 4 The Potential Equation Poisson Equation Many problems in engineering and physics require the solution of the Poiss ...

4.3 Further Examples for a Rectangle 269 Thus, we may setu(x,y)=v(x)+w(x,y)and determine thatwis a solution of the problem ∂^2 w ...

270 Chapter 4 The Potential Equation 6.Explain the difference between the cosine series in Example 1 and the co- sine series for ...

4.4 Potential in Unbounded Regions 271 In order to make the separation of variables work, we must break this up into two problem ...

272 Chapter 4 The Potential Equation The solution of the second problem is somewhat different. Again we seek solutions in the pr ...

4.4 Potential in Unbounded Regions 273 EXERCISES Find a formula for the constantsanin Eq. (7). Ve r i f y t h a tu 1 (x,y)in th ...

274 Chapter 4 The Potential Equation that cos(a/ 2 )=0: v(x,y)= cos ( x−^12 a ) cos ( 1 2 a ) e−y. What partial differential eq ...

4.5 Potential in a Disk 275 15.Find product solutions of the potential equation in the half-planey>0: ∂^2 u ∂x^2 +∂ (^2) u ∂y ...

276 Chapter 4 The Potential Equation The Dirichlet problem on a disk can now be stated as 1 r ∂ ∂r ( r ∂v ∂r ) + 1 r^2 ∂^2 v ∂θ^ ...

4.5 Potential in a Disk 277 Knowing thatλ^2 n=n^2 ,wecaneasilyfindR(r). The equation forRbecomes r^2 R′′+rR′−n^2 R= 0 , 0 <r& ...

278 Chapter 4 The Potential Equation ThesolutionisgivenbyEq.(10),providedthatthecoefficientsarechosen accordingtoEq.(11).Sincef( ...

4.5 Potential in a Disk 279 principle,whas maximum and minimum values 0, and thereforewis identi- cally 0 throughoutR.Inotherwor ...

280 Chapter 4 The Potential Equation e.add up the series (see Section 1.10, Exercise 5a). Thenv(r,θ)is given by the single integ ...

4.6 Classification and Limitations 281 whereA,B,C,and so forth are, in general, functions ofξandη.(WeuseGreek letters for the in ...

282 Chapter 4 The Potential Equation homogeneous or “homogeneous-like” conditionsonoppositesidesofagener- alized rectangle. Exam ...

4.7 Comments and References 283 u(x, 0 )=f(x), ∂∂uy(x, 0 )=0, 0 <x<1, u( 0 ,y)=0, u( 1 ,y)=0, 0 <y; c. ∂ (^2) u ∂x^2 =∂ ...

284 Chapter 4 The Potential Equation Figure 3 Streamlines (solid) and equipotential curves (dashed) for flow in a cor- ner. The ...

Miscellaneous Exercises 285 to get a rough graphical solution of the potential equation for hydrodynamics problems. Where a phys ...

286 Chapter 4 The Potential Equation 5.Same as Exercise 3, but the boundary conditions are u( 0 ,y)= 1 , u(a,y)= 1 , 0 <y< ...

Miscellaneous Exercises 287 12.Apply the following formula (see Section 4.4, Exercise 16) for the solu- tion of the potential pr ...