1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

408 Chapter 7 Numerical Methods ∂ (^2) u ∂x^2 =∂u ∂t , u( 0 ,t)=0, ∂u ∂x ( 1 ,t)+u( 1 ,t)=1, u(x, 0 )=0. 9. ∂^2 u ∂x^2 = ∂u ∂t ...

7.3 Wave Equation 409 valid fori= 1 , 2 ,...,n−1. Naturally, the boundary conditions, Eq. (2), carry over asu 0 (m)=0,un(m)=0. I ...

410 Chapter 7 Numerical Methods i m 01234 0 00 .51 0. 50 1 0 0. 50. 50. 5 0 2 0 0000 3 0 − 0. 5 − 0. 5 − 0. 5 0 4 0 − 0. 5 − 1 − ...

7.3 Wave Equation 411 i m 01234 0 00 .51 0. 50 1 0 0 .50 0. 5 0 2 0 − 1. 51 − 1. 5 0 3 0 4. 5 − 84. 5 0 4 0 − 23. 533 − 23. 5 0 ...

412 Chapter 7 Numerical Methods i m 01234 0 00000 1 0 0. 50. 50. 5 0 2 0 1. 21 1. 71 1. 21 0 3 0 1. 21 1. 91 1. 21 0 4 0 0. 00 0 ...

7.3 Wave Equation 413 Compare the results of Exercise 1 with the d’Alembert solution. Obtain an approximate solution of Eqs. (1 ...

414 Chapter 7 Numerical Methods 7.4 Potential Equation In this section, we will be concerned with approximate solutions of the p ...

7.4 Potential Equation 415 Figure 1 Pointion a square mesh and its four neighbors. Figure 2 Numbering for mesh points, and value ...

416 Chapter 7 Numerical Methods Figure 3 Numerical solution of Eqs. (3)–(6). Example. Set up the replacement equations for the p ...

7.4 Potential Equation 417 Figure 4 Mesh numbering forL-shaped region. tem of equations to be solved will be rather less regular ...

418 Chapter 7 Numerical Methods Iterative Methods Systems of up to 10 equations, such as those in the foregoing examples, can re ...

7.4 Potential Equation 419 Figure 5 Regions and mesh numbering for Exercises 5–9. EXERCISES Set up and solve replacement equatio ...

420 Chapter 7 Numerical Methods and the upper halves of the vertical sides), andu=0 on the lower half. Take x= y= 1 /3. See Fig. ...

7.5 Two-Dimensional Problems 421 Figure 6 Mesh numbering for numerical solution of Eqs. (2)–(5). We t a k e x= y= 1 /4 and numbe ...

422 Chapter 7 Numerical Methods i m 1256 01111 1 12 34 34 1 (^238169121316) (^3176416725643964) Table 9 Numerical solution of Eq ...

7.5 Two-Dimensional Problems 423 Figure 7 Mesh numbering for numerical solution of Eqs. (9)–(11). Because x= y= 1 /5, the parame ...

424 Chapter 7 Numerical Methods i m 1234567 f(tm) 00000000 0 10000000 1. 0 20. 50. 25 0. 25 0. 50. 50. 25 0 2. 0 31. 22 0. 75 0. ...

7.5 Two-Dimensional Problems 425 The running equation is Eq. (20), which, withρ^2 = 1 /2, simplifies to ui(m+ 1 )= 1 2 [ uE(m)+u ...

426 Chapter 7 Numerical Methods Figure 8 Displacements of the square membrane. Numbers shown are ui(m)×64. The regionRis an inve ...

7.5 Two-Dimensional Problems 427 Figure 9 Regions for Exercises 1–3. Same as Exercise 2, except that the region is a cross. (Se ...