1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

388 Chapter 6 Laplace Transform EXERCISES 1.Find the persistent part of the solution of the heat problem ∂^2 u ∂x^2 = ∂u ∂t,^0 & ...

6.5 Comments and References 389 a. Solve ∂^2 u ∂x^2 =∂u ∂t , 0 <x< 1 , 0 <t, u( 0 ,t)= 0 , u( 1 ,t)= 1 −e−at, 0 <t ...

390 Chapter 6 Laplace Transform 2.Find the “persistent part” of the solution of ∂^2 u ∂x^2 = ∂u ∂t,^0 <x<^1 ,^0 <t, ∂u ...

Miscellaneous Exercises 391 9.Find the transform of the solution of ∂^2 u ∂x^2 =∂u ∂t , 0 <x, 0 <t, u( 0 ,t)= 0 , 0 <t, ...

392 Chapter 6 Laplace Transform 16.Suppose that the functionf(t)is periodic with period 2a. Show that the Laplace transform offi ...

Miscellaneous Exercises 393 ∂^2 u ∂x^2 =∂ (^2) u ∂t^2 , 0 <x, 0 <t, u( 0 ,t)=h(t), 0 <t, u(x, 0 )= 0 , ∂u ∂t(x,^0 )=^0 ...

394 Chapter 6 Laplace Transform Next, the equations are made dimensionless by introducing new vari- ables: C ̄=C−Ca A , x ̄= Vx ...

Miscellaneous Exercises 395 so that α^2 −β^2 + 2 iαβ= 1 4 +iω, or { α^2 −β^2 =^14 , 2 αβ=ω. (To solve these equations: (i) solve ...

This page intentionally left blank ...

Numerical Methods CHAPTER 7 7.1 Boundary Value Problems More often than not, significant practical problems in partial — and eve ...

398 Chapter 7 Numerical Methods x:0. 00. 20. 40. 60. 81. 0 u(x):1. 00. 643 0. 302 − 0. 026 − 0. 406 − 1. 0 Table 1 Approximate s ...

Chapter 7 Numerical Methods 399 of Eq. (3) are 25 (u 2 − 2 u 1 +u 0 )−^125 u 1 =− 1 , 25 (u 3 − 2 u 2 +u 1 )− 24 5 u^2 =−^1 , 25 ...

400 Chapter 7 Numerical Methods We need to knowu 0 ,u 1 ,...,un. The derivative boundary condition atx= 1 forces us to includeun ...

Chapter 7 Numerical Methods 401 x:00. 25 0. 50. 75 1 u(n=4): 1 2. 174 4. 707 7. 057 7. 567 u(n=100): 1 2. 155 4. 729 7. 125 7. 6 ...

402 Chapter 7 Numerical Methods Thus, the values ofuatx 0 ,x 1 ,...,xn, which satisfy Eq. (19) exactly, will nearly satisfy Eq. ...

7.2 Heat Problems 403 Usen=3andn=4. Sketch the results and explain why they vary so much. In Exercises 8–11, set up and solve re ...

404 Chapter 7 Numerical Methods The spatial derivatives in a heat problem will be replaced by difference quo- tients, as before: ...

7.2 Heat Problems 405 i m 01234 0 00. 25 0. 50. 75 1 1 0 0. 25 0. 50. 75 0 2 0 0. 25 0. 50. 25 0 3 0 0. 25 0. 25 0. 25 0 4 0 0. ...

406 Chapter 7 Numerical Methods i m 01 2 34 0 00 .25 0.50 0.75 1 1 0 0. 25 0. 50 0. 75 0 2 0 0. 25 0. 50 − 0. 25 0 3 0 0. 25 − 0 ...

7.2 Heat Problems 407 the replacement equations are found to be (forn=4) u 1 (m+ 1 )=ru 0 (m)+( 1 − 2 r)u 1 (m)+ru 2 (m), u 2 (m ...