1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

308 Chapter 5 Higher Dimensions and Other Coordinates 6.Supposethat,insteadofboundaryconditionsEqs.(2)and(3),wehave u(x, 0 ,t)=f ...

5.4 Problems in Polar Coordinates 309 If we consider now the vibrations of a circular membrane or heat conduc- tion in a circula ...

310 Chapter 5 Higher Dimensions and Other Coordinates R(r)Q(θ ). After some algebra, we find that (rR′)′ rR + Q′′ r^2 Q=−λ (^2) ...

5.5 Bessel’s Equation 311 Suppose the problems originally stated were to be solved in the half-disk 0 <r<a,0<θ <π, ...

312 Chapter 5 Higher Dimensions and Other Coordinates When the differentiations in Eq. (1) are carried out and the equation is m ...

5.5 Bessel’s Equation 313 and so forth. Allc’s with odd index are zero, since they are all multiples ofc 1. The general formula ...

314 Chapter 5 Higher Dimensions and Other Coordinates Figure 7 Graphs of Bessel functions of the first kind. Also see the CD. (a ...

5.5 Bessel’s Equation 315 Using the same method as in the preceding, an infinite series can be developed for the solutions (see ...

316 Chapter 5 Higher Dimensions and Other Coordinates 5.By using Exercise 4 and Rolle’s theorem, and knowing thatJ 0 (x)=0 for a ...

5.6 Temperature in a Cylinder 317 of the angular coordinateθ.(Wewritev(r,t)then.) As a consequence of this assumption, the two-d ...

318 Chapter 5 Higher Dimensions and Other Coordinates The functionφ(r)=J 0 (λr)is a solution of Eq. (6), and we wish to chooseλ ...

5.6 Temperature in a Cylinder 319 Theorem.If f(r)is sectionally smooth on the interval 0 <r<a, then at every point r on th ...

320 Chapter 5 Higher Dimensions and Other Coordinates n αn J 1 (αn) α^2 nJ 1 (αn) 1 2.405 + 0. 5191 + 1. 6020 2 5.520 − 0. 3403 ...

5.7 Vibrations of a Circular Membrane 321 EXERCISES Use Eq. (18) to find an expression for the functionv( 0 ,t)/T 0 .Evaluateth ...

322 Chapter 5 Higher Dimensions and Other Coordinates v(r, 0 )=f(r), 0 <r<a, (3) ∂v ∂t (r, 0 )=g(r), 0 <r<a. (4) We ...

5.7 Vibrations of a Circular Membrane 323 The rest of our problem can now be dispatched easily. Returning to Eq. (5), we see tha ...

324 Chapter 5 Higher Dimensions and Other Coordinates u(r,θ, 0 )=f(r,θ), 0 <r<a, (15) ∂u ∂t (r,θ, 0 )=g(r,θ), 0 <r<a ...

5.7 Vibrations of a Circular Membrane 325 In order for the boundedness condition in Eq. (20) to be fulfilled,Dmust be zero. Then ...

326 Chapter 5 Higher Dimensions and Other Coordinates several series to form the combination: u(r,θ,t)= ∑ n a 0 nJ 0 (λ 0 nr)cos ...

5.7 Vibrations of a Circular Membrane 327 There are two other relations like this one involving functions from two dif- ferent s ...