1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

208 Chapter 2 The Heat Equation u( 0 ,t)=T 0 ,0<t, u(x, 0 )= { 0 , 0 <x<a, T 0 , a<x. 13. ∂^2 u ∂x^2 = 1 k ∂u ∂t,0&l ...

Miscellaneous Exercises 209 are solutions of the heat equation. (These are sometimes called heat polynomials.) Find a linear com ...

210 Chapter 2 The Heat Equation 24.Solve the eigenvalue problem by settingφ(ρ)=ψ(ρ)/ρ: 1 ρ^2 ( ρ^2 φ′ )′ +λ^2 φ= 0 , 0 <ρ< ...

Miscellaneous Exercises 211 30.Prove the following identity: √^1 4 πkt ∫a b exp [ −(ξ−x) 2 4 kt ] dξ=^12 [ erf ( b√−x 4 kt ) −er ...

212 Chapter 2 The Heat Equation In these equations,y(x,t)is the water table elevation above sea level, h(x,t)is water table elev ...

Miscellaneous Exercises 213 P.W. Carr [Fourier analysis of the transient response of potentiomet- ric enzyme electrodes,Analytic ...

214 Chapter 2 The Heat Equation Definingu=S+P, find the boundary and initial conditions foru,and solve completely. Then findP(x, ...

The Wave Equation CHAPTER 3 3.1 The Vibrating String A simple and historically important example of a problem that includes the ...

216 Chapter 3 The Wave Equation Figure 1 String fixed at the ends. Figure 2 Section of string showing forces exerted on it. The ...

Chapter 3 The Wave Equation 217 When these expressions are substituted into Eq. (2), we have −Ttan ( φ(x,t) ) +Ttan ( φ(x+ x,t) ...

218 Chapter 3 The Wave Equation the string is ∂^2 u ∂x^2 =^1 c^2 ∂^2 u ∂t^2 , 0 <x<a, 0 <t, (7) u( 0 ,t)= 0 , u(a,t)= 0 ...

3.2 Solution of the Vibrating String Problem 219 ∂^2 u ∂x^2 =^1 c^2 ∂^2 u ∂t^2 , 0 <x<a, 0 <t, (1) u( 0 ,t)= 0 , u(a,t) ...

220 Chapter 3 The Wave Equation whereanandbnarearbitrary.(Inotherwords,therearetwoindependentsolu- tions.) Note, however, that t ...

3.2 Solution of the Vibrating String Problem 221 and bn nπ ac= 2 a ∫a 0 g(x)sin (nπx a ) dx or bn=nπ^2 c ∫a 0 g(x)sin ( nπx a ) ...

222 Chapter 3 The Wave Equation By applying the trigonometric identity sin(A)cos(B)=^1 2 [ sin(A−B)+sin(A+B) ] we can expressu(x ...

3.2 Solution of the Vibrating String Problem 223 Figure 3 On the left are the graphs off ̄o(x+ct)(solid) and ̄fo(x−ct)(dashed) f ...

224 Chapter 3 The Wave Equation tare frequencies, in radians per unit time;λnc/ 2 πare frequencies in cycles per unit time (or H ...

3.2 Solution of the Vibrating String Problem 225 u(x,t)=^1 2 ( ̄ fo(x−ct)+f ̄o(x+ct) ) +^1 2 ( ̄ Ge(x+ct)−G ̄e(x−ct) ) . Here, ̄ ...

226 Chapter 3 The Wave Equation Figure 4 Shapes of car antenna. eachequalto256cyclespersecond.Thedifferenceinthesetoffrequen- ci ...

3.3 d’Alembert’s Solution 227 Solve the eigenvalue problem, sketch the first two eigenfunctions, and compare them to the figure. ...