1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

348 Chapter 5 Higher Dimensions and Other Coordinates Figure 15 Graphs of the solution of the example problem, withu(φ, 0 )posit ...

5.10 Some Applications of Legendre Polynomials 349 Again we see that the ratio containingmust be constant, say,−μ^2 .Hence, we ...

350 Chapter 5 Higher Dimensions and Other Coordinates sin(λa)/λa=0, soλm=mπ/aform= 1 , 2 ,....Forothern’s, solutions of Jn+ 1 / ...

5.10 Some Applications of Legendre Polynomials 351 Solve the potential equation in a hemisphere, 0<ρ<1, 0<φ<π/2, su ...

352 Chapter 5 Higher Dimensions and Other Coordinates ifλis the second positive solution ofJ 5 / 2 (λ)= 0. 8.Solve the potential ...

5.11 Comments and References 353 5.11 Comments and References We have seen just a few problems in two or three dimensions, but t ...

354 Chapter 5 Higher Dimensions and Other Coordinates vibrating membrane, will be found inThe Physics of Musical Instruments,by ...

Miscellaneous Exercises 355 both directly and by assuming that bothu(r)and the constant function 1 haveBesselseriesontheinterval ...

356 Chapter 5 Higher Dimensions and Other Coordinates hasasitssolution φ(x)= ∫x 0 1 1 −y^2 ∫ 1 y f(z)dz dy, provided that the fu ...

Miscellaneous Exercises 357 15.Solve the following potential problem in a cylinder: 1 r ∂ ∂r ( r ∂u ∂r ) + ∂^2 u ∂z^2 =^0 ,^0 &l ...

358 Chapter 5 Higher Dimensions and Other Coordinates 20.Observe that the functionφin Exercise 19 is the difference of two diffe ...

Miscellaneous Exercises 359 wheregis the acceleration of gravity andUis mean depth. The tidal motion of the sea is represented b ...

360 Chapter 5 Higher Dimensions and Other Coordinates Here,DrandDzare the diffusion constants in the radial and axial di- rectio ...

Miscellaneous Exercises 361 andthecoefficientsshouldbedeterminedby a 0 = 2 ∫ 1 0 f(ρ)ρdρ, an= ∫ 1 (^0) ∫f(ρ)J^0 (λnρ)ρdρ 1 0 J^2 ...

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Laplace Transform CHAPTER 6 6.1 Definition and Elementary Properties The Laplace transform serves as a device for simplifying or ...

364 Chapter 6 Laplace Transform Theorem. Let f(t)be sectionally continuous in every finite interval 0 ≤t<T. If, for some cons ...

Chapter 6 Laplace Transform 365 Then by definition L ( f′(t) ) = ∫∞ 0 e−stf′(t)dt. Integrating by parts, we get L ( f′(t) ) =e−s ...

366 Chapter 6 Laplace Transform L(f)=F(s)= ∫∞ 0 e−stf(t)dt L(cf(t))=cL(f(t)) L(f(t)+g(t))=L(f(t))+L(g(t)) L(f′(t))=−f( 0 )+sF(s) ...

Chapter 6 Laplace Transform 367 f(t) F(s) f(t) F(s) 00 tk skk+! 1 (^1) s^1 ebtcos(ωt) s (^2) − 2 bss−+bb (^2) +ω 2 eat s−^1 a eb ...