1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

428 Chapter 7 Numerical Methods 11.Approximate the solution of the wave equation in a semi-infinite strip 3 units wide. Assumeu= ...

Miscellaneous Exercises 429 3.By means of the transformation mentioned in Exercise 2, a heat problem on an annular ring is conve ...

430 Chapter 7 Numerical Methods with the solution of the problem consisting of the equation ∂^2 u ∂x^2 − 16 u=∂u ∂t , 0 <x< ...

Miscellaneous Exercises 431 14.The analytical solution of the problem in Exercise 13 is u(x,y)= ( 2 /π )tan−^1 (y/x). Compare yo ...

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Bibliography Abramowitz, M., and I. Stegun (eds).Handbook of Mathematical Functions, 10th ed. Washington, DC, National Bureau of ...

434 Bibliography Fletcher, N.H., and T.D. Rossing.The Physics of Musical Instruments,2nded. New York, Springer-Verlag, 1998. Ise ...

Appendix: Mathematical References Trigonometric Functions sin(A±B)=sin(A)cos(B)±cos(A)sin(B) cos(A±B)=cos(A)cos(B)∓sin(A)sin(B) ...

436 Appendix: Mathematical References Hyperbolic Functions cosh(A)= 1 2 ( eA+e−A ) , sinh(A)= 1 2 ( eA−e−A ) dcosh(u)=sinh(u)du, ...

Appendix: Mathematical References 437 b. ∫a a f(x)dx= 0 c. ∫b a f(x)dx=− ∫a b f(x)dx d. ∫b a f(x)dx= ∫c a f(x)dx+ ∫b c f(x)dx 3. ...

438 Appendix: Mathematical References e.Integrated Bessel function IJ(x)= ∫x 0 J 0 (z)dz Table of Integrals Any letter exceptxre ...

Appendix: Mathematical References 439 3.2 ∫ xekxdx= kx− 1 k^2 e kx 3.3 ∫ sinh(kx)dx= cosh(kx) k 3.4 ∫ cosh(kx)dx=sinh(kx) k 3.5 ...

440 Appendix: Mathematical References 4.13 ∫ ekxsin(λx)dx=e kx(ksin(λx)−λcos(λx)) k^2 +λ^2 4.14 ∫ ekxcos(λx)dx= ekx(kcos(λx)+λsi ...

Answers to Odd-Numbered Exercises Chapter 0 Section 0.1 1.φ(x)=c 1 cos(λx)+c 2 sin(λx). The equation has constant coefficientsk ...

442 Answers to Odd-Numbered Exercises Roots of characteristic equation: m=−α±iβ, β= √ σ^2 −α^2. Solution of differential equat ...

Answers to Odd-Numbered Exercises 443 c.λ=± nπ a,n=^0 ,^1 ,^2 ,.... 5.c=−a/2,c′=h−^1 μ cosh (μa 2 ) . 7.u(x)=T+c 1 cosh(γx)+c 2 ...

444 Answers to Odd-Numbered Exercises 7.u(r)= 325 + 104 ( 0. 25 −r^2 )/4;u( 0 )=950. 9.u(x)=T 0 +AL^2 ( 1 −e−x/L). Section 0.5 1 ...

Answers to Odd-Numbered Exercises 445 (iv) This is true becauseu 1 (x)andu 2 (x)are solutions of the homoge- neous equation. Cha ...

446 Answers to Odd-Numbered Exercises Finallyu(x)=(U−^3 /^2 +( 3 / 2 ) √ 2 γ^2 / 5 x)−^2 /^3. 459.77 rad/s. 29.u(x)=C 0 e−ax. ...

Chapter 1 447 b. 4 π [ sin (πx 2 ) + 1 3 sin ( 3 πx 2 ) + 1 5 sin ( 5 πx 2 ) +··· ] ; c. 121 −π^12 [ cos( 2 πx)−^14 cos( 4 πx)+^ ...