Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 7 docx

... ≤ b can be expanded in a series of eigenfunctions f(x) ∼ ∞  n=1 c n φ n (x), 144 7 Now consider the case λ = 1 /4. A set of solutions is  (x + 1) (1+ √ 1 4 )/2 , (x + 1) (1− √ 1 4 )/2  . We ... Find the eigenvalues and eigenfunctions for: d 4 φ dx 4 = λφ,  φ(0) = φ  (0) = 0, φ(1) = φ  (1) = 0. Hint, Solution 144 2 (b) We substitute x = 1/2 into the cosine series. 1 4 =...

Ngày tải lên: 06/08/2014, 01:21

... 1)c n+2 + c n−2 = 0, for n ≥ 2 c n +4 = − c n (n + 4) (n + 3) For our first solution we have the difference equation a 0 = 1, a 1 = 0, a 2 = 0, a 3 = 0, a n +4 = − a n (n + 4) (n + 3) . For our second solution, b 0 = ... polynomial are α 1 = 1 +  1 − 3 /4 2 = 3 4 , α 2 = 1 −  1 − 3 /4 2 = 1 4 . Thus our two series solutions will be of the form w 1 = z 3 /4 ∞  n=0 a n z n , w...

Ngày tải lên: 06/08/2014, 01:21

... transform of ˆy(s) by first finding its partial fraction expansion. ˆy(s) = s/3 s 2 + 1 − s/3 s 2 + 4 + s s 2 + 1 = − s/3 s 2 + 4 + 4s/3 s 2 + 1 y(t) = − 1 3 cos(2t) + 4 3 cos(t) Example 31 .4. 3 ... transform methods and show that i 1 = E 0 2R + E 0 2ωL e −αt sin(ωt) where α = R 2L and ω 2 = 2 LC − α 2 . Hint, Solution Exercise 31.23 Solve the initial value problem, y  + 4y  + 4...

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... function at the point x = p is G(p|ξ) = − 1 4 |p − ξ| + const. 1 849 40 .4 Hints Hint 40 .1 Hint 40 .2 Hint 40 .3 Hint 40 .4 Hint 40 .5 Hint 40 .6 Hint 40 .7 Hint 40 .8 1 846 ... y) = 0 and u(x, 0) = 0, u(x, 1) =  sin(nπx). Is this well posed? Exercise 40 .4 Use the fundamental solutions for the Laplace equation ∇ 2 G = δ(x − ξ) 1 843 We seek a solution of the form, u(x...

Ngày tải lên: 06/08/2014, 01:21

... − 1 2 048 + ı √ 3 2 048 2. (11 + 4) 2 = 105 + ı88 Solution 6 .4 1.  2 + ı ı6 − (1 − ı2)  2 =  2 + ı −1 + ı8  2 = 3 + 4 −63 − ı16 = 3 + 4 −63 − ı16 −63 + ı16 −63 + ı16 = − 253 42 25 − ı 2 04 4225 215 2. (1 ... (polar), form.  1 + ı √ 3  −10 =  2 e ıπ/3  −10 = 2 −10 e −ı10π/3 = 1 10 24  cos  − 10π 3  + ı sin  − 10π 3  = 1 10 24  cos  4 3  − ı sin  4 3  = 1 10 2...

Ngày tải lên: 06/08/2014, 01:21

... − z 4 6 + ··· 1 − z 4 90 + ··· = 1 z 5  1 − z 4 6 + ···  1 + z 4 90 + ···  = 1 z 5  1 − 7 45 z 4 + ···  = 1 z 5 − 7 45 1 z + ··· Thus we see that the residue is − 7 45 . N ow we can evaluate ... − z 2 2 + z 4 24 − ···  1 + z 2 2 + z 4 24 + ···  z 3  z − z 3 6 + z 5 120 − ···  z + z 3 6 + z 5 120 + ···  = 1 − z 4 6 + ··· z 3  z 2 + z 6  −1 36 + 1 60...

Ngày tải lên: 06/08/2014, 01:21

... Equations 10 17 19.6 Hints The Constant Coefficient Equation Normal Form Hint 19.1 Transform the equation to normal form. Transformations of the Independent Variable Integral Equations Hint 19.2 Transform ... equation to normal form an d then apply the scale transformation x = λξ + µ. Hint 19.3 Transform the equation to normal form an d then apply the scale transformation x = λξ. Hint 19 .4 M...

Ngày tải lên: 06/08/2014, 01:21

... =  1 −1 1 4  9x 4 − 6x 2 + 1  dx =  1 4  9x 5 5 − 2x 3 + x  1 −1 = 2 5  1 −1 P 3 (x)P 3 (x) dx =  1 −1 1 4  25x 6 − 30x 4 + 9x 2  dx =  1 4  25x 7 7 − 6x 5 + 3x 3  1 −1 = 2 7 Solution ... absolute convergence and uniform convergence. What is the relationship between the two? Solution Problem 23 .4 Write the geometric series and the function to which it conv...

Ngày tải lên: 06/08/2014, 01:21

... 21) π 4 P 4 (x) = 105 8π 4 [(315 − 30π 2 )x 4 + ( 24 2 − 270 )x 2 + ( 27 − 2π 2 )] The cosine and this polynomial are plotted in the second graph in Figure 25.1. The le ast squares fit method uses information ... x 3 , x 4 } is independent, but not orthogonal in the interval [−1, 1]. Using Gramm-Schmidt orthogo- 12 84 1 2 3 4 5 6 -60 -40 -20 Figure 24. 3: log(error in approx...

Ngày tải lên: 06/08/2014, 01:21

... the formula to obtain information about the eigenvalues before we solve a problem. Example 27 .4. 2 Consider the self-adjoint eigenvalue problem −y  = λy, y(0) = y(π) = 0. 1322 Example 27 .4. 1 ... eigenfunction φ. Green’s formula states φ|L[φ] − L[φ]|φ = 0 φ|λφ − λφ|φ = 0 (λ − λ)φ|φ = 0 Since φ ≡ 0, φ|φ > 0. Thus λ = λ and λ is real. 1319 27. 7 Hints Hint 27. 1 13 27...

Ngày tải lên: 06/08/2014, 01:21