Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 5 pdf

... boundary condition, φ(0) − φ  (0) = 0. The first few are λ 1 ≈ 0 .67 8298 λ 2 ≈ 7.27931 λ 3 ≈ 24.9302 λ 4 ≈ 54 . 259 3 λ 5 ≈ 95. 3 057 The eigenfunctions are, φ n (x) = v  (1; λ n )u(x; λ n ) − u  (1; ... equation is φ(x) =        a + bx for λ = 0 a cos  √ λx  + b sin  √ λx  for λ > 0 a cosh  √ −λx  + b sinh  √ −λx  for λ < 0 We see that for λ = 0 and λ &l...

Ngày tải lên: 06/08/2014, 01:21

... + √ 87  2/3 − 6 −1/3  9 + √ 87  1/3  2   ≈ (0 .58 9 755 , 0, 0.34781). The closest point is shown graphically in Figure 5. 10. -1 -0 .5 0 0 .5 1 -1 -0 .5 0 0 .5 1 0 0 .5 1 1 .5 2 -1 -0 .5 0 0 .5 1 0 0 .5 1 1 .5 2 Figure ... (See Figure 5. 12.) We find the volume obtained by rotating the 172 Set x = 2 and x = −2 to solve for a and b. Hint 4. 16 Expanding the integral...

Ngày tải lên: 06/08/2014, 01:21

... problem y  + 5y  + 6y = 0; y(0) = 1, y  (0) = V. For what values of V does y(t) remain nonnegative for all t > 0? 952 3 4 5 6 -10 10 20 30 40 50 Figure 17.2: The solution of y  − 2y  + 5y = 0, ... b,   x ξ  −a  1 + a ln x ξ  ,  x ξ  −a ξ ln x ξ  if a 2 = b. 943 1 2 3 4 5 -30 - 25 -20 - 15 -10 -5 Figure 17.1: The solution of 6y  − 5y  + y = 0, y(0) = 4, y...

Ngày tải lên: 06/08/2014, 01:21

... sin(nπx) dx = 4 (nπ) 2 sin(nπ/2) =  4 (nπ) 2 (−1) (n−1)/2 for odd n 0 for even n. 13 56 -3 -2 -1 1 2 3 -1 .5 -1 -0 .5 0 .5 1 1 .5 Figure 28 .5: Fourier Sine Series. The functions {. . . , e −ıx , 1, e ıx , e ı2x , ... -1 -0 .5 0 .5 1 -1 -0 .5 0 .5 1 -1 -0 .5 0 .5 1 -0.4 -0.2 0.2 0.4 Figure 28.7: Three Term Approximation for a Function with Jump Discontinuities and a Conti...

Ngày tải lên: 06/08/2014, 01:21

... . . . . . . . . 155 9 32 .5 Solving Differential Equations with the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 1 56 0 32 .6 The Fourier Cosine and Sine Transform . . . . . . ... . . . . . . . . . . . 1 56 2 32 .6. 1 The Fourier Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 56 2 32 .6. 2 The Fourier Sine Transform . . . ....

Ngày tải lên: 06/08/2014, 01:21

... , converges for α > 1 and diverges for α ≤ 1. Hint, Solution 56 4 Example 12.3.2 Convergence and Uniform Convergence. Consider the series log(1 − z) = − ∞  n=1 z n n . This series converges for |z| ... large. Thus this series is not uniformly convergent in the domain |z| ≤ 1, z = 1. The series is uniformly convergent for |z| ≤ r < 1. 54 5 12.2.2 Uniform Convergence and...

Ngày tải lên: 06/08/2014, 01:21

... ···  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  + ···  = 1 z 5 1 − 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 the integral.  C cot...

Ngày tải lên: 06/08/2014, 01:21

... continuous. -1 -0 .5 0 .5 1 -1 -0 .5 0 .5 1 -1 -0 .5 0 .5 1 -1 -0 .5 0 .5 1 Figure 25. 1: Polynomial Approximations to cos(πx). 12 86 0 1 2 3 4 5 0. 25 0 .5 0. 75 1 1. 25 1 .5 1. 75 2 Figure 24.1: Plot of K 0 (x) and it’s ... assumptions, u   x −2 → u   5 16 x −2 u   x −1 → (u  ) 2  5 16 x −2 . Thus we obtain −2x 1/2 u  + 5 16 x −2 ∼ 0 u  ∼ 5 32 x 5/...

Ngày tải lên: 06/08/2014, 01:21

... = ξ and time t = τ. The disturbance influences the solution for all ξ − ct < x < ξ + ct and t > τ. -1 -0 .5 0 0 .5 1 x 0 0.2 0.4 0 .6 0.8 1 t 0 0.2 0.4 -1 -0 .5 0 0 .5 1 x 0 0.2 0.4 0 .6 0.8 1 t Figure ... < x − t Figure 45. 1 shows the solution at t = 1/2 and t = 2. -2 -1 1 2 0.1 0.2 0.3 0.4 0 .5 -4 -2 2 4 0.1 0.2 0.3 0.4 0 .5 Figure 45. 1: The solution at t = 1/2...

Ngày tải lên: 06/08/2014, 01:21

... 0 and n > 0 are respectively, {1, t},  cos  nπct L  , sin  nπct L  . 2028 0. 05 0.1 0. 15 0.2 0. 25 0 0.2 0.4 0 .6 0.8 1 0 1 2 0. 05 0.1 0. 15 0.2 0. 25 0. 05 0.1 0. 15 0.2 0. 25 0 0.2 0.4 0 .6 0.8 1 0 1 2 0. 05 0.1 0. 15 0.2 0. 25 Figure ... cos(7nπ). Note that the summand is nonzero only for even terms. u(3/4, 7/2) = 53 15 − 3 16 4 ∞  n=1 1 n 4 cos(3nπ) cos...

Ngày tải lên: 06/08/2014, 01:21