Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 9 pot

... = 0 for all x > 0, φ(0, t) = f(t) for all t > 0, where f(t) is a given function. 1. Obtain the formula for the Laplace transform of φ(x, t), Φ(x, s) and use the convolution theorem for ... c 1  1/(2 √ ξ) e −t 2 dt + c 2 f(ξ) = c 1 erf  1 2 √ ξ  + c 2 1 896 Chapter 44 Transform Methods 44.1 Fourier Transform for Partial Differential Equations Solve Laplace’s equation in...

Ngày tải lên: 06/08/2014, 01:21

... the one, two and three term approximations give relative errors of 0.01, 0.0006 and 0.00006, respectively. 16 79 Part V Partial Differential Equations 1680 Solution 34.8 For this part we will use ... the signs in the quadratic formulas to get different solutions for ξ and ψ. Now we have first order quasi-linear partial differential equations for the coordinates ξ and ψ. We solv...

Ngày tải lên: 06/08/2014, 01:21

... Write out an explicit formula for the value of the function on this branch. Figure 7.33: Four candidate sets of branch cuts for ((z − 1)(z − 2)(z − 3)) 1/2 . Hint, Solution 294 change if we walk ... Hints Cartesian and Modulus-Argument Form Hint 7.1 Hint 7.2 Trigonometric Functions Hint 7.3 Recall that sin(z) = 1 ı2 ( e ız − e −ız ). Use Result 6.3.1 to convert between Cartesian and...

Ngày tải lên: 06/08/2014, 01:21

... positive, real-valued for a ∈ (−1 . . . 1) and there are branch cuts on the real axis on the intervals: (−∞. . . − 1] and [1 . . . ∞). Result 13 .9. 1 For evaluating integrals of the form  a+2π a F ... ıα dx. The integral exists for α arbitrarily close to zero, but diverges when α = 0. Plot the real and imaginary part of the integrand. If one were to assign meaning to the integ...

Ngày tải lên: 06/08/2014, 01:21

... integral converges for (a) < 1. 7 25 Figure 13.8: The real and imaginary part of the integrand for several values of α. Note that the integral exists for all nonzero real α and that lim α→0 +  1 −1 1 x ... real and imaginary parts to solve for the desired integrals.  ∞ 0 ln 2 x 1 + x 2 dx = π 3 8  ∞ 0 ln x 1 + x 2 dx = 0 Solution 13. 25 We consider the branch of the f...

Ngày tải lên: 06/08/2014, 01:21

... < 0, and define the solution, y, to be y(x) =  y + (x), for x ≥ 0, y − (x), for x ≤ 0. The initial condition for y − demands that the solution be continuous. Solving the two problems for positive ... =      −1 for x < 0 0 for x = 0 1 for x > 0. Since sign x is piecewise defined, we solve the two problems, y  + + y + = 0, y + (1) = 1, for x > 0 y  − − y − =...

Ngày tải lên: 06/08/2014, 01:21

... A = −2 For positive A, the solution is bounded at the origin only for c = 0. For A = 0, there are no bounded solutions. For negative A, the solution is bounded there for any value of c and thus ... infinity. du dζ − 1 ζ 2 1 (1/ζ) 2 + 9 u = 0 du dζ − 1 9 2 + 1 u = 0 Since the equation for u has a ordinary point at ζ = 0, z = ∞ is a ordinary point of the equation for w. 8 15...

Ngày tải lên: 06/08/2014, 01:21

... the form: d dx [P (x)y  + f(x)y] = 0 for some function f(x). We carry out the differentiation to write the equation in standard form: P (x)y  + (P  (x) + f(x)) y  + f  (x)y = 0 (16 .5) We ... 0 e 3t   894 Particular Solutions. Any function, y p , that satisfies the inhomogeneous equation, L[y p ] = f (x), is called a particular solution or particular integral of the equation. Note...

Ngày tải lên: 06/08/2014, 01:21

... the problem for u, (and hence the problem for y). As a check, then general solution for y is y = − 1 3 cos 2x + c 1 cos x + c 2 sin x. 11 15 We guess a particular solution of the form y p = t e −t (a ... t, where c 1 and c 2 are arbitrary constants and a and b are any conveniently chosen points. 2. Using the result of part (a) show that the solution satisfying the initial cond...

Ngày tải lên: 06/08/2014, 01:21

... inverse Laplace transform for t > 0. 14 85 • t e 2t is of exponential order α for any α > 2. • e t 2 is not of exponential order α for any α. • t n is of exponential order α for any α > 0. • ... transform as the integral diverges. Example 31.1.1 Consider the Laplace transform of f(t) = 1. Since f(t) = 1 is of exponential order α for any α > 0, the Laplace transform inte...

Ngày tải lên: 06/08/2014, 01:21