60 sample documents related to MATH 5587

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  Differentiation with variables Notes part 1

  UCF MATH 5587
  

  Remark : On a connected domain R 2 , all harmonic conjugates to a given function u(x, y) only differ by a constant: v(x, y) = v(x, y) + c; see Exercise . Although most harmonic functions have harmonic conjugates, unfortunately this is not always the case.
   
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  Integration with variables Notes_Part_1

  UCF MATH 5587
  

  Chapter 7 Complex Analysis and Conformal Mapping The term \"complex analysis\" refers to the calculus of complex-valued functions f (z) depending on a single complex variable z. To the novice, it may seem that this subject should merely be a simple reworkin
   
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  Integration with variables Notes_Part_2

  UCF MATH 5587
  

  1 Re z Figure 7.1. 1 Im z 1 Real and Imaginary Parts of f (z) = z . Therefore, if f (z) is any complex function, we can write it as a complex combination f (z) = f (x + i y) = u(x, y) + i v(x, y), of two inter-related real harmonic functions: u(x, y) = Re
   
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  Integration with variables Notes_Part_3

  UCF MATH 5587
  

  Figure 7.4. Real and Imaginary Parts of z. also have complex logarithms! On the other hand, if z = x < 0 is real and negative, then log z = log | x | + (2 k + 1) i is complex no matter which value of ph z is chosen. (This explains why one avoids defining
   
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  Integration with variables Notes_Part_4

  UCF MATH 5587
  

  The proof of the converse - that any function whose real and imaginary components satisfy the CauchyRiemann equations is differentiable - will be omitted, but can be found in any basic text on complex analysis, e.g., [3, 65, 118]. Remark : It is worth poi
   
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  Integration with variables Notes_Part_5

  UCF MATH 5587
  

  is analytic everywhere except for singularities at the points z = 3 and z = -1, where its denominator vanishes. Since f (z) = h1 (z) , z-3 where h1 (z) = ez (z + 1)2 1 is analytic at z = 3 and h1 (3) = 16 e3 = 0, we conclude that z = 3 is a simple (order
   
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  Integration with variables Notes_Part_7

  UCF MATH 5587
  

  if and only if it has vanishing divergence: v = u v + = 0. x y (7.36) Incompressibility means that the fluid volume does not change as it flows. Most liquids, including water, are, for all practical purposes, incompressible. On the other hand, the flow is
   
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  Integration with variables Notes_Part_8

  UCF MATH 5587
  

  Using formula (7.19) for the complex derivative, d = -i = u - i v, dz x y so = u, x = v. y Thus, = v, and hence the real part (x, y) of the complex function (z) defines a velocity potential for the fluid flow. For this reason, the anti-derivative (z) is k
   
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  Integration with variables Notes_Part_9

  UCF MATH 5587
  

  g D Figure 7.14. Mapping to the Unit Disk. Remark : In this section, we have focused on the fluid mechanical roles of a harmonic function and its conjugate. An analogous interpretation applies when (x, y) represents an electromagnetic potential function;
   
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  Integration with variables Notes_Part_10

  UCF MATH 5587
  

  Figure 7.16. The Effect of = z 2 on Various Domains. obtained by cutting the complex plane along the negative real axis. On the other hand, vertical lines Re z = a are mapped to circles | | = ea . Thus, a vertical strip a < Re z < b is mapped to an annulu
   
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  Integration with variables Notes_Part_11

  UCF MATH 5587
  

  z ph z Figure 7.18. Complex Curve and Tangent. notation x(t) = ( x(t), y(t) ) to complex notation z(t) = x(t)+ i y(t). All the usual vectorial curve terminology - closed, simple (non-self intersecting), piecewise smooth, etc. - is employed without modific
   
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  Integration with variables Notes_Part_12

  UCF MATH 5587
  

  Center: .1 Radius: .5 Center: .2 + i Radius: 1 Center: 1 + i Radius: 1 Center: -2 + 3 i Radius: 3 2 4.2426 Center: .2 + i Radius: 1.2806 Figure 7.21. Center: .1 + .3 i Radius: .9487 Center: .1 + .1 i Radius: 1.1045 Center: -.2 + .1 i Radius: 1.2042 Airfoi
   
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  Integration with variables Notes_Part_13

  UCF MATH 5587
  

  Example 7.35. The goal of this example is to construct an conformal map that takes a half disk D+ = | z | < 1, Im z > 0 (7.73) to the full unit disk D = cfw_ | | < 1 . The answer is not = z 2 because the image of D+ omits the positive real axis, resulting
   
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  Integration with variables Notes_Part_14

  UCF MATH 5587
  

  7.5. Applications of Conformal Mapping. Let us now apply what we have learned about analytic/conformal maps. We begin with boundary value problems for the Laplace equation, and then present some applications in fluid mechanics. We conclude by discussing h
   
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  Integration with variables Notes_Part_15

  UCF MATH 5587
  

  Figure 7.25. A NonCoaxial Cable. Example 7.39. A non-coaxial cable. The goal of this example is to determine the electrostatic potential inside a non-coaxial cylindrical cable, as illustrated in Figure 7.25, with prescribed constant potential values on th
   
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  Integration with variables Notes_Part_16

  UCF MATH 5587
  

  0 Figure 7.29. 15 Fluid Flow Past a Tilted Plate. 30 Note that = ( 1, 0 ), and hence this flow satisfies the Neumann boundary conditions (7.95) on the horizontal segment D = . The corresponding complex potential is (z) = z, with complex velocity f (z) = (
   
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  Integration with variables Notes_Part_17

  UCF MATH 5587
  

  on the unit disk D for an impulse concentrated at the origin; see Section 6.3 for details. How do we obtain the corresponding solution when the unit impulse is concentrated at another point = + i D instead of the origin? According to Example 7.25, the lin
   
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  Integration with variables Notes_Part_18

  UCF MATH 5587
  

  as long as n = -1. Therefore, we can use the Fundamental Theorem of Calculus (which works equally well for real integrals of complex-valued functions), to evaluate n+1 1 -1 = n = 2 k + 1 odd, 0, 2 t + i (t - 1) 2 z n dz = = , n = 2 k even. n+1 P t = -1 n+
   
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  Integration with variables Notes_Part_19

  UCF MATH 5587
  

  Figure 7.32. Orientation of Domain Boundary. Theorem 7.48. If f (z) is analytic on a bounded domain C, then f (z) dz = 0. (7.118) Proof : If we apply Green\'s Theorem to the two real line integrals in (7.109), we find u dx - v dy = - u v - x y = 0, v dx +
   
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  Integration with variables Notes_Part_20

  UCF MATH 5587
  

  Proof : Note that the integrand f (z) = 1/(z - a) is analytic everywhere except at z = a, where it has a simple pole. If a is outside C, then Cauchy\'s Theorem 7.48 applies, and the integral is zero. On the other hand, if a is inside C, then Proposition 7.
   
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  Integration with variables Notes_Part_21

  UCF MATH 5587
  

  0 Figure 7.36. 15 Kutta Flow Past a Tilted Airfoil. 30 which remains asymptotically 1 at large distances. By Cauchy\'s Theorem 7.48 coupled with formula (7.123), if C is a curve going once around the disk in a counter-clockwise direction, then i 1 dz = - 2
   
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  Integration with variables Notes_Part_22

  UCF MATH 5587
  

  is analytic in the disk | z | 2 since its only singularity, at z = 3, lies outside the contour C. Therefore, by Cauchy\'s formula (7.135), we immediately obtain the integral ez dz = z2 - 2 z - 3 f (z) i dz = 2 i f (-1) = - . z+1 2e C C Note: Path independe
   
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  Partial Differentials Notes_Part_1

  UCF MATH 5587
  

  Chapter 12 Dynamics of Planar Media In previous chapters we studied the equilibrium configurations of planar media - plates and membranes - governed by the two-dimensional Laplace and Poisson equations. In this chapter, we analyze their dynamics, modeled
   
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  Partial Differentials Notes_Part_2

  UCF MATH 5587
  

  In this manner, we arrive at the basic conservation law relating the heat energy density and the heat flux vector w. As in our one-dimensional model, cf. (4.3), the heat energy density (t, x, y) is proportional to the temperature, so (t, x, y) = (x, y) u(
   
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  Partial Differentials Notes_Part_3

  UCF MATH 5587
  

  for the diffusion equation. See [35; p. 369] for a precise statement and proof of the general theorem. Qualitative Properties Before tackling examples in which we are able to construct explicit formulae for the eigenfunctions and eigenvalues, let us see w
   
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  Partial Differentials Notes_Part_4

  UCF MATH 5587
  

  Theorem 12.1. Suppose u(t, x, y) is a solution to the forced heat equation ut = u + F (t, x, y), for (x, y) , 0 < t < c, where is a bounded domain, and > 0. Suppose F (t, x, y) 0 for all (x, y) and 0 t c. Then the global maximum of u on the set cfw_ (t, x
   
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  Partial Differentials Notes_Part_5

  UCF MATH 5587
  

  so there are no non-separable eigenfunctions . As a consequence, the general solution to the initial-boundary value problem can be expressed as a linear combination u(t, x, y) = m,n = 1 cm,n um,n (t, x, y) = m,n = 1 cm,n e- m,n t vm,n (x, y) (12.41) of
   
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  Partial Differentials Notes_Part_6

  UCF MATH 5587
  

  Let us start with the equation for q(). The second boundary condition in (12.50) requires that q() be 2 periodic. Therefore, the required solutions are the elementary trigonometric functions q() = cos m or sin m , where = m2 , (12.53) with m = 0, 1, 2, .
   
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  Partial Differentials Notes_Part_7

  UCF MATH 5587
  

  15 10 5 -4 -2 -5 -10 -15 2 4 Figure 12.3. The Gamma Function. Thus, at integer values of x, the gamma function agrees with the elementary factorial. A few other values can be computed exactly. One important case is when x = 1 . Using 2 the substitution t
   
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  Partial Differentials Notes_Part_8

  UCF MATH 5587
  

  Remark : The definition of a singular point assumes that the other coefficients do not both vanish there, i.e., either q(x0 ) = 0 or r(x0 ) = 0. If all three functions happen to vanish at x0 , we can cancel any common factor (x - x0 )k , and hence, withou
   
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  Partial Differentials Notes_Part_9

  UCF MATH 5587
  

  we find that the only non-zero coefficients un are when n = 3 k +1. The recurrence relation u3 k+1 = u3 k-2 (3 k + 1)(3 k) yields u3 k+1 = 1 . (3 k + 1)(3 k)(3 k - 2)(3 k - 3) 7 6 4 3 The resulting solution is x3 k+1 . (3 k + 1)(3 k)(3 k - 2)(3 k - 3) 7 6
   
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  Partial Differentials Notes_Part_10

  UCF MATH 5587
  

  two different Frobenius expansions. Usually, this expectation is valid, but there is an important exception, which occurs when the indices differ by an integer. The general result is summarized in the following list: (i ) If r2 - r1 is not an integer, the
   
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  Partial Differentials Notes_Part_11

  UCF MATH 5587
  

  We have thus found the series solution (-1)k xm+2k . 22k k(k - 1) 3 2 (r + k)(r + k - 1) (r + 2)(r + 1) k=0 k=0 (12.93) So far, we not paid attention to the precise values of the indices r = m. In order to continue the recurrence, we need to ensure that t
   
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  Partial Differentials Notes_Part_12

  UCF MATH 5587
  

  where h0 = 0, while = lim k hk = 1 + 1 1 1 + + + , 2 3 k (12.102) hk - log k .5772156649 . . . is known as Euler\'s constant. All Bessel functions of the second kind have a singularity at the origin x = 0; indeed, by inspection of (12.101), we find that th
   
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  Partial Differentials Notes_Part_13

  UCF MATH 5587
  

  of the Bessel boundary value problem (12.5455) are the squares of the roots of the Bessel function of order m. The corresponding eigenfunctions are wm,n (r) = Jm (m,n r) , n = 1, 2, 3, . . . , m = 0, 1, 2, . . . , (12.112) defined for 0 r 1. Combining (12
   
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  Partial Differentials Notes_Part_14

  UCF MATH 5587
  

  t=0 t = .04 t = .08 t = .12 Figure 12.6. t = .16 Heat Diffusion in a Disk. t = .2 12.5. The Fundamental Solution of the Heat Equation. As we learned in Section 4.1, the fundamental solution to the heat equation measures the temperature distribution result
   
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  Partial Differentials Notes_Part_15

  UCF MATH 5587
  

  for the planar heat equation is given by the linear superposition formula u(t, x, y) = 1 4 t f (, ) e- [ (x-) 2 +(y-)2 ]/(4 t) d d. (12.125) We can interpret the solution formula (12.125) as a two-dimensional convolution u(t, x, y) = F (t, x, y) f (x, y)
   
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  Partial Differentials Notes_Part_16

  UCF MATH 5587
  

  Vibration of a Rectangular Drum Let us first consider the vibrations of a membrane in the shape of a rectangle R= 0 < x < a, 0 < y < b , with side lengths a and b, whose edges are fixed to the (x, y)plane. Thus, we seek to solve the wave equation utt = c2
   
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  Partial Differentials Notes_Part_17

  UCF MATH 5587
  

  A table of their values (for the case c = 1) can be found in the preceding section. The Bessel roots do not follow any easily discernible pattern, and are not rational multiples of each other. This result, known as Bourget\'s hypothesis, [142; p. 484], was
   
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  Partial Differentials Notes_Part_18

  UCF MATH 5587
  

  following table, we display a list of all relative vibrational frequencies (12.158) that are < 6. Once the lowest frequency 0,1 has been determined - either theoretically, numerically or experimentally - all the higher overtones m,n = m,n 0,1 are simply o
   
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  Partial Differentials Notes_Part_19

  UCF MATH 5587
  

  For example, on a unit square R = 0 < x, y < 1 , an accidental degeneracy occurs whenever m2 + n2 = k 2 + l2 (12.163) for distinct pairs of positive integers (m, n) = (k, l). The simplest possibility arises whenever m = n, in which case we can merely reve
   
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  ee

  UCF MATH 5587
  

  Chapter 9 Linear and Nonlinear Evolution Equations In this chapter, we analyze several of the most important evolution equations, both linear and nonlinear, involving a single spatial variable. Our first stop is to revisit the heat equation. We introduce
   
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  fsp

  UCF MATH 5587
  

  Chapter 3 Fourier Series Just before 1800, the French mathematician/physicist/engineer Jean Baptiste Joseph Fourier made an astonishing discovery. Through his deep analytical investigations into the partial differential equations modeling heat propagation
   
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  ft

  UCF MATH 5587
  

  Chapter 8 Fourier Transforms Fourier series and their ilk are designed to solve boundary value problems on bounded intervals. The extension of Fourier methods to the entire real line leads naturally to the Fourier transform, an extremely powerful mathemat
   
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  gf

  UCF MATH 5587
  

  Chapter 6 Generalized Functions and Green\'s Functions Boundary value problems, involving both ordinary and partial differential equations, can be profitably viewed as the infinite-dimensional function space versions of finite dimensional systems of linear
   
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  hkpd1

  UCF MATH 5587
  

  Math 5587 September 8, 2011 Homework #1 Problems: Chapter 1: 1.1ae, 1.2b,d, 1.5a,e, 1.6, 1.12a, 1.16ad, 1.18, 1.19, 1.20, 1.24. Chapter 2: 2.1 2, 3c,e, 4, 6. Due: Thursday, September 15
   
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  hkpd2

  UCF MATH 5587
  

  Math 5587 September 20, 2011 Homework #2 Problems: Chapter 2: 2.2 2.3 2a, 3b, 9, 17, 26, 27. 2, 5, 14, 15. Due: Thursday, September 29 First Midterm: Tuesday, October 11 Will cover chapters 1 11\" sheet of notes. Note
   
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  hkpd3

  UCF MATH 5587
  

  Math 5587 September 29, 2011 Homework #3 Problems: Chapter 2: 2.4 2, 3, 4c,d, 8, 11, 12. Also, in 2.4.8, determine the domain of influence of the point (0,2) and the domain of dependence of the point (3,-1). Discuss what these tell you about the solution.
   
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  hkpd4

  UCF MATH 5587
  

  Math 5587 October 13, 2011 Homework #4 Problems: Chapter 3: 3.1 3.2 2b, 5. 1, 2g, 3a, 5, 6a,g, 15a,d, 16a,d, 24, 25, 34, 35, 41b, 52, 53. Due: Thursday, October 20
   
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  hkpd5

  UCF MATH 5587
  

  Math 5587 October 25, 2011 Homework #5 Problems: Chapter 3: 3.3 1, 2, 8. 3.4 2b, 3c, 7, 9 (just use one of the two methods). 3.5 2b,c,d, 4, 8, 11a,b,c. Due: Tuesday, November 1 Second Midterm: Thursday, November 17 Will cover chapters 3 & 4. You will be a
   
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  hkpd6

  UCF MATH 5587
  

  Math 5587 November 3, 2011 Homework #6 Problems: Chapter 3: 3.5 13, 21c,e, 22b,c, 27b,d, 30, 31, 35a, 42. Chapter 4: 4.1 2, 4c, 10, 17a,b. Due: Thursday, November 10 Second Midterm: Thursday, November 17 Will cover chapters 3 & 4. You will be allowed to u
   
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  hkpd7

  UCF MATH 5587
  

  Math 5587 November 10, 2011 Homework #7 Problems: Chapter 4: 4.2 3a, 4b,e, 8, 14a,d,e, 26. 4.3 4, 7, 10c, 11, 12a, 16, 24a, 29, 31. Due: Tuesday, November 22 Second Midterm: Thursday, November 17 Will cover chapters 3 & 4. You will be allowed to use one 8
   
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  hkpd8

  UCF MATH 5587
  

  Math 5587 December 6, 2011 Homework #8 Problems: Chapter 4: 4.4 2a,e,f, 12a,e,f, 13, 17a,b. Chapter 6: 6.1 1b,d, 2d, 3, 5b, 8, 13, 19, 35. 6.2 2, 6. 6.3 1, 2, 6. Due: Tuesday, December 13 Final Exam: Take Home, to be handed out on Tuesday, December 13 and
   
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  lnw

  UCF MATH 5587
  

  Chapter 2 Linear and Nonlinear Waves Our exploration of the vast mathematical continent that is partial differential equations will begin with simple first order equations. In applications, first order partial differential equations are most commonly used
   
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  nfd

  UCF MATH 5587
  

  Chapter 5 Numerical Methods: Finite Differences As you know, the differential equations that can be solved by an explicit analytic formula are few and far between. Consequently, the development of accurate numerical approximation schemes is essential for
   
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  nfe

  UCF MATH 5587
  

  Chapter 11 Numerical Methods: Finite Elements In Chapter 5, we introduced the first, the oldest, and in many ways the simplest class of numerical algorithms for approximating the solutions to partial differential equations: finite differences. In the pres
   
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  pam

  UCF MATH 5587
  

  Chapter 10 A General Framework for Linear Partial Differential Equations Before pressing on to the higher dimensional forms of the heat, wave, and Laplace/ Poisson equations, it is worth taking some time to develop a general, abstract, linear algebraic fr
   
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  ppz

  UCF MATH 5587
  

  Chapter 12 Partial Differential Equations in Space At last we have reached the ultimate rung of the dimensional ladder (at least for those of us living in a three-dimensional universe): partial differential equations in physical space. As in the one- and
   
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  sv

  UCF MATH 5587
  

  Chapter 4 Separation of Variables There are three paradigmatic linear second order partial differential equations that have collectively driven the development of the entire subject. The first two we have already encountered: The wave equation describes v
   
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  wpd

  UCF MATH 5587
  

  Chapter 1 What are Partial Differential Equations? Let us begin by specifying our object of study. A differential equation is an equation that relates the derivatives of a (scalar) function depending on one or more variables. For example, d4 u du + u2 = c