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### Numerical Solution of Scalar Equations

Course: MATH 5485, Fall 2009
School: UCF
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Word Count: 9219

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Lecture AIMS Notes 2006 Peter J. Olver 2. Numerical Solution of Scalar Equations Most numerical solution methods are based on some form of iteration. The basic idea is that repeated application of the algorithm will produce closer and closer approximations to the desired solution. To analyze such algorithms, our first task is to understand general iterative processes. 2.1. Iteration of Functions. Iteration,...

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UCF - MATH - 5485
Chapter 20 Nonlinear Ordinary Differential EquationsThis chapter is concerned with initial value problems for systems of ordinary differential equations. We have already dealt with the linear case in Chapter 9, and so here our emphasis will be on nonline
UCF - MATH - 5485
Chapter 18 Partial Differential Equations in ThreeDimensional SpaceAt last we have ascended the dimensional ladder to its ultimate rung (at least for those of us living in a three-dimensional universe): partial differential equations in physical space. A
UCF - MATH - 5485
Orthogonal Bases and the QR Algorithmby Peter J. Olver University of Minnesota1. Orthogonal Bases.Throughout, we work in the Euclidean vector space V = R n , the space of column vectors with n real entries. As inner product, we will only use the dot pr
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver14. Finite ElementsIn this part, we introduce the powerful finite element method for finding numerical approximations to the solutions to boundary value problems involving both ordinary and partial differential equa
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver12. MinimizationIn this part, we will introduce and solve the most basic mathematical optimization problem: minimize a quadratic function depending on several variables. This will require a short introduction to pos
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.
UCF - MATH - 5587
Chapter 7 Complex Analysis and Conformal MappingThe term &quot;complex analysis&quot; 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
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
UCF - MATH - 5587
Figure 7.4.Real and Imaginary Parts ofz.also have complex logarithms! On the other hand, if z = x &lt; 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
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
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)21 is analytic at z = 3 and h1 (3) = 16 e3 = 0, we conclude that z = 3 is a simple (order
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
UCF - MATH - 5587
Using formula (7.19) for the complex derivative, d = -i = u - i v, dz x y so = u, x = v. yThus, = 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
UCF - MATH - 5587
gDFigure 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;
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 &lt; Re z &lt; b is mapped to an annulu
UCF - MATH - 5587
zph zFigure 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
UCF - MATH - 5587
Center: .1 Radius: .5Center: .2 + i Radius: 1Center: 1 + i Radius: 1Center: -2 + 3 i Radius: 3 2 4.2426Center: .2 + i Radius: 1.2806 Figure 7.21.Center: .1 + .3 i Radius: .9487Center: .1 + .1 i Radius: 1.1045Center: -.2 + .1 i Radius: 1.2042Airfoi
UCF - MATH - 5587
Example 7.35. The goal of this example is to construct an conformal map that takes a half disk D+ = | z | &lt; 1, Im z &gt; 0 (7.73) to the full unit disk D = cfw_ | | &lt; 1 . The answer is not = z 2 because the image of D+ omits the positive real axis, resulting
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
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
UCF - MATH - 5587
0 Figure 7.29.15 Fluid Flow Past a Tilted Plate.30Note 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) = (
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
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+
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 +
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.
UCF - MATH - 5587
0 Figure 7.36.15 Kutta Flow Past a Tilted Airfoil.30which 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
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 2eCCNote: Path independe
UCF - MATH - 5587
Chapter 12 Dynamics of Planar MediaIn 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
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(
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
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 &lt; t &lt; c,where is a bounded domain, and &gt; 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
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 = 1cm,n um,n (t, x, y) =m,n = 1cm,n e- m,n t vm,n (x, y)(12.41)of
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, .
UCF - MATH - 5587
15 10 5 -4 -2 -5 -10 -15 2 4Figure 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
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
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 3The resulting solution isx3 k+1 . (3 k + 1)(3 k)(3 k - 2)(3 k - 3) 7 6
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
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
UCF - MATH - 5587
where h0 = 0, while = limkhk = 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
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
UCF - MATH - 5587
t=0t = .04t = .08t = .12 Figure 12.6.t = .16 Heat Diffusion in a Disk.t = .212.5. 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
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)
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 &lt; x &lt; a, 0 &lt; y &lt; 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
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
UCF - MATH - 5587
following table, we display a list of all relative vibrational frequencies (12.158) that are &lt; 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
UCF - MATH - 5587
For example, on a unit square R = 0 &lt; x, y &lt; 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
UCF - MATH - 5587
Chapter 9 Linear and Nonlinear Evolution EquationsIn 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
UCF - MATH - 5587
Chapter 3 Fourier SeriesJust 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
UCF - MATH - 5587
Chapter 8 Fourier TransformsFourier 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
UCF - MATH - 5587
Chapter 6 Generalized Functions and Green's FunctionsBoundary 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
UCF - MATH - 5587
Math 5587 September 8, 2011Homework #1Problems: 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
UCF - MATH - 5587
Math 5587 September 20, 2011Homework #2Problems: 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 &amp; 2. You will be allowed to use one 8&quot; 11&quot; sheet of notes. Note
UCF - MATH - 5587
Math 5587 September 29, 2011Homework #3Problems: 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.
UCF - MATH - 5587
Math 5587 October 13, 2011Homework #4Problems: 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
UCF - MATH - 5587
Math 5587 October 25, 2011Homework #5Problems: 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 &amp; 4. You will be a
UCF - MATH - 5587
Math 5587 November 3, 2011Homework #6Problems: 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 &amp; 4. You will be allowed to u
UCF - MATH - 5587
Math 5587 November 10, 2011Homework #7Problems: 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 &amp; 4. You will be allowed to use one 8
UCF - MATH - 5587
Math 5587 December 6, 2011Homework #8Problems: 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
UCF - MATH - 5587
Chapter 2 Linear and Nonlinear WavesOur 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
UCF - MATH - 5587
Chapter 5 Numerical Methods: Finite DifferencesAs 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
UCF - MATH - 5587
Chapter 11 Numerical Methods: Finite ElementsIn 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