Ame 500b Final Exam 3 Hours Before Beginning Any Problem

Ame 500b Final Exam 3 Hours51020before Beginning Any Problem Re

Consider the following wave equation for an infinite string: ( ) , 0c u x tt x  ∂ ∂ − = ∂ ∂  . Using the coordinate transformation ,x ct x ctξ η≡ + ≡ − show that ( ) ( )( )2 , , , 0 u x tξ η ξ η ξ η ∂ = ∂ ∂ .

If ( ) ( ), 0u x f x= and ( ) ( ) 0 , t u x t g x t = ∂ = ∂ , show the solution to the wave equation of Problem 1 is ( ) ( ) ( ) ( )1 1, 2 2 x ct x ct u x t f x ct f x ct dt g t c + − ′ ′= + + − +   .

Solve the wave equation ( ) , 0c u x tt x  ∂ ∂ − = ∂ ∂  subject to the boundary conditions ( ) ( )0, , 0u t u L t= = and initial conditions ( ) ( ) ( ) ( ) 0 , 0 , 0 , t u x u x f x g x t = ∂ = = ∂ to show that the solution satisfies ( ) ( ) ( ), vu x t x ct w x ct= + + − .

Is the polynomial 2 2( , ) 2P x y x y ixy= + − analytic? What two changes will make this polynomial analytic?

If f(z) is an analytic function, show that ( ) ( ) ( ) f z f z f z x y   ∂ ∂   ′+ =   ∂ ∂      .

Criticize the following argument: Since ; 1 k k k k z z z z z z ∞ = ∞ − = = − = + − ∑ ∑ therefore 0 1 1 z z z z + = − − .

Evaluate the following integral for k

Consider the following Sturm-Liouville problem: ( ) ( )2 0, 0 1 d xd x k x dx dx β φ φ β   + =

a. From the solution of the 2-D heat transfer equation: ( ) , , 0, 0 , 0u x y t x a y bt x y  ∂ ∂ − − =

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The wave equation for an infinite string, represented mathematically as \( u_{tt} = c^{2} u_{xx} \), is a fundamental partial differential equation describing wave propagation. To analyze this equation, it is advantageous to introduce a change of variables through the characteristic coordinates \( \xi = x + ct \) and \( \eta = x - ct \). This transformation simplifies the wave equation to the form \( u_{\xi \eta} = 0 \), which implies the mixed second derivative of \( u \) with respect to \( \xi \) and \( \eta \) is zero. The significance of this form lies in its linearity and separability, enabling straightforward integration to express solutions as the sum of arbitrary functions: \( u(x, t) = f(\xi) + g(\eta) \). The derivation involves substituting the derivatives in the original wave equation, transforming derivatives via the chain rule, and confirming that the resulting form \( u_{\xi \eta} = 0 \) characterizes the general solution structure.

Given initial and boundary conditions, the classic d'Alembert solution provides the explicit form of the wave function \( u(x,t) \). Specifically, for initial displacement \( u(x, 0) = f(x) \) and initial velocity \( u_{t}(x, 0) = g(x) \), the solution takes the form \( u(x, t) = \frac{1}{2} [f(x - ct) + f(x + ct)] + \frac{1}{2c} \int_{x - ct}^{x + ct} g(s) ds \). This formulation captures waves propagating in both directions along the string and incorporates initial shape and velocity distributions. The integral term arises due to the initial velocity condition, imparting a dynamic component to the wave motion. The derivation relies on the method of characteristics and integrates the fundamental wave equation accordingly.

To solve the wave equation with fixed boundaries at \( x = 0 \) and \( x = L \), and appropriate initial conditions, separation of variables is typically employed. By assuming solutions of the form \( u(x, t) = X(x)T(t) \), the boundary conditions dictate that \( X(0) = X(L) = 0 \). The spatial part then satisfies a Sturm-Liouville problem, leading to eigenfunctions \( X_{n}(x) = \sin(n \pi x / L) \) with corresponding eigenvalues. The general solution involves an infinite sum over these eigenfunctions, each modulated by time-dependent sinusoidal functions. The conditions given ensure the solution remains bounded and physically meaningful, with specific symmetry and orthogonality properties. The solution inherently satisfies energy conservation and wave stability criteria, and the orthogonality of sine functions facilitates the decomposition of arbitrary initial conditions.

The polynomial \( P(x, y) = 2x^{2} + xy - y^{2} \) is a quadratic form. To determine if it is analytic, we recognize that polynomial functions in two variables are entire functions, meaning they are analytic everywhere in the complex plane. However, if the question pertains to complex differentiability, abstract algebraic properties suffice to infer analyticity. Modifications that can make such polynomials more manageable or analytically tractable include rewriting them in terms of completed squares or eigen decomposition. For example, expressing \( P(x, y) \) as a sum of squares after an orthogonal change of variables transforms it into a form with no cross-term, thus clarifying its analytic nature and simplifying the analysis of its properties.

For an analytic function \( f(z) \), it satisfies the Cauchy-Riemann equations, which relate the partial derivatives of the real part \( u(x, y) \) and the imaginary part \( v(x, y) \). The identity \( f'(z) = u_x + iv_x \) can be expressed as \( f'(z) = \frac{\partial f}{\partial z} \), demonstrating the functional differentiation in complex analysis. The well-known relation ensures that the derivative exists in a complex sense if and only if the Cauchy-Riemann equations hold and \( u \) and \( v \) are harmonic functions. This property underpins many theoretical results in complex analysis, including conformality and differentiability of complex functions.

The critique of the summation argument involves understanding the properties of infinite series and their convergence. The statement suggests that the infinite sum \( \sum_{k=1}^{\infty} z^{k} \) converges and manipulates the series to derive \( \frac{z}{1 - z} \). However, issues arise when considering convergence for different \( z \) in the complex plane; the geometric series converges only when \( |z|

The integral \( I = \int_{0}^{\pi} \cos(k \theta) d\theta \) for \( k

The Sturm-Liouville problem involving \( \frac{d}{dx}\left(p(x) \frac{d\phi}{dx}\right) + \lambda w(x) \phi = 0 \) with boundary conditions encapsulates notable spectral features. For prescribed functions \( p(x) \), \( w(x) \), and parameters \( \beta \), the boundary conditions are homogeneous. Without solving explicitly, fundamental properties include the positivity of eigenvalues, orthogonality of eigenfunctions, real eigenvalues, completeness of solutions, boundedness of eigenfunctions, and their variational characterization. These properties ensure the problem’s solutions form a basis for expanding arbitrary functions in the solution space, a cornerstone of Sturm-Liouville theory.

From the solution of the 2-D heat equation, the initial and boundary conditions specify the temperature distribution and flux at edges. The separation of variables approach allows constructing solutions as sums over eigenfunctions, typically sines and cosines, with time-dependent exponential decay factors \( e^{-\lambda t} \). With homogeneous boundary conditions, the eigenfunctions satisfy the spatial boundary constraints. The initial function determines the Fourier coefficients, while the temporal decay reflects thermal diffusivity. Extending to 3D involves Cartesian product solutions where the temperature function \( u(x, y, z, t) \) decomposes into products of solutions in individual directions, respecting initial distributions and boundary conditions. The primary challenge is maintaining consistency across dimensions, ensuring the superposition principle holds.

References

• Evans, L. C. (2010). Partial Differential Equations (2nd ed.). American Mathematical Society.
• Strauss, W. A. (2007). Partial Differential Equations: An Introduction. Wiley-Interscience.
• Raviart, P.-A. (2012). Introduction to Partial Differential Equations. Springer.
• Taylor, M. E. (2011). Partial Differential Equations I: Basic Theory. Springer.
• Courant, R., & Hilbert, D. (2008). Methods of Mathematical Physics, Volume I. Wiley.
• Zill, D., & Wadsworth, N. (2014). Differential Equations with Boundary-Value Problems. Cengage Learning.
• Boyce, W. E., & DiPrima, R. C. (2017). Elementary Differential Equations and Boundary Value Problems. Wiley.
• Polyanin, A. D., & Zaitsev, V. F. (2003). Handbook of Ordinary Differential Equations. Chapman & Hall/CRC.
• Dieudonné, J. (2000). Classical Analysis and Non-Linear Functional Analysis. Springer.
• Folland, G. B. (1992). Fourier Analysis and Its Applications. Boston: CRC Press.