Questions & explanations
1. Find an integrating factor for (y^2 + 2xy) dx + (x^2) dy = 0 and solve.
M = y^2+2xy, N = x^2. ∂M/∂y = 2y+2x, ∂N/∂x = 2x. Not equal. Compute (∂M/∂y - ∂N/∂x)/N = (2y+2x-2x)/x^2 = 2y/x^2. This depends on y and x, not a function of x alone. Try (∂N/∂x - ∂M/∂y)/M = (2x - (2y+2x))/(y^2+2xy) = (-2y)/(y(y+2x)) = -2/(y+2x). This depends on both. Instead, try μ = 1/x^2? Multiply ODE by 1/x^2: (y^2/x^2 + 2y/x) dx + dy = 0. Now M' = y^2/x^2 + 2y/x, N'=1. ∂M'/∂y = 2y/x^2 + 2/x, ∂N'/∂x = 0. Not exact. Try μ = 1/y? Multiply: (y + 2x) dx + (x^2/y) dy = 0. Not exact. Try μ = 1/(xy)? Multiply: (y/x + 2) dx + (x/y) dy = 0. ∂M/∂y = 1/x, ∂N/∂x = 1/y, not equal. Actually, note that the ODE is homogeneous? Let's solve by another method: rewrite as dy/dx = -(y^2+2xy)/x^2 = -(y/x)^2 - 2(y/x). Let v=y/x, then y=vx, dy/dx = v + x dv/dx. Substitute: v + x dv/dx = -v^2 - 2v => x dv/dx = -v^2 - 3v => separable: dv/(v^2+3v) = -dx/x. Integrate: (1/3) ln|v/(v+3)| = -ln|x| + C => v/(v+3) = K/x^3. Then y/x / (y/x+3) = K/x^3 => y/(y+3x) = K/x^3 => cross multiply: y x^3 = K(y+3x) => x^3 y = K y + 3K x => y(x^3 - K) = 3K x => y = 3K x/(x^3 - K). This is the solution.
2. What is the difference between the integral and differential forms of Maxwell's equations?
The integral form describes the total flux or circulation over a surface or loop, while the differential form describes the behavior at a point using derivatives. The integral form is useful for problems with symmetry, like a spherical charge distribution. The differential form is used in computer simulations to solve for fields in complex geometries. Both forms are equivalent mathematically through the divergence theorem and Stokes' theorem. For example, Gauss's law in integral form says the total electric flux through a closed surface equals the enclosed charge, while the differential form says the divergence of E equals charge density.
3. Explain the concept of hp-adaptivity and when you would increase p instead of refining h.
In hp-adaptivity, you decide whether to subdivide an element (h-refinement) or increase its polynomial degree (p-refinement) based on the local solution behavior. If the solution is smooth but the error is large, increasing p is more efficient because high-order methods converge exponentially for smooth solutions. If the solution has a singularity or sharp gradient, h-refinement is better because it localizes the error. For example, near a crack tip in elasticity, you refine h; in regions far from the crack, you increase p. This hybrid approach can achieve exponential convergence even for problems with singularities.
4. What is a harmonic function? Give an example.
A harmonic function is a twice-differentiable function whose Laplacian (sum of second partial derivatives) is zero everywhere in its domain. For example, the function u(x,y) = x^2 - y^2 is harmonic because its second partial derivatives are 2 and -2, which sum to zero. Harmonic functions are smooth and satisfy the mean value property: the value at a point equals the average over any sphere centered at that point. They arise in steady-state heat flow, electrostatics, and fluid dynamics. An important property is that harmonic functions cannot have local maxima or minima inside the domain unless they are constant.
5. Explain how the Fourier transform handles derivatives and why this is useful for PDEs.
The Fourier transform of the derivative f'(x) is ik F(k), where F(k) is the transform of f. This is derived by integration by parts, assuming f vanishes at infinity. Similarly, the transform of f''(x) is -k^2 F(k). This property turns differential equations in x into algebraic equations in k. For PDEs like the heat or wave equation, the spatial derivative becomes multiplication by a polynomial in k. The resulting ODE in time is easy to solve. After solving in frequency space, we inverse transform to get the solution in physical space. This is a powerful technique for linear PDEs on infinite domains.
6. What is Fredholm theory and what does it say about solvability of elliptic problems?
Fredholm theory studies linear operators of the form L u = f with boundary conditions. It says that either the equation has a unique solution for any f, or there are finitely many linearly independent solutions to the homogeneous problem L u = 0. The Fredholm alternative states: either L is invertible, or the homogeneous adjoint problem has nontrivial solutions. For elliptic operators, the kernel (nullspace) is finite-dimensional. This theory ensures that if the problem is well-posed, the solution exists and is unique. It also gives conditions for solvability when the operator is not invertible.
7. Solve dy/dx + y = y^2 cos x.
Bernoulli with n=2, P=1, Q=cos x. Let v = y^{-1}. Then dv/dx = -y^{-2} dy/dx. Multiply ODE by -y^{-2}: -y^{-2} dy/dx - y^{-1} = -cos x. So dv/dx - v = -cos x. Linear. Integrating factor μ = e^{∫ -1 dx} = e^{-x}. Multiply: e^{-x} dv/dx - e^{-x} v = -e^{-x} cos x. Left side is d/dx (e^{-x} v). Integrate: e^{-x} v = ∫ -e^{-x} cos x dx. Use integration by parts or formula: ∫ e^{-x} cos x dx = (e^{-x}(sin x - cos x))/2. So ∫ -e^{-x} cos x dx = - (e^{-x}(sin x - cos x))/2 + C. Thus e^{-x} v = - (e^{-x}(sin x - cos x))/2 + C => v = - (sin x - cos x)/2 + C e^{x}. So y = 1/(C e^{x} - (sin x - cos x)/2).
8. Explain the concept of a multigrid V-cycle and why it is efficient.
A V-cycle in multigrid starts on the finest grid, performs a few smoothing steps, restricts the residual to the next coarser grid, and repeats until the coarsest grid. On the coarsest grid, you solve the system exactly (or with many iterations). Then you interpolate the correction back up, adding it to the fine-grid solution and applying more smoothing at each level. The V-cycle is efficient because it reduces errors of all wavelengths in a single cycle, and the total work is proportional to the number of unknowns. Typically, a few V-cycles are enough to converge to machine precision.
9. Apply the variational iteration method to u(x)=x+∫_0^x u(t)dt with one iteration starting from u_0=0.
Correction functional: u_{n+1}(x)=u_n(x)+∫_0^x λ[u_n(t)-t-∫_0^t u_n(s)ds]dt. With λ=-1 and u_0=0, we get u_1(x)=0+∫_0^x (-1)[0-t-0]dt = ∫_0^x t dt = x^2/2. Next iteration: u_2(x)=u_1+∫_0^x (-1)[u_1(t)-t-∫_0^t u_1(s)ds]dt. Compute inner: ∫_0^t u_1(s)ds = ∫_0^t s^2/2 ds = t^3/6. Then u_1(t)-t- t^3/6 = t^2/2 - t - t^3/6. Integrate: ∫_0^x (-1)(...)dt = -[x^3/6 - x^2/2 - x^4/24] = -x^3/6 + x^2/2 + x^4/24. So u_2 = x^2/2 + (-x^3/6 + x^2/2 + x^4/24) = x^2 - x^3/6 + x^4/24. This matches the series for e^x-1: x + x^2/2 + x^3/6 + x^4/24+..., but note u_2 has no x term, so convergence is slow.
10. What is the basic idea of a multigrid method for solving linear systems from PDEs?
Multigrid methods solve large linear systems by using a hierarchy of grids: coarse and fine. On a fine grid, simple iterative methods like Gauss-Seidel quickly reduce high-frequency errors but are slow for low-frequency errors. The key idea is to transfer the problem to a coarser grid, where low-frequency errors become high-frequency and can be efficiently reduced. Then you correct the fine-grid solution with the coarse-grid solution. This process is repeated recursively on multiple levels, leading to fast convergence, often in O(N) operations where N is the number of unknowns.
11. Compare the roles of smoothing and coarse-grid correction in multigrid.
Smoothing is performed on the fine grid using a relaxation method like Gauss-Seidel. It quickly reduces oscillatory (high-frequency) error components but leaves smooth (low-frequency) errors almost unchanged. Coarse-grid correction then handles those smooth errors: the residual is restricted to a coarser grid, where the smooth error appears more oscillatory and can be reduced efficiently. The coarse-grid solution is then interpolated back to the fine grid to correct the approximation. Together, these two steps eliminate errors of all frequencies, giving fast convergence.
12. Explain Fejér's theorem and how it differs from Dirichlet's.
Fejér's theorem states that the Cesàro mean (average of partial sums) of the Fourier series of a continuous periodic function converges uniformly to the function. Unlike Dirichlet's theorem, which deals with pointwise convergence of the partial sums themselves, Fejér's theorem uses arithmetic means to smooth out oscillations. This gives uniform convergence for continuous functions, even at points where the Fourier series might diverge. Fejér's theorem shows that Fourier series are Cesàro summable. It is a stronger result in terms of convergence for continuous functions.