Partial Differential Equations: Comprehensive Course

Review the gradient as a three-dimensional vector of partial derivatives in x, y, z, and the Laplacian as the divergence of the gradient, a core PDE operator.

Solve partial differential equations with Fourier transforms in electrostatic settings, linking electric field divergence to charge density and the Laplace equation for the potential under far-field boundary conditions.

Apply the three-dimensional fourier transform to del^2 phi = F(x) with phi and its derivatives vanishing at infinity. Show that -k^2 phi_hat = F_hat and outline inverse transform for phi.

Derive the electric field of a point charge at the origin from a delta function charge density, compute the potential via an integral, and use E = -grad V.

Explore the diffusion problem by modeling a particle distribution F(x,t) in three-dimensional space and deriving the governing partial differential equation with boundary conditions.

The lecture derives the diffusion equation df/dt = D ∇^2 f by a Taylor expansion in independent x, y, z directions and second-order terms.

Apply Fourier transforms to the diffusion equation in time and space, derive the transformed equation, and handle the initial Dirac delta and boundary behavior with complex calculus.

Solve the diffusion equation using Fourier transforms, converting x to k and omega, deriving F double tilde, and outlining inverse transforms to time and space domains.

Compute the inverse transform of f tilde of k for the diffusion equation using complex calculus and residues, evaluating the contour integral and obtaining e^{-k^2 d}.

Derive the diffusion equation solution as a gaussian distribution with time-dependent variance by completing the square and then apply convolution with a delta initial condition for general initial functions.

Learn to solve a damped second-order linear ODE with a forcing term using the Fourier transform, yielding a time-domain solution from initial conditions plus a forced response from f(t).

Apply the method of characteristics to solve u_t = x u_x - u with u(x,0)=f(x) and |x|<1; get u(x,t)=e^{-t} f(x e^{t}), which vanishes as t grows.

Solve the two-dimensional Laplace equation in Cartesian coordinates on a rectangle by separation of variables. Build a cosine-series solution in x with y-dependent exponentials, applying Neumann and step-function boundary conditions.

Solves the Laplace equation outside a circle with polar coordinates using separation of variables, yielding a solution comprising a constant and a cos(3θ)/r^3 term that matches the boundary cos(3θ)+log 2.

Solve Laplace's equation in a semicircular region using polar coordinates, with u(a, θ)=g(θ) on the circle and an insulated diameter; derive a Fourier cosine series for u(r, θ).

Solve Laplace’s equation in an annulus using polar coordinates, enforce boundary data at a and B, and determine coefficients via orthogonality and linear systems.

Examine streamlines as particle trajectories in an incompressible, divergence-free velocity field, where velocity is tangential to lines of constant U. A Laplace equation solution yields circular streamlines.

Solve the nonhomogeneous heat equation by superposing a homogeneous solution with a forcing term, using Fourier series, separation of variables, and integrating factors for zero boundary and sine initial data.

Transform a nonhomogeneous heat equation with nonhomogeneous boundary and initial conditions into a homogeneous problem using a tilde function, then solve with W and V decomposition.

Split the nonhomogeneous heat equation into a homogeneous part and a nonhomogeneous part, solve each (via separation of variables as needed), and sum to get U = W + V.

Apply the D'Alembert formula to a nonhomogeneous wave equation by splitting into two problems - a homogeneous part and a forcing part - and summing their solutions.

Apply the D'Alambert formula to solve the homogeneous wave equation with given boundary and initial conditions, using the element representation and recalling the required sine terms.

Prove the energy conservation law for the homogeneous wave equation with normal boundary conditions, showing that the energy E(t) = 1/2 ∫0^l [u_t^2 + c^2 u_x^2] dx remains constant.