Introduction to Partial Differential Equations

Learn how the Laplace transform converts PDEs in unbounded domains into simpler ODEs, using linearity, differentiation properties, and convolution to solve transport problems and recover solutions.

Learn to solve the heat equation on a semi-infinite rod with the Laplace transform, derive fundamental solution and delta data, and use the er function with convolution for boundary conditions.

Explore the Fourier sine and cosine transforms on the semi-infinite axis, derive forward and inverse transforms, and apply them to solve the heat equation with boundary conditions, illustrating Green's representation.

Explains Fourier transform and inverse Fourier transform and how they turn the heat equation into an ODE, then solves via convolution with the initial temperature and the fundamental solution.

The method of images for the heat equation on a semi-infinite rod yields a green function as a difference of fundamental solutions.

Explore d'Alembert's formula for the wave equation on an infinite string, decomposing initial displacement and velocity into left and right moving traveling waves at speed c.

Derives Poisson's formula for the Laplace equation in the half-plane via Fourier transforms, yielding the Poisson kernel and a convolution representation.

Apply Laplace transforms to turn PDEs into solvable ODEs, solving a non-homogeneous wave equation with initial and boundary conditions, and a convection–diffusion heat equation via the fundamental solution and convolution.

Learn to solve heat and wave equations in high dimensions by extending separation of variables to multiple Fourier series, building product solutions on rectangular regions and determining eigenvalues and coefficients.

Solve Laplace's equation in a disk by transforming to polar coordinates and separating variables, yielding a Fourier series in theta and a Poisson integral from boundary data.

Explore the Bessel equation and its Bessel functions of the first kind, including half-integer cases, and see how the gamma function helps reveal their properties in circular domains.

Explore solving the two-dimensional wave equation on a unit disk using separation of variables in polar coordinates, yielding eigenfunctions with Bessel functions and a center regularity condition.

Solve wave and electrostatic problems in cylindrical domains by separation of variables, yielding Bessel and modified Bessel eigenfunctions with grounded boundary conditions.

Explore Legendre functions and polynomials, including associated Legendre forms, as eigenfunction solutions to the Legendre equation, yielding bounded, orthogonal functions on [-1,1] for spherical PDEs.

Solve the Laplace equation inside a ball using spherical coordinates and boundary data, then obtain solutions through separation of variables and spherical harmonics, guided by eigenvalue problems.

Solve the heat equation in a unit ball by separation of variables, using spherical harmonics and spherical Bessel functions, to form a complete initial-value solution with zero boundary conditions.

Solve partial differential equations using special functions, via separation of variables, as shown in two problems. Explore the circular drumhead with fixed boundary and the grounded spherical shell electrostatics problem.