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PHYS-216: Mathematical methods (for SPH)

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Lectures in this course (63)

Bessel equation: Bessel functions of the 1st kind

Explores the Bessel equation and the derivation of Bessel functions of the 1st kind.

Bessel Equation: Bessel Functions of the 2nd Kind

Explores the Bessel equation and its solutions, focusing on the Bessel functions of the 2nd kind and their properties.

Linear 2nd Order ODEs: Boundary Value Problems

Explores linear 2nd order ODEs as boundary value problems with various types of conditions, ensuring unique solutions.

Sturm-Liouville Problem: Introduction

Introduces the Sturm-Liouville problem, covering ODE solutions, boundary conditions, and eigenvalue problems.

Sturm-Liouville Problem: Homogeneous Boundary Conditions Essential Role

Explores the Sturm-Liouville eigenvalue problem, emphasizing the essential role of boundary conditions in ensuring self-adjointness and forming an orthogonal basis.

Sturm-Liouville Problem: Legendre Equation

Explores the Sturm-Liouville problem linked to the Legendre equation and its significance in physics.

Legendre Functions of 2nd Kind & Spherical Harmonics

Explores Legendre functions of the second kind and spherical harmonics properties and relationships.

Linear Algebra Concepts: Motivation for Studying Eigenmodes in Physical Systems

Explores the motivation for studying linear algebra concepts in eigenmodes of physical systems and their central role in quantum mechanics.

Casting ODEs into Sturm-Liouville form

Explores casting ODEs into self-adjoint form and the role of boundary conditions in ensuring self-adjointness.

Unbounded and Bounded Systems: Fourier Methods

Explores the spectral properties of unbounded and bounded systems using Fourier methods and emphasizes the importance of choosing the correct representation for different boundary conditions.

Spectral Properties of Fourier Series and Fourier Transforms

Explores the motivation behind Fourier series and transforms, their fundamentals, and applications in solving differential equations.

Basis Decomposition of PDEs: 1D and Multidimensional Problems

Explores solving PDEs using eigenmodes decomposition and Fourier series representation of solutions.

Properties of Fourier and inverse Fourier transforms

Explores the properties of Fourier and inverse Fourier transforms, analyzing finite wave trains, Gaussian functions, and 3D space transformations.

Boundary Conditions and Basis Decomposition

Explores boundary conditions, basis functions, and PDE solutions using Fourier and Bessel series.

Fourier Transforms: Dirac Delta, Fourier Integral, and Transform

Explores the Dirac delta, Fourier integral, and Fourier transform applications in solving PDE problems.

Fourier Transforms: Properties and Applications

Explores Fourier transforms, including properties, convolution, Parseval's theorem, and energy spectral density for non-periodic functions.

Solving unbounded PDE's using Fourier Transforms and Convolution: Obtaining the d'Alembert solution of the wave equation

Explores elementary properties of Fourier Transforms, convolution, Parseval's Theorem, and the d'Alembert solution of the wave equation using Fourier Transforms and convolution.

Dispersion and Schrödinger Equation Solution

Explores the solution of the 1D Schrödinger equation using Fourier transforms and convolution techniques.

Function Spaces and Hilbert Spaces

Introduces function spaces and Hilbert spaces, discussing inner product spaces and the importance of completeness in Hilbert spaces.

Orthogonal/Orthonormal Bases and Polynomials

Explores orthogonal and orthonormal bases, Gram-Schmidt process, and orthogonal polynomials in physics.

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