Introduction To Partial Differential Equations

Partial Differential Equations (PDEs) involve partial derivatives of multivariable functions, essential in various scientific fields.

Definition

A Partial Differential Equation (PDE) involves partial derivatives of a multivariable function, typically expressed in a general format.

First-Order Pdes

First-order partial differential equations (PDEs) are equations involving the first derivatives of a function with respect to multiple variables, and this section covers their formulation and solutions using Lagrange's method.

Linear Pdes (Lagrange's Method)

This section discusses Linear Partial Differential Equations and introduces Lagrange's method for their solutions using auxiliary equations.

Second-Order Linear Pdes

This section focuses on the general form, classification, and solution methodologies for second-order linear partial differential equations (PDEs).

General Form

This section discusses the general form of Partial Differential Equations (PDEs) and their classification based on order and behavior.

Classification

This section outlines the classification of second-order linear partial differential equations (PDEs) into elliptic, parabolic, and hyperbolic categories based on the discriminant B² - 4AC.

Cf & Pi Method (Complementary Function And Particular Integral)

This section focuses on the CF & PI method for solving second-order linear partial differential equations, highlighting the importance of complementary functions and particular integrals.

Initial And Boundary Conditions

This section provides an overview of initial and boundary conditions essential for solving partial differential equations (PDEs).

Initial Conditions

Initial conditions are essential for solving partial differential equations (PDEs), as they specify the solution and its derivatives at the start time.

Boundary Conditions

Boundary conditions define how a solution behaves at the boundaries of a given domain in PDEs.

Wave Equation And D'alembert's Solution

The section presents the wave equation for one-dimensional wave propagation and introduces D'Alembert's solution, which involves arbitrary functions derived from initial conditions.

D'alembert's Solution

D'Alembert's solution presents a method for solving the one-dimensional wave equation using arbitrary functions based on initial conditions.

Duhamel's Principle

Duhamel's Principle is a method used to solve non-homogeneous wave equations by superposition of solutions.

Diffusion And Vibration Problems

This section introduces the heat equation in the context of diffusion problems and discusses the vibration of strings using the wave equation.

Heat Equation (Diffusion)

The Heat Equation describes how heat diffuses through a given medium over time.

Vibration Of A String

This section introduces the vibration of strings modeled by the wave equation and the formation of standing waves under specific boundary conditions.

Separation Of Variables

The Separation of Variables technique transforms a PDE into two ODEs, which can be solved independently.

Laplacian In Different Coordinates

This section covers the expression of the Laplacian operator in Cartesian, cylindrical, and spherical coordinate systems.

Cartesian

This section introduces the Laplacian operator in Cartesian coordinates, detailing its formulation and application in partial differential equations (PDEs).

Cylindrical

This section introduces the Laplacian operator in cylindrical coordinates, focusing on its formulation and applications.

Spherical

This section covers the Laplacian operator in spherical coordinates, essential for problems involving three-dimensional space.

Special Function Solutions

This section discusses special function solutions to partial differential equations, specifically Bessel and Legendre functions.

Bessel Functions

Bessel functions are essential solutions to differential equations that arise in cylindrical coordinate systems, commonly applied in physics and engineering.

Legendre Functions

Legendre functions are solutions to the Legendre differential equation, important in solving problems in spherical coordinates.

One-Dimensional Diffusion (Heat) Equation

This section introduces the one-dimensional diffusion equation, which describes how heat diffuses over time in a given medium.