Partial differential equation problem

Thread starter sam topper.
Start date Oct 26, 2013
Tags

Differential Differential equation Partial

In summary, a partial differential equation (PDE) is a mathematical equation that involves multiple independent variables and their partial derivatives. They are important because they have widespread applications in various fields such as physics, engineering, economics, and biology, helping to model and understand complex systems. Common methods for solving PDEs include separation of variables, method of characteristics, and various numerical methods. Boundary conditions are essential for obtaining a unique solution to a PDE problem. Real-life examples of PDEs include heat conduction, fluid flow, population dynamics, electromagnetic fields, and financial mathematics and image processing.

Oct 26, 2013

sam topper.

Homework Statement

using the method of separation of variables, solve ∂u/∂x=4∂u/∂y, where [itex]u=3e^-y - e^-5y[/itex] when x=0.

Homework Equations

The Attempt at a Solution

let u(x,y)=X(x)Y(y)
=XY.

Physics news on Phys.org

Oct 26, 2013

tiny-tim

Science Advisor

Homework Helper

hi sam!

sam topper. said:

let u(x,y)=X(x)Y(y)
=XY.

good

what next?

What is a partial differential equation?

A partial differential equation (PDE) is a mathematical equation that involves multiple independent variables and their partial derivatives. It is used to describe the relationship between a function and its partial derivatives in terms of the independent variables.

Why are partial differential equations important?

Partial differential equations are important because they have widespread applications in various fields such as physics, engineering, economics, and biology. They help in modeling and understanding complex systems and phenomena, making them a powerful tool in scientific research and problem-solving.

What are some common methods for solving partial differential equations?

Some common methods for solving partial differential equations include separation of variables, method of characteristics, finite difference methods, finite element methods, and numerical methods such as the Runge-Kutta method and the shooting method.

What are the boundary conditions in a partial differential equation problem?

Boundary conditions are the conditions that must be satisfied by the solution to a partial differential equation at the boundaries of the domain in which it is defined. These conditions specify the behavior of the solution at the boundaries and are essential for obtaining a unique solution to the problem.

What are some real-life examples of partial differential equations?

Partial differential equations can be used to model various phenomena in the real world, such as heat conduction, fluid flow, population dynamics, and electromagnetic fields. They are also used in financial mathematics to model stock prices and in image processing to enhance and analyze images.