Solving PDE: A Step-by-Step Guide

Thread starter stanley.st
Start date Oct 1, 2011
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Pde Specific

In summary, a PDE, or partial differential equation, is a mathematical equation used to describe physical phenomena in various fields. The process for solving a PDE involves identifying its type, converting it to standard form, and using appropriate techniques such as separation of variables or numerical methods. The type of PDE can be determined by looking at the highest-order derivatives present in the equation. Common techniques for solving PDEs include separation of variables, method of characteristics, and finite difference methods. To check the accuracy of a PDE solution, you can compare it with other known solutions, use convergence analysis, or calculate error norms.

Oct 1, 2011

stanley.st

Hello,

I derived a model in the form

[tex]\begin{array}{rcl}\frac{\partial U(\vec{x},t)}{\partial t}&=&\gamma^2\Vert\nabla U(\vec{x},t)\Vert,\\\int_{\Omega}U(\vec{x},t)\, d \Omega&=&U_0,\quad\forall t\\U(\vec{x},0)&=&f(\vec{x}).\end{array}[/tex]

I don't know to solve that.

THanks for help.

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Oct 3, 2011

Eynstone

This looks like the heat equation.So, I suggest seperating variables.

1. What is a PDE?

A PDE, or partial differential equation, is a type of mathematical equation that involves multiple independent variables and their partial derivatives. It is commonly used to describe physical phenomena in fields such as physics, engineering, and economics.

2. What is the process for solving a PDE?

The process for solving a PDE involves several steps, including identifying the type of PDE, converting it to standard form, determining boundary and initial conditions, and using appropriate techniques such as separation of variables or numerical methods to find a solution.

3. How do I determine the type of PDE?

The type of PDE can be determined by looking at the highest-order derivatives present in the equation. If the highest-order derivative is with respect to only one independent variable, it is a first-order PDE. If it is with respect to two independent variables, it is a second-order PDE, and so on.

4. What are some common techniques for solving PDEs?

Some common techniques for solving PDEs include separation of variables, method of characteristics, finite difference methods, and numerical methods such as finite element analysis or finite volume method. The choice of technique depends on the type of PDE and its boundary conditions.

5. How can I check the accuracy of my PDE solution?

One way to check the accuracy of a PDE solution is to compare it with other known solutions or analytical solutions. Another way is to use convergence analysis, where the solution is calculated for different grid sizes or time steps to see if it approaches a particular value. You can also use error norms to quantify the difference between the approximate and exact solution.