Parabolic Equations Calculator

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Example

Created on 2024-06-20Asked by Ella Hill (Solvelet student)

Solve the parabolic equation

u_t = 3u_{xx}

with initial condition

u(x,0) = e^{-x^2}

Solution

To solve the parabolic equation

u_t = 3u_{xx}

with initial condition

u(x,0) = e^{-x^2}

: 1. The general solution of the heat equation

u_t = \alpha u_{xx}

is given by:

u(x,t) = \frac{1}{\sqrt{4\pi \alpha t}} \int_{-\infty}^{\infty} e^{-\frac{(x-y)^2}{4\alpha t}} u(y,0) \, dy.

2. For

\alpha = 3

and

u(y,0) = e^{-y^2}

u(x,t) = \frac{1}{\sqrt{12\pi t}} \int_{-\infty}^{\infty} e^{-\frac{(x-y)^2}{12t}} e^{-y^2} \, dy.

3. Simplify the integral:

u(x,t) = \frac{1}{\sqrt{12\pi t}} \int_{-\infty}^{\infty} e^{-\left( \frac{(x-y)^2}{12t} + y^2 \right)} \, dy.

Therefore, the solution to the parabolic equation is given by the integral formula above. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Aiden Jones on Solvelet

1. Solve the heat equation

\frac{{\partial u}}{{\partial t}} = \frac{{\partial^2 u}}{{\partial x^2}}

subject to given initial and boundary conditions.2. Analyze the stability of the solution to the parabolic equation

u_t = u_{xx}

using von Neumann stability analysis.

DefinitionParabolic equationsIt is a kind of Partial Differential Equation (PDE) that describes the diffusion processes. For u the desired function, these are ut=kuxx. For instance, the heat equation ut=αuxx is used to describe the flow of heat in an area over time.

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