Partial Differential Equations Calculator

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Example

Created on 2024-06-20Asked by Theodore Lopez (Solvelet student)

Solve the partial differential equation

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

subject to the initial condition

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

Solution

To solve the partial differential equation

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

subject to the initial condition

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

: 1. The general solution of the heat equation

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

is given by:

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

2. For

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

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

3. Simplify the integral:

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

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

Some of the related questions asked by Mateo Anderson on Solvelet

1. Solve the wave equation

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

subject to given initial and boundary conditions.2. Analyze the existence and uniqueness of solutions to the heat equation

u_t = u_{xx}

with given initial and boundary conditions.

DefinitionThe method of partial fraction is used to simplify rational functions by breaking it down in simpler fractions. This helps in integrating and solving differential equations easily. For instance: So (x+1)(x+2)2x+3 is x+1A+x+2B.

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