Partial differential equations

Partial differential equations is the study of functions with partial derivatives, and two or more independent variables. PDE’s are incredibly useful models, and appear in almost every natural science.

The way that heat is transferred through material is given by a PDE, known as the Heat transfer equation

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

The equation of the displacment in a wave over time can be given by a PDE,

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

And the Poisson equation, used in a wide variety of fields in physics.

\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = P(x,y)

Exam 3 review

• Fourier Series

– Derivatives of fourier

– Sin and cosine

– Integrals of fourier series

• Wave equation

– Find solutions using sep of var

– Boundary conditions

– Initial conditions from solution

– Solutions from int. con

u(x,t) = \sum^\infty_{n=1} \sin\left(\frac{n\pi x}{L}\right) \left( A_n\cos\left( \frac{n\pi ct}{L} \right) + B_n \sin \left( \frac{n\pi ct}{L} \right)\right)

We convert

_n\cos\left( \frac{n\pi ct}{L} \right) + B_n \sin \left( \frac{n\pi ct}{L} \right)

To

R\sin \left( \frac{n\pi ct}{L} + \theta \right)

Where R = \sqrt{A^2_N + B^2_n} and \theta = \arctan(A_n/B_n).

Note that

\sin\left( \frac{n\pi x}{L} \right)\left( R\sin \left( \frac{n\pi ct}{L} + \theta \right) \right) = n^{th} \text{ node}

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Where a node

= A_n\sin \left( \frac{n\pi x}{L} \right)\cos \left( \frac{n\pi ct}{L} \right) + B_n \sin \left( \frac{n\pi x}{L} \right)\sin \left( \frac{n\pi ct}{L} \right)

Using the formula

\cos \alpha - \cos \beta = -2\sin\left( \frac{\alpha + \beta}{2} \right)\sin\left( \frac{\alpha - \beta}{2}\right)

\frac{1}{2}\cos \beta - \frac{1}{2} \cos \alpha = \sin\left( \frac{\alpha + \beta}{2} \right)\sin\left( \frac{\alpha - \beta}{2}\right)

So we set

\frac{\alpha + \beta}{2} = \frac{n\pi x}{L} \quad \text{and} \quad \frac{\alpha - \beta}{2} = \frac{n\pi ct}{L}

Adding them together we get

\frac{\alpha + \beta}{2} + \frac{\alpha - \beta}{2} = \alpha = \frac{n\pi}{L}(x + ct)

Subtracting gives used

\frac{\alpha + \beta}{2} - \frac{\alpha - \beta}{2} = \beta = \frac{n\pi}{L}(x - ct)

Thus we get

\sin \left( \frac{n\pi x}{L} \right)\sin \left( \frac{n\pi ct}{L} \right) = \frac{1}{2}\cos \left( \frac{n\pi}{L}(x-ct) \right) - \frac{1}{2} \cos \left( \frac{n\pi}{L} (x+ct) \right)

When t = 0, we get

\frac{1}{2}\cos \left( \frac{n\pi}{L}(x-ct) \right) - \frac{1}{2} \cos \left( \frac{n\pi}{L} (x+ct) \right) = 0

The solution of the wave equation is the sum of two traveling waves, one to the left and one to the right.

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By conservation of energy we have

\frac{d^2u}{dt^2} = c^2\frac{d^2u}{dx^2}

We have

\int^L_0 \frac{\partial^2u}{\partial t^2} \cdot \frac{\partial u}{\partial t} \, dx = \int^L_0 c^2\frac{\partial^2u}{\partial x^2} \cdot \frac{\partial u}{\partial t} \, dx

Note that

\frac{\partial^2u}{\partial t^2} \cdot \frac{\partial u}{\partial t} = \frac{\partial}{\partial t}\left( \frac{1}{2} \left(\frac{\partial u}{\partial t}\right)^2\right)

Thus

\int^L_0 \frac{\partial^2u}{\partial t^2} \cdot \frac{\partial u}{\partial t} \, dx = \int^L_0 \frac{\partial}{\partial t}\left( \frac{1}{2} \left(\frac{\partial u}{\partial t}\right)^2\right) \, dx

And

\int^L_0 c^2\frac{\partial^2u}{\partial x^2} \cdot \frac{\partial u}{\partial t} \, dx = \frac{\partial u}{\partial x} \frac{\partial u}{\partial t}

The solution is also the infinite sum of some

\text{Solution } = \sum^\infty_{n=1} \text{ Nodes}

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