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Natural Sciences Notes Mathematics for Natural Sciences Notes

Partial Differential Equations Notes

Updated Partial Differential Equations Notes

Mathematics for Natural Sciences Notes

Mathematics for Natural Sciences

Approximately 176 pages

Notes for every topic covered in Mathematics for Natural Sciences, Part IA. The notes emphasise the most important aspects of each topic - specifically the material which is commonly addressed in examination questions.

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Notes for Partial Differential Equations

General properties of PDEs:

A partial differential equation (PDE) is an equation relating a function f(x,y,…) of more than one variable and its partial derivatives with respect to these variables. It therefore has the form:

$$F\left( x,y,\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\ldots,\frac{\partial^{2}f}{\partial x^{2}},\frac{\partial^{2}f}{\partial y^{2}},\frac{\partial^{2}f}{\partial x\partial y},\ldots \right) = const.$$

Where, in general, PDEs which represent physical systems never exceed the second order. A PDE is considered to be ‘linear’ if it is of the form Ly=const., where L is a linear operator on differentiable functions of y.

Parallels between ODEs and PDEs:

The boundary conditions required to specify unique solutions to a PDE can be compared to those required to specify unique solutions to an ODE:

Ordinary differential equation (ODE): Partial differential equation (PDE):
Information in the form of values of function f and/or values of its ordinary derivatives at points in (x,y,…) space is required. Information in the form of values of function f and/or values of its partial derivatives on surfaces in (x,y,…) space is required.
The boundary conditions fix the arbitrary constants of integration, thus one is needed for a first-order equation, and two are needed for a second-order equation. The boundary conditions fix the arbitrary functions of integration. Usually, if a solution is sought for variables (x,y,…) in some region 𝔻, then the ‘boundary conditions’ need be given on all or part of the region boundary 𝔻. It is generally difficult to work out how much information is sufficient.

Physically-important PDEs:

Wave equation: Diffusion equation: LaPlace’s equation:
$$\boxed{\frac{\mathbf{\partial}^{\mathbf{2}}\mathbf{\psi}}{\mathbf{\partial}\mathbf{x}^{\mathbf{2}}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{c}^{\mathbf{2}}}\frac{\mathbf{\partial}^{\mathbf{2}}\mathbf{\psi}}{\mathbf{\partial}\mathbf{t}^{\mathbf{2}}}}$$ $$\boxed{\frac{\mathbf{\partial}\mathbf{\Phi}}{\mathbf{\partial t}}\mathbf{= k}\frac{\mathbf{\partial}^{\mathbf{2}}\mathbf{\Phi}}{\mathbf{\partial}\mathbf{x}^{\mathbf{2}}}}$$ $$\boxed{\frac{\mathbf{\partial}^{\mathbf{2}}\mathbf{\phi}}{\mathbf{\partial}\mathbf{x}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{\partial}^{\mathbf{2}}\mathbf{\phi}}{\mathbf{\partial}\mathbf{y}^{\mathbf{2}}}\mathbf{= 0}}$$

Physical interpretation:

The curvature of the function in space is the same as its curvature in time. That is, taking a snapshot of the wave at a particular time produces the same type of functional form as examining the behaviour of a particular particle over a given time period. This defines ‘wavelike behaviour’.

Physical interpretation:

The rate of flow of heat is equivalent to the curvature of the heat distribution in space. If the spatial heat distribution at a point is a well (2Φ/x2>0), then the heat flows towards that point with time (Φ/t>0), whereas if it is a maximum, then the heat flows away from that point with time (Φ/t<0).

Physical interpretation:

ϕ(x,y) describes the temperature of a 2-dimensional body in equilibrium. This means that heat is flowing into the body at the same rate as it is flowing out, such that the heat gradient across the body is constant. Thus, the curvature of the heat function in space is zero, in both dimensions.

Types of partial differential equation:

Partial differential equations are classified as elliptic, parabolic or hyperbolic, depending on the values of the coefficients in the general form:

$$a\frac{\partial^{2}\psi}{\partial x^{2}} + 2b\frac{\partial^{2}\psi}{\partial x\partial y} + c\frac{\partial^{2}\psi}{\partial y^{2}} + f\frac{\partial\psi}{\partial x} + g\frac{\partial\psi}{\partial y} + h\psi = 0$$

Elliptic: Parabolic: Hyperbolic:
b2<ac b2=ac b2>ac

LaPlace’s equation is elliptic:

$$\frac{\partial^{2}\phi}{\partial x^{2}} + \frac{\partial^{2}\phi}{\partial y^{2}} = 0$$

The heat equation is parabolic:

$$\frac{\partial\Phi}{\partial t} = k\frac{\partial^{2}\Phi}{\partial x^{2}}$$

The wave equation is hyperbolic:

$$\frac{\partial^{2}\psi}{\partial x^{2}} = \frac{1}{c^{2}}\frac{\partial^{2}\psi}{\partial t^{2}}$$

Method of the separation of variables:

The separation of variables is the most prolific method for solving PDEs for which b=0. That is, PDEs of the form:

$$a\frac{\partial^{2}\psi}{\partial x^{2}} + c\frac{\partial^{2}\psi}{\partial y^{2}} + f\frac{\partial\psi}{\partial x} + g\frac{\partial\psi}{\partial y} + h\psi = 0$$

We find a solution of the separable form ψ(x,y)=X(x)Y(y), which is not the most general solution, and not the only solution, but is the easiest to find because it reduces the PDE to an equivalent number of ODEs. The general steps involved are as follows:

1

Assume a general solution of the form:

ψ(x,y)=X(x)Y(y)

This leads to an equation of ordinary derivatives:

$$aY\frac{d^{2}X}{dx^{2}} + cX\frac{d^{2}Y}{dy^{2}} + fY\frac{dX}{dx} + gX\frac{\partial Y}{\partial y} + hXY = 0$$
2 Divide by XY and rearrange to give two expressions in terms of X and Y, separately.

$$\frac{1}{X}\left\lbrack a\frac{d^{2}X}{dx^{2}} + f\frac{dX}{dx} + hX \right\rbrack = - \frac{1}{Y}\left\lbrack c\frac{d^{2}Y}{dy^{2}} + g\frac{dY}{dy} \right\rbrack = const.$$

Where in order for these expressions in different variables to be equal, they must both equal a constant.
3 Choose a positive, negative or zero constant, depending on whether a sinusoidal, exponential or linear solution is required in either x or y by the boundary conditions.

If the expression in x is, for example:

$$\frac{1}{X}\frac{d^{2}X}{dx^{2}} = const.$$

Then a positive constant gives exponential solutions, a negative constant gives sinusoidal solutions and a zero constant gives linear solutions.
4 Solve the ODEs in x and y for X and Y, respectively, to give the general...

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