3 Approximating partial derivatives

Earlier, in HELM booklet  31.3, we saw methods for approximating first and second derivatives of a function of one variable. We review some of that material here. If $y=y\left(x\right)$ then the forward and central difference approximations to the first derivative are:

$\frac{dy}{dx}\approx \frac{y\left(x+\delta x\right)-y\left(x\right)}{\delta x},\frac{dy}{dx}\approx \frac{y\left(x+\delta x\right)-y\left(x-\delta x\right)}{2\delta x}$

and the central difference approximation to the second derivative is:

$\frac{d^{2}y}{d{x}^2}\approx \frac{y\left(x+\delta x\right)-2y\left(x\right)+y\left(x-\delta x\right)}{{\left(\delta x\right)}^2}$

in which $\delta x$ is a small $x$ -increment. The quantity $\delta x$ is what we previously referred to as $h$ , but it is now convenient to use a notation which is more closely related to the independent variable (in this case $x$ ). (Examples implementing the difference approximations for derivatives can be found in HELM booklet  31.)

We now return to the subject of this Section, that of partial derivatives. The PDE $u_t=\alpha u_{xx}$ involves the first derivative $\frac{\partial u}{\partial t}$ and the second derivative $\frac{{\partial }^{2}u}{\partial x^2}$ . We now adapt the ideas used for functions of one variable to the present case involving $u=u\left(x,t\right)$ .

Let $\delta t$ be a small increment of $t$ , then the partial derivative $\frac{\partial u}{\partial t}$ may be approximated by:

$\frac{\partial u}{\partial t}\approx \frac{u\left(x,t+\delta t\right)-u\left(x,t\right)}{\delta t}$

Let $\delta x$ be a small increment of $x$ , then the partial derivative $\frac{{\partial }^{2}u}{\partial x^2}$ may be approximated by:

$\frac{{\partial }^{2}u}{\partial x}\approx \frac{u\left(x+\delta x,t\right)-2u\left(x,t\right)+u\left(x-\delta x,t\right)}{{\left(\delta x\right)}^2}$

The two difference approximations above are the ones we will use later in this Section. Example 14 below refers to these and others.

Example 14

Consider the function $u$ defined by

$u\left(x,t\right)=sin\left(x^2+2t\right)$

Using increments of $\delta x=0.004$ and $\delta t=0.04$ , and working to 8 decimal places, approximate

1. $u_x\left(2,3\right)$ with a one-sided forward difference
2. $u_{xx}\left(2,3\right)$ with a central difference
3. $u_t\left(2,3\right)$ with a one-sided forward difference
4. $u_t\left(2,3\right)$ with a central difference.

Enter your approximate derivatives to 3 decimal places.

Solution

The evaluations of $u$ we will need are $u\left(x,t\right)=-0.54402111$ , $u\left(x+\delta x,t\right)=-0.55738933$ , $u\left(x-\delta x,t\right)=-0.53054047$ , $u\left(x,t+\delta t\right)=-0.60933532$ , $u\left(x,t-\delta t\right)=-0.47522703.$ It follows that

1. $u_x\left(2,3\right)\approx \frac{-0.55738933+0.54402111}{0.004}=-3.342$
2. $u_{xx}\left(2,3\right)\approx \frac{-0.55738933+2\times 0.54402111-0.53054047}{0.004^2}=7.026$
3. $u_t\left(2,3\right)\approx \frac{-0.60933532+0.54402111}{0.04}=-1.633$
4. $u_t\left(2,3\right)\approx \frac{-0.60933532+0.47522703}{2\times 0.04}=-1.676$

to 3 decimal places. (Workings shown to 8 decimal places.)