A partial differential equation is one of the types of differential equations, in which the equation contains unknown multi variables with their partial derivatives. It is a special case of an ordinary differential equation.
Why should we learn about Partial Differentiation?
Let us look at some of the applications before actually diving into the pool of Partial differential equation.
Calculating Errors:
It is used to estimate errors in calculated quantities that depend on more than one uncertain experimental.
Thermodynamics:
Thermodynamic energy functions (enthalpy, Gibb’s free energy, Hellholtz free energy) are function of two or more variables. Most thermodynamic quantities (temperature, entrophy, heat capacity) can be expressed as derivatives of these functions.
Financial Engineering:
Financial engineers use partial derivatives to assess a portfolio’s sensitivity to changes in market conditions (interest rates, volatility). Then can hedge against risk by designing portfolio’s respect to market values.
Partial Differential Equations:
Many laws of nature are best expressed as relations between the partial derivatives of one or more quantities.
\( ih \frac {\partial \psi} {\partial t}= \frac {-h^{2}}{2m} \triangledown^{2} \psi+\triangledown \psi\)
and the Navier-Stokes equation describes all fluid motion.
Important properties of functions that we encounter in engineering are checked by using partial derivatives For instance, the Laplace’s equation
\(\frac {\partial ^{2} u} {\partial x^{2}}+ \frac{\partial^{2}u}{\partial y^{2}}=0\)
is used to check if the function \(u = u(x, y)\) is harmonic.
Harmonic functions have few properties:
Theorem (Maximum principle): Suppose \(u = u(x, y)\) is harmonic on a open region \(Z\). And, \(z_{0}\) is in \(Z\) region . If \(u\) has a relative maximum or minimum at \(z_0\) then, \(u\) is constant on a disk centered at \(z_0\). Harmonic functions are shown to have several more useful properties which make them well suited for robotics applications.
Getting started with partial derivatives
Recall that given a function of one variable, \(f(x)\), the derivative, \(f_0 (x)\), represents the rate of change of the function as \(x\) changes. This is an important interpretation of derivatives and we are not going to want to lose it with functions of more than one variable.
What do we do if we only want one of the variables to change, or if we want more than one of them to change?
Suppose we are designing a vehicle and we want to find out the effect of temperature on mileage.
We know that mileage depends on factors other than temperature – the pressure, velocity with which the vehicle is running, the terrain etc.
Then how can we go about to find out what we want?
What we do is, we keep the other factors (other than temperature, here) a constant ?
How do we do it?
If we know for what type of usage, the terrain etc.
That we are designing the vehicle – for example, it can be a mountainous terrain with an average velocity of \(50km/hr\) – then we can suppress the other factors and observe how changes in temperature affects the mileage.
This is exactly what we are going to do here. We will consider a function of several variables and find out the rate at which the function changes when exactly one of the variables is changing and the remaining variables are kept constant.
In fact, if we are going to allow more than one of the variables to change there are then going to be an infinite amount of ways for them to change. For instance, one variable could be changing faster than the other variable(s) in the function. Notice as well that it will be completely possible for the function to be changing differently depending on how we allow one or more of the variables to change.
If we consider an example from medicine:
An individual’s health depends on various factors like blood pressure, glucose level, age, type of profession etc.
That is, \(h = h(x, y, z, w)\) where \(h\) denotes health which is a function of \(x\) → blood pressure, \(y\) → glucose level, \(z\) → age and \(w\) → profession.
Therefore, if a researcher wants to check if a drug is effective in lowering blood pressure, then she will be able to interpret the results properly only if she keeps the remaining factors such as glucose level, age and profession as constant for the individuals she is testing.
This is because, the changes that one sees in the blood sugar levels can be attributed to (younger) age or absence of diabetes etc.
Hence, if she observes the changes in blood pressure for a sample with similar glucose level, age and profession, then the effect of the drug can be correctly interpreted for blood pressure.
Partial Differentiation
A partial derivative of a function of several variables is the ordinary derivative w.r.t. one of the variables are held constant. Partial differentiation is the process of finding partial derivatives. All the rules of differentiation applicable to function of a single independent variable are also applicable in partial differentiation with the only difference that while differentiating (partially) w.r.t. one variable, all the other variables are treated (temporarily) as constants.
Consider a function \(u\) of three independent variables \(x,y,z,u = f(x,y,z)\).
Keeping \(y,z\) constant and varying only \(x\), the partial derivative of \(u\) w.r.t. \(x\) is denoted by \(\frac{\partial u}{\partial x}\) and is defined as the limit
\(\lim_{\triangle x \rightarrow 0} \frac {\partial u}{\partial x}= \frac {f(x+\triangle x,y,z)- f(x,y,z)}{\triangle x}\)
Partial derivatives of \(u\) w.r.t. \(y\) and \(z\) can defined similarly and are denoted by \(\frac {\partial u}{\partial y}+ \frac {\partial u}{\partial z}\)
Notation: The partial derivative \(\frac {\partial u}{\partial x}\) or \(f_{x}\) or \(f_{x}(x,y,z)=D_{x}f\).
Thus, we can have \( \frac {\partial u}{\partial y}= \frac {\partial f}{\partial y}=f_{y}=D_{y}f\)
The value at a partial derivative at a point \((a,b,c)\) is denoted by,
\(\frac {\partial u}{\partial x} \mid _{x=a,y=b, z=c} = \frac {\partial u}{\partial x} \mid _{(a,b,c)}=f_{x}(a,b,c)\)
Geometrical Interpretation
The partial derivative of a function of two variables \(z = f(x, y)\) represents the equation of a surface in \(xyz\) co-ordinate system.
Let \(AP B\) be the curve, which is cut by a plane through any point \(P\) on the surface parallel to the \(xz\)-plane.
As the point \(P\) moves along this curve \(AP B\), its co-ordinates \(z\) and \(x\) vary while \(y\) remains constant.
The slope of the tangent line at \(P\) to \(AP B\) represents the rate at which \(z\) changes w.r.t. to \(x\).
Thus, \(\frac {\partial z}{\partial x}= tan(\alpha)\)=slope of the curve of \(AP B\) at the point \(P\).
Similarly, \(\frac {\partial z}{\partial y}= tan(\beta)\)=slope of the curve of \(CPD\) at the point \(P\).
Partial Differential Equation
A Partial Differential Equation (PDE) is a differential equation containing partial derivatives of the dependent variable (one or more) with more than one independent variable.
Partial differential equations (PDE’s) are equations that involve rates of change with respect to continuous variables.
A PDE for a function \(u(x_{1},……x_{n})\) is an equation of the form
\(f(x_{1}..x_{n};\frac{\partial u}{\partial x_{1}},..,\frac{\partial u}{\partial x_{n}}; \frac{\partial ^{2} u}{\partial x_{1}\partial x_{1} },..,\frac{\partial ^{2} u}{\partial x_{n}\partial x_{n}};…)=0\)
The PDE is said to be linear if \(f\) is a linear function of \(u\) and its derivatives. The simple PDE is given by
\(\frac{\partial u}{\partial x}(x,y)= 0\)
This relation implies that the function \(u(x, y)\) is independent of \(x\). However, the equation gives no information on the function’s dependence on the variable \(y\).
Hence, the general solution of this equation is \(u(x, y) = f(y)\) where \(f\) is an arbitrary function of \(y\).
The analogous ordinary differential equation is \(\frac{\partial u}{\partial x}(x)= 0\) which has the solution \(u(x) = c\), where \(c\) is a constant value.