Partial Differential Equations

  • Introduction

So far, while studying calculus, we have dealt with functions of single variable, i.e. {y=f(x)}. {sin (x^2), ln \ x, e^{cos \ (tan \ x)}} are few examples. Irrespective of their complexity, the variable {y} always depended on the value of independent variable {x}. We also defined the derivatives and integrals of {f(x)} and studied few applications of them.

More often than not, we encounter situations, where a function {f} needs more than 1 independent variable specified for its definition. Such functions are known as functions of several variables. e.g. a function of 2 variables is

{f(x,y) = sin (x) e^y \times xy^{3/2}}

Thus, without knowing values of both {x} and {y} simultaneously, we cannot get a unique value of {f(x,y)}.

One can define a function of as many variables as one wants. (Of course, it should make some sense.) In many of the problems in mechanical engineering, the functions are of at the most 4 independent variables; viz. 3 space variables, {(x,y,z)} and a time variable {t}.

The partial differentiation involves obtaining the derivatives of functions of several variables.

  • Definition and Rules

Let {z} be a function of 2 independent variables {x} and {y}. To differentiate {z} partially w.r.t {x}, we treat {y} as a constant and follow the usual process of differentiation. Thus,

{\frac {\partial z}{\partial x} = \lim \limits_{\delta x \to 0} \frac {f(x + \delta x, y) - f(x,y)}{\delta x}}

Similarly,

{\frac {\partial z}{\partial y} = \lim \limits_{\delta y \to 0} \frac {f(x, y+ \delta y) - f(x,y)}{\delta y}}

Thus, the definition is similar to that of ordinary differentiation. The condition of existence of the limit is necessary.

Note that we use the letter {\partial} for partial derivatives and the letter {d} for ordinary derivatives.

The rules of for differentiation of addition, subtraction, multiplication, division are same as ordinary differentiation.

  • Derivatives of Higher Order

Having obtained the first order derivatives {\frac {\partial z}{\partial x}} and {\frac {\partial z}{\partial y}}, we now define the second order derivatives, i.e.

{\frac {\partial}{ \partial x} \Big ( \frac {\partial z}{\partial x} \Big) , \frac {\partial}{ \partial y} \Big ( \frac {\partial z}{\partial x} \Big), \frac {\partial}{ \partial y} \Big ( \frac {\partial z}{\partial x} \Big), \frac {\partial}{ \partial y} \Big ( \frac {\partial z}{\partial y} \Big)}

For a function of 2 variables, four 2nd order derivatives are possible. These are sometimes written as

{\frac {\partial^2 z}{ \partial x^2} = z_{xx}, \ \frac {\partial^2 z}{\partial y \partial x} = z_{yx},\ \frac {\partial^2 z}{\partial x \partial y} = z_{xy}, \ \frac {\partial^2 z}{\partial y^2} = z_{yy}}

If the function and its derivatives are continuous, then we have

{\frac {\partial^2 z}{\partial y \partial x}= \frac {\partial^2 z}{\partial y \partial x}}

One can define derivatives of order {> 2} by following the same procedure.

  • Types of Problems (Crucial from exam point of view)

I) Based on the definition and the commutative property of partial differentiation

II) Based on the concept of composite functions (Mostly involve the relations between cartesian and polar coordinates)

  • Homogeneous Functions

When the sum of indices of the variables in a function is same for all terms, the function is said to be homogeneous of degree equal to the sum.

{6x^3y^2 + x^5 - xy^4}

is an example. (Degree {= 5})

Note that each term must be explicitly of the form $a x^m y^n$. Thus, $sin (6x^3y^2 + x^5 – xy^4)$ is NOT a homogeneous function.

  • Euler’s Theorem (by Leonhard Euler)

For a homogeneous function {z=f(x,y)} of degree {n},

{x \frac {\partial z}{\partial x} + y \frac {\partial z}{\partial y} = nz}

As a consequence of this,

{x^2 \frac {\partial^2 z}{ \partial x^2} + 2xy \frac {\partial^2 z}{\partial x \partial y} + y^2 \frac {\partial^2 z}{ \partial y^2} = n (n-1)z}

Similarly, if {u =f(x,y,z)} is a homogeneous function of 3 independent variables of degree {n}, then

{x \frac {\partial u}{\partial x} + y \frac {\partial u}{\partial y} + z \frac {\partial u}{\partial z}= nu}

  • Total Derivatives

Consider a function {z = f(x,y)}. If it so happens that {x} and {y} themselves are functions of another variable {t}, then the total derivative of {z} w.r.t. {t} is defined as

{\frac {dz}{dt} = \frac {\partial z}{\partial x} \times \frac {dx}{dt} + \frac {\partial z}{\partial y} \times \frac {dy}{dt}}

Thus, if we are given a function {z = g(t)}, we would differentiate it w.r.t. {t}, thus getting {\frac {dz}{dt}}. Instead, if {z} is expressed as {f(x,y)} and {x= \phi (t)} and {y = \psi (t)}, then obtaining the total derivative of {f(x,y)} will be equivalent to getting {\frac {d}{dt} g(t)}

  • Applications

We will discuss the applications of partial differentiation in the next blogpost.

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