Introduction to Partial Differential Equation

A differential equation is an equation that relates the derivatives of a (scalar) function depending on one or more variables. A differential equation is called ordinary if a function say,  u, f, depends on only a single variable,say x,or y , and partial if it depends on more than one variable say, (x,y), (x,t) …

 

 

In other words, ODEs evolve from single variable functions i.e u(x) = some function of x,   f(x) = some function of x,…u(y) = some function of y,   f(t) = some function of t,…

While PDEs evolve from functions of several variables i.e

u(x,y) = some function of x and y,  

f(x,y,t) = some function of x,y and t,…

u(y,s) = some function of y and s, 

f(w,x,y,t) = some function of w,x,y and t,…

In the above illustrations, u and f are dependent variables whiles s,t,w,x,y are the independent variables.

 

A partial differential equation (PDE) is one which involves one dependent variable and two or more independent variables.The independent variables may be space variables only or one or more space variables and time.

 

Before you can study PDEs, you should be able to

i. carry out some partial differentiation

ii. solve ODEs

 

Since you might have forgotten most of what you studied in your ODE, we shall take our time to revise the relevant ODEs before progressing in each section.

 

Section Warm Up

1. Find the partial derivatives fx and fy for the function   f(x, y) = 3x – x2y2 + 4xy2.

Solution

Treating y as a constant and differentiating with respect to x produces

f(x, y) = 3x – x2y2 + 4xy2                Write original function. 

  fx(x, y) = 3 – 2xy2 + 4y2.            Partial derivative with respect to x.

 

 

Treating x as a constant and differentiating with respect to y produces

f(x, y) = 3x – x2y2 + 4xy2                Write original function. 

  fy(x, y) =  – 4x2y + 8xy.            Partial derivative with respect to y

 

2. Given that f(x, y, z) = xy + yz2 + xz. Find fz     

 

Solution

To find the partial derivative of f(x, y, z) = xy + yz2 + xz  with respect to z, consider x  and y to be constant and obtain

fz(x, y, z) = 2yz + x 

3. Find the partial derivative of f(x, y, z) = z sin(xy2 + 2z) with respect to z.

Solution

Consider x  and y to be constant. Then, using the Product Rule, you obtain

fz(x, y, z) = (z)fz[sin(xy2 + 2z)] + sin(xy2 + 2z)fz[z]

              = (z)cos(xy2 + 2z)(2) + sin(xy2 + 2z)

              = 2zcos(xy2 + 2z) + sin(xy2 + 2z)

 

 

 

 

            

 

 

 

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