Contents

1. Partial Differentiation and Partial Differential Equation...................... 1-37 2. Partial Differential Equations............................................................. 38-85 3. Fourier Series....................................................................................86-113 4. Laplace Transformation....................................................................114-152 5. Numerical Techniques. ..................................................................... 153-184 6. Numerical Methods for Solution .

of

Partial Differential Equation. 185-221

Chapter-1

1.1 Functions of Two or More Variables A symbol z which has a definite value for every pair of values of x and y is called a function of two independent variables x and y and is written as z f x, y or x, y 1.2 Limits The function f x, y is said to tend to limit l as x a and y b if and only if the limit l is independent of the path followed by the point x, y as x a and y b and we write lim f x , y l

x y

a b

Or, in circular neighbourhood we define the limit as The function f x, y defined in a region R, is said to tend to the limit l as x a and y b if and only if corresponding to a positive number , another positive number such that

f x , y

l

e for 0 x a 2 y b

for every point x, y in R .