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Classification of 2 Order PDE: Calculus

This document discusses the classification and solution methods of second order partial differential equations (PDEs). It states that second order linear homogeneous PDEs can be classified as parabolic, elliptic, or hyperbolic based on the value of B^2 - 4AC. The separation of variables method is introduced to find solutions, where the dependent variable u is written as the product of functions of the independent variables x and y. Some examples are given to demonstrate the classification and solution process.

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398 views2 pages

Classification of 2 Order PDE: Calculus

This document discusses the classification and solution methods of second order partial differential equations (PDEs). It states that second order linear homogeneous PDEs can be classified as parabolic, elliptic, or hyperbolic based on the value of B^2 - 4AC. The separation of variables method is introduced to find solutions, where the dependent variable u is written as the product of functions of the independent variables x and y. Some examples are given to demonstrate the classification and solution process.

Uploaded by

balaji
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Calculus

Classification of 2nd Order PDE

Classification of Second Order Homogenous Linear Equation” A second order


linear homogeneous PDE of the form

∂ 2φ ∂ 2φ ∂ 2φ ∂φ ∂φ
A ∂x 2 + B ∂x∂y + C ∂y 2 + D ∂x + E ∂y + Fφ (x, y) = 0 ------ (1)

Where A, B, C, D, E and F are either functions of x and y only or constants is called


(i) a parabolic equation, if B2 − 4AC = 0
(ii) an elliptic equation, if B2 − 4AC < 0
(iii) a hyperbolic equation, if B2 − 4AC > 0

For Example
(1) consider the one dimensional heat equation
∂u ∂ 2u
2
∂t = c ∂x 2
∂ 2u ∂u
⇒ c ∂x 2 2
− ∂t = 0
Comparing it with (1), we have
A = c2, B = 0 and C = 0
∴ B2 – 4AC = 02 − 4 × c2 × 0 = 0
∴ One dimensional heat equation is parabolic
Similarly, it can be easily observed that

(2) one dimensional wave equation


∂2y 2
∂2y 2
∂t
2 = c ∂x 2 is hyperbolic (B − 4AC > 0) and (3). The laplace equation

∂ 2u ∂ 2u
∂x 2
+ ∂y 2 = 0 is elliptic (B2 − 4AC < 0)

Method of Separation of Variables: Consider a PDE involving a dependent variable u


and two independent variables x and y. In the method of separation of variables, we
find a solution of the PDE in the form of a product of a function of x and a function of y
Calculus

i.e. we write u(x, y) = X(x). Y(y) ----- (1)


∂u ∂ ∂u ∂
then ∂x
= ∂x (xy) = x1y; ∂y
= ∂y
(xy) = xy1

∂ 2u ∂ 2u 1 1 ∂ u
2
= x y, ∂x∂y = x y , ∂y2 = xy11
11
and so on
∂x 2

dX dY d2 X d2 Y
1 1 11 11
Here x = dx ; y = dy ; x = ;y =
dx 2 dy 2

Substituting these in the given PDE, separating x and its derivatives from y and its
derivatives, finding solutions for x and y and substituting them in (1), we get the
solution of the given PDE. This is best explained through the examples given

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