classification of second order partial differential equation

1. Active learning Assignment
Topic : CLASSIFICATION OF SECOND ORDER PARTIAL
DIFFERENTIAL EQUATION
BRANCH : ELECTRICAL ENGINEERING
BATCH : B1
SUBJECT : ADVANCED ENGINEERING MATHEMATICS
Prepared By : JIGAR METHANIYA(150120109021)
Guided By : Prof. MIHIR SUTHAR
1

2. • The general form of a non Homogeneous second order
P.D.E is
• 𝐴 𝑥, 𝑦
𝜕2 𝑢
𝜕𝑥2+B 𝑥, 𝑦
𝜕2 𝑢
𝜕𝑥𝜕𝑦
+C 𝑥, 𝑦
𝜕2 𝑢
𝜕𝑦2+f
𝑥, 𝑦, 𝑢,
𝜕𝑢
𝜕𝑥
,
𝜕𝑢
𝜕𝑦
=F 𝑥, 𝑦 ………..(1)
• Equation (1) is said to be
• Elliptic , if 𝐵2
-4AC < 0
• Parabolic , if 𝐵2
-4AC = 0
• Hyperbolic , if 𝐵2-4AC > 0
• CLASSIFICATION OF SECOND-ORDER
PARTIAL DIFFERENTIAL EQUATION

3. • EXAMPLE:-1
Classify the Following P.D.E
𝜕𝑢
𝜕𝑡
=
𝜕2 𝑢
𝜕𝑥2
Ans:- Comparing this equation with (1) we get
A=1 , B=0 , C=0
So , 𝐵2
-4AC = 0
Hence given P.D.E. is parabolic.

4. • EXAMPLE:-2
Classify the following P.D.E
𝜕2 𝑢
𝜕𝑥2 +
𝜕2 𝑢
𝜕𝑦2=0
Ans:- Comparing this given P.D.E with (1) we get
A=C=1 , B=0
So , 𝐵2-4AC = -4<0
Hence , given P.D.E is elliptic.

5. • EXAMPLE:- 3
Classify the Following P.D.E
𝜕2 𝑢
𝜕𝑥2 + 3
𝜕2 𝑢
𝜕𝑥𝜕𝑡
+
𝜕2 𝑢
𝜕𝑡2 =0
Ans:- Comparing this given P.D.E with (1) we get
A=1 , B=3 , C=1
So , 𝐵2
-4AC = 9-4 = 5>0
Hence the given P.D.E is hyperbolic.