This document presents information on partial differential equations (PDEs) including non-linear PDEs and their applications. It discusses standard types of non-linear PDEs like Clairaut's form where the PDE is formed as z=px+qy+f(p,q). It also provides examples of PDE applications in fields like engineering, physics, finance, fluid dynamics and heat transfer. Some limitations of PDEs are also outlined such as complexity, boundary/initial conditions, nonlinearity and limited applicability. In conclusion, the analysis of PDEs plays a crucial role in understanding multi-variable functions and has revolutionized industries through applications in areas like fluid dynamics and electromagnetics.

1. SUBJECT NAME: Linear Algebra and Partial
Differential Equations
SUBJECT CODE:191MAB303T
DATE: 6th
November 2023
Presented by,
GROUP-11
MOHAMMED AADIL.A-083
RASHINA BANU.S-117
MOHANA PRIYA.R-085
MOHAMMED RAIYAN ALAM.N-084
MANI BHARATHI.S-077

2. Standard Types of
Non-Linear PDE

3. TABLE OF CONTENTS
 Introduction to partial differential equations
 Types of standard partial differential equations
 Formulae derivations
Real time applications of PDE
 Limitations of PDE
 Conclusion

4.  PDEs computes a function between various partial derivatives of a
multivariable function.
 PDEs are mathematical equations used to describe how quantities change
in relation to multiple variables
 PDEs are used in image processing to smooth out noisy images or to
extract features from images
WHAT IS PDE?

5. STANDARD NON-LINEAR PDE’S:
f(p,q) = 0
The solution is z = 𝑎𝑥 + 𝑏𝑦 + c
let
𝜕𝑧
𝜕𝑥
= 𝑎 and
𝜕𝑧
𝜕𝑦
= 𝑏
So if f(p,q) = 0 we get f(𝑎, 𝑏) = 0
So to obtain solution lets take it in “𝑏” terms
We get 𝑏 = 𝜓(𝑎)
Then the required solution is
𝑧 = 𝑎𝑥 + 𝛹(𝑎)𝑦 + 𝑐
This type is also called as Clairaut’s form
The PDE’s is formed as 𝑧 = 𝑝𝑥 + 𝑞𝑦 +
𝑓(𝑝, 𝑞)
The solution is z = 𝑎𝑥 + 𝑏𝑦 + c
From comparing both c = 𝑓(𝑝, 𝑞)
Let a = p =
𝜕𝑧
𝜕𝑥
, b = q =
𝜕𝑧
𝜕𝑦
Substituting a , b in z we get
𝑧 = 𝑎𝑥 + 𝑏𝑦 + 𝑓(𝑎, 𝑏)
Type-1 Type-2

6. STANDARD NON-LINEAR PDE’S:
𝑓 𝑥, 𝑝 = 𝑔 𝑦, 𝑞
𝑙𝑒𝑡 𝑓 𝑥, 𝑝 = 𝑎 = 𝑔 𝑦, 𝑞
To find p and q in terms of a
𝑝 = 𝑓 𝑥, 𝑎 , 𝑞 = 𝑔 𝑦, 𝑎
𝑑𝑧 = 𝑝𝑑𝑥 + 𝑞𝑑𝑦 substitute p,q in 𝑑𝑧
𝑑𝑧 = 𝑓 𝑥, 𝑎 𝑑𝑥 + 𝑔 𝑦, 𝑎 𝑑𝑦
By Integrating 𝑑𝑧 we get
𝑑𝑧 = 𝑓(𝑥, 𝑎)𝑑𝑥 + 𝑔(𝑦, 𝑎)𝑑𝑦
𝑧 = 𝑓(𝑥, 𝑎)𝑑𝑥 + 𝑔(𝑦, 𝑎)𝑑𝑦
Type-3 Type-4
𝑓 𝑧, 𝑝, 𝑞 = 0
The PDE is formed as z = 1 + 𝑝2
+ 𝑞2
Let 𝑝 =
𝑑𝑧
𝑑𝑢
, 𝑞 = 𝑎
𝑑𝑧
𝑑𝑢
, 𝑞 = 𝑝𝑎 , 𝑝 = 𝑔 𝑎, 𝑧
Let 𝑢 = 𝑥 + 𝑎𝑦
Substitute p , q in z we get
𝑑𝑧 = 𝑝𝑑𝑥 + 𝑞𝑑𝑦 => 𝑝𝑑𝑥 + 𝑎𝑝𝑑𝑦
𝑔 𝑎, 𝑧 𝑑𝑥 + 𝑎𝑑𝑦 ⇒
dz
g a,z
= 𝑑𝑥 + 𝑎𝑑𝑦

7. APPLICATIONS OF PDE:
Engineering:
 To analyze and optimize structural designs ,
stimulate fluid flow in pipes and channels
 We can predict the behavior of materials under
various conditions
 Makes informed decisions, improved designs and
ensures safety and efficiency in engineering
systems

8. APPLICATIONS OF PDE:
PHYSICS:
 Describes the behavior of physical systems
for example: Quantum mechanics
Fluid dynamics
Electromagnetism
 Serves as a powerful tool in unraveling the
mysteries of the universe

9. APPLICATIONS OF PDE:
FINANCE:
 Used to model and analyze complex financial
systems
for example: Option pricing
Risk management
Portfolio optimization
 To make informed decisions , assess market trends
,etc…
 Provides valuable insights into market dynamics

10. APPLICATIONS OF PDE:
Fluid dynamics:
 To model the behavior of fluids in real-time systems
 Describes fluid flow phenomena , turbulence , wave
propagation
 Stimulate and optimize complex fluid systems leading
to advancements in aerospace engineering , weather
prediction and environment studies

11. APPLICATIONS OF PDE:
Heat transfer:
 Accurately models the transfer of heat in real-time
systems
 In designing efficient cooling systems , analyze
thermal behavior and optimize energy consumption
 By solving through PDE researchers gain insight into
heat distribution , temperature gradients and thermal
stability

12. APPLICATIONS OF PDE:
Electromagnetic wave propagation:
 Accurately describe the behavior of light and
electromagnetic waves for the engineers to design &
optimize devices such as antennas , optical fiber and
wireless systems
 Develops innovative solutions in fields like
telecommunications and photonics

13. LIMITATIONS OF PDE:
Complexity:
PDEs can become extremely complex, especially in real-
world applications where multiple factors and boundary
conditions are involved
Boundary and Initial Conditions:
PDEs require appropriate boundary and initial conditions
to be well-posed problems. Choosing the right conditions
is critical.

14. LIMITATIONS OF PDE:
Nonlinearity:
Many real-world problems involve nonlinear PDEs, which
can exhibit behaviors that are difficult to predict and
analyze.
Limited Applicability:
PDEs may not always accurately model certain physical phenomena,
especially at very small scales (such as quantum mechanics) or very
high speeds (such as relativistic effects), where other theories like
quantum mechanics and general relativity are more appropriate.

15. CONCLUSION
 The analysis of standard partial differentiation equation plays a crucial
role in understanding the behavior of functions with multiple variable
 Continual research and advancements in this area will undoubtedly
lead to further discoveries and practical solutions
 Their applications in fluid dynamics , heat transfer and electromagnetic
wave propagation have revolutionized industries and led to significant
advancements

16. THANK
YOU

17. Example for application wave equation
A tightly stretched string with fixed end points x = 0 & x = ℓ is initially at rest in its equilibrium
position . If it is set vibrating by giving to each of its points a velocity , find the vibrational wave of
the string
We know that the boundary points are
I. 𝑦 0, 𝑡 = 0 , 𝑓𝑜𝑟 𝑡 ≥ 0 = 𝑥
II. 𝑦 𝑙, 𝑡 = 0 , 𝑓𝑜𝑟 𝑡 ≥ 0 , 𝑥 = 𝑙
III. 𝑦 𝑥, 0 = 0 , 𝑓𝑜𝑟 𝑡 = 0 , 0 ≤ 𝑥 ≤ 𝑙
IV.
𝑑𝑦
𝑑𝑡
= 𝑘𝑥 𝑙 − 𝑥 , 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 𝑙
Since the string is in periodic therefore the solution is in the form of
𝑦 𝑥, 𝑡 = 𝐴𝑐𝑜𝑠𝜑𝑥 + 𝐵𝑠𝑖𝑛𝜑𝑥 (𝑐𝑐𝑜𝑠𝜑𝑎𝑡 + 𝑑𝑠𝑖𝑛𝜑𝑎𝑡)
The above problem will be solved by the presenter