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
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
