LAPDE ..PPT (STANDARD TYPES OF PDE).pptx
- 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