Uploaded bySanthanam Krishnan
Solution to pde
The document discusses concepts related to solutions of partial differential equations (PDEs), differentiating between complete, particular, singular, and general solutions. It outlines procedures for finding complete integrals, singular integrals, and general integrals based on the forms of the equations involved. Various cases and methods for reducing equations and solving them are also presented.
More Related Content
More from Santhanam Krishnan
Solution to pde
- 1.
- 2. • In asolution to a PDE, if the number of arbitrary constants and the number of independent variables are equal, then the solution is called complete solution or complete integral.
- 3. : • In aComplete Integral (Complete Solution) , if a particular values found for the arbitrary constants , then it is called particular Solution or Particular Integral. • (i.e) A solution obtained by giving particular values to the arbitrary constants in a complete integral is called particular Integral or Particular Solution.
- 4. • A solutionwhich contains no arbitrary terms is called Singular Solution (or) Singular Integral.
- 5. • A solutionwhich contains only arbitrary function but no arbitrary constants is called General Solution or General Integral. • (i.e) the solution contains maximum possible number of arbitrary functions
- 11. Take 𝑧= 𝑓 𝑢 , where 𝑢 = 𝑥 + 𝑎𝑦 -------- (1) Differentiate (1) partially with respect to x & y , we will get 𝛿𝑢 𝛿𝑥 = 1 & 𝛿𝑢 𝛿𝑦 = 𝑎 But W.K.T 𝛿𝑧 𝛿𝑥 = 𝑝 & 𝛿𝑧 𝛿𝑦 = 𝑞 Also W.K.T, 𝑝 = 𝜕𝑧 𝜕𝑥 = 𝑑𝑧 𝑑𝑢 𝜕u 𝜕x = dz du and 𝑞 = 𝜕𝑧 𝜕𝑦 = 𝑑𝑧 𝑑𝑢 𝜕u 𝜕y = a dz du (∵ 𝛿𝑢 𝛿𝑥 = 1 & 𝛿𝑢 𝛿𝑦 = 𝑎) Substitute 𝑝 = dz du & q = a dz du in the given pde, we will get 𝑓 𝑧, 𝑑𝑧 𝑑𝑢 = 0 , which is a ODE of first order. Solving the ODE by using variable separable method, we will get C.I of the given pde
- 12. In this type,there is no Singular Solution like TYPE I
- 13. Put 𝑐 =∅(𝑎) in C.I and proceed same like previous Type
- 14. The pde isin the form f(p, q, x) = 0 (PDE contains p, q & x)
- 15. Take 𝑞= 𝑎, Substitute 𝑞 = 𝑎 in the given equation f(p,q,x) = 0 Then find p W.K.T dz = p dx +q dy & substitute the values of p & q in this equation and integrate, we will get C.I
- 17. Procedure to findGeneral Integral (General Solution): Put 𝑐 = ∅(𝑎) in C.I and proceed same like previous Type
- 18. Type IV The pdeis in the form f(p, q, y) = 0 (PDE contains p, q & y)
- 19. Procedure to findSingular Integral (Singular Solution) S.I: • In this type, there is no Singular Solution like TYPE I
- 20. Procedure to findGeneral Integral (General Solution): Put 𝑐 = ∅(𝑎) in C.I and proceed same like previous Type
- 21. The pdeis in the form z = px +qy + f(p,q)
- 22. Procedure to findComplete Integral (Complete Solution) C.I: Take 𝑝 = 𝑎, & 𝑞 = 𝑏 Substitute 𝑝 = 𝑎 & 𝑞 = 𝑏 in the given equation 𝑧 = 𝑝𝑥 + 𝑞𝑦 + 𝑓(𝑝, 𝑞) 𝑧 = 𝑎𝑥 + 𝑏𝑦 + 𝑓(𝑎, 𝑏), which is the C.I
- 23. Procedure to findSingular Integral (Singular Solution) S.I: Consider the C.I 𝑧 = 𝑎𝑥 + 𝑏𝑦 + 𝑓(𝑎, 𝑏) -- (2) Differentiate (2) partially w.r.to ‘a’, we get x + 𝜕𝑓 𝜕𝑎 = 0 –(3) Differentiate (2) partially w.r.to ‘b’, we get y + 𝜕𝑓 𝜕𝑏 = 0 - (4) Eliminating ‘a’ & ‘b’ from (2) , (3) & (4) gives Singular Integral
- 24. Procedure to findGeneral Integral (General Solution):
- 25. The pde isin the form 𝒇 𝟏 𝒙, 𝒑 = 𝒇 𝟐 𝒚, 𝒒 𝒐𝒓 𝒇 𝒙, 𝒚, 𝒑, 𝒒 = 𝟎 (The pde contains only x, y, p & q and does not contain z)
- 26. Procedure to findComplete Integral (Complete Solution) C.I: If the given function is in the form 𝑓 𝑥, 𝑦, 𝑝, 𝑞 = 0, then separate this into 𝑓1 𝑥, 𝑝 = 𝑓2(𝑦, 𝑞) Take 𝑓1 𝑥, 𝑝 = 𝑓2 𝑦, 𝑞 = 𝑎 From 𝑓1 𝑥, 𝑝 = 𝑎 , find p and from 𝑓2 𝑦, 𝑞 = 𝑎, find q W.K.T dz = p dx +q dy Substitute the values of p & q in the above and integrate both sides , we will get C.I
- 27. • There isno Singular Solution
- 28. Procedure to findGeneral Integral (General Solution): Put 𝑐 = ∅(𝑎) in C.I and proceed same like previous Type
- 29. Type VII -Equation reducible to standard form The pde is in the form 𝒇 𝒙 𝒎 𝒑 , 𝒚 𝒏 𝒒 = 𝟎 and 𝒇 𝒛, 𝒙 𝒎 𝒑, 𝒚 𝒏 𝒒 = 𝟎
- 30. Procedure to findComplete Integral (Complete Solution) C.I: Case (i) If 𝒎 ≠ 𝟏 and 𝒏 ≠ 𝟏 Put 𝑋 = 𝑥1−𝑚 & 𝑌 = 𝑦1−𝑛 Substitute 𝑥 𝑚 𝑝 = 𝑃 1 − 𝑚 & 𝑦 𝑛 𝑞 = 𝑄(1 − 𝑛) in the given equation , where P = 𝜕𝑧 𝜕𝑋 , 𝑄 = 𝜕𝑧 𝜕𝑌 After substitution, the given equation reduced to the equation of the form 𝑓 𝑃, 𝑄 = 0 𝑜𝑟 𝑓 𝑧, 𝑃, 𝑄 = 0, as per the previous procedure find the C.I Case(ii) If 𝒎 = 𝟏 𝒂𝒏𝒅 𝒏 = 𝟏 Put 𝑋 = log 𝑥 & 𝑌 = log 𝑦 Substitute 𝑥𝑝 = 𝑃 & 𝑦𝑞 = 𝑄 in the given equation where P = 𝜕𝑧 𝜕𝑋 , 𝑄 = 𝜕𝑧 𝜕𝑌 After substitution, the given equation reduced to the equation of the form 𝑓 𝑃, 𝑄 = 0 𝑜𝑟 𝑓 𝑧, 𝑃, 𝑄 = 0, as per the previous procedure find the C.I
- 31. There is noSingular Solution
- 32. Procedure to findGeneral Integral (General Solution): Put 𝑐 = ∅(𝑎) in C.I and proceed same like previous Type
- 33. Type VIII -Equation reducible to standard form The pde is in the form 𝒇 𝒛 𝒎 𝒑 , 𝒛 𝒎 𝒒 = 𝟎 and 𝒇 𝒙, 𝒛 𝒎 𝒑, 𝒛 𝒎 𝒒 = 𝟎
- 34. Procedure to findComplete Integral (Complete Solution) C.I Case (i) If 𝒎 ≠ −𝟏 Put 𝑍 = 𝑧 𝑚+1 , 𝑃 𝑚+1 = 𝑧 𝑚 𝑝 & 𝑄 𝑚+1 = 𝑧 𝑚 𝑞 Where 𝑃 = 𝜕𝑍 𝜕𝑥 & 𝑄 = 𝜕𝑍 𝜕𝑦 Substitute in given equation, After substitution, the given equation reduced to the equation of the form 𝑓 𝑃, 𝑄 = 0 𝑜𝑟 𝑓1 𝑥, 𝑃 = 𝑓2 𝑦, 𝑄 , as per the previous procedure find the C.I Case(ii) If 𝒎 = −𝟏 Put 𝑍 = 𝑙𝑜𝑔𝑧 Substitute 𝑧𝑃 = 𝑝 & 𝑧𝑄 = 𝑞 in the given equation , where 𝑃 = 𝜕𝑍 𝜕𝑥 & 𝑄 = 𝜕𝑍 𝜕𝑦 After substitution, the given equation reduced to the equation of the form 𝑓 𝑃, 𝑄 = 0 𝑜𝑟 𝑓1 𝑥, 𝑃 = 𝑓2 𝑦, 𝑄 , as per the previous procedure find the C.I
- 35. Procedure to findSingular Integral (Singular Solution) S.I • There is no Singular Solution
- 36. Procedure to findGeneral Integral (General Solution): Put c = ∅(a) in C.I and proceed same like previous Type