A new approach to get series solution of nonlinear PDEs

1. A new approach to get series solution of nonlinear PDEs Dr. Shijun Liao School of Naval Architecture, Ocean and Civil Engineering Shanghai Jiao Tong University, China Int. Conf. Nonlinear PDEs and applications June 2007, Dong Hua University

2. Outline • Motivation • Purpose • Basic ideas  • Some applications • Chance and challenge

3. 1. Motivation Previous analytic techniques: Perturbation method: Straightforward expansion method Matching expansion method Multiple scale method,…. Non-perturbation techniques: Lyapunov’s artificial parameter method Adomian’s decomposition Method Delta-expansion method

4. 1. Motivation Limitations of previous analytic methods: • dependent upon small/large parameters; • valid only for weakly nonlinear problems; • no convenient way to control and adjust the convergence region; • no freedom to choose basis functions to more efficiently express solutions

5. 1. Motivation All of these limitations greatly restrict the applications of previous analytic methods.

6. One famous example : viscous flow past a sphere A difficult, classical problem: Stokes (1849) Oseen (1910) Goldstein (1929) Proudman & Pearson (1957) Chester & Breach (1967) Valid Region: Re <3

7. One Example:viscous flow past a sphere

8. F.M. White: “The idea of using creeping flow to expand into the high Reynolds number region has not been successful.” (Viscous Fluid Flow, McGraw-Hill, New York, 1991)

9. 2. Purpose Develop a new analytic method that • is independent upon small/large physical parameters; • is valid for highly nonlinear problems; • provides a convenient way to control and adjust the convergence region of approximation series; • provides us with freedom to choose base functions so as to more efficiently approximate solutions.

10. 2. Purpose In Short,develop amore powerful, easy-to-use, analytic tool valid foras manynonlinear problems in science and engineering as possible

11. 3. Basic ideas Concept of the traditional homotopy: Thus, as increase from 0 to 1, varies from to .

12. 3. Basic ideas Applications of traditional homotopy Pure mathematicians: existence, uniqueness of solutions Applied mathematicians: Numerical techniques such as continuation method, homotopy continuation method (by Li and Yorke)

13. 3. Basic ideas If the traditional concept of homotopy is applied to develop analytic techniques, we can not overcome the limitations of previous analytic techniques mentioned before.

14. 3. Basic ideas Concept of generalized homotopy

15. 3. Basic ideas The traditional homotopy is only a special case of the generalized homotopy when Thus, the generalized concept of the homotopy contains the traditional one and is more general.

16. 3. Basic ideas Consider a nonlinear equation Where is a nonlinear operator. ( the boundary/initial conditions can be treated in the similar way )

17. 3. Basic ideas Construct a family of equations

18. 3. Basic ideas So, as increases from 0 to 1, varies from to

19. 3. Basic ideas Expand in Taylor series with respect to the embedding parameter q, i.e.

20. 3. Basic ideas Setting , we have the solution series: where is governed by a linear equation

21. 3. Basic ideas In this way, we transfer any a nonlinear problem into an infinite number of linear sub-problems. Our transformation need not any small parameters!

22. 3. Basic ideas We proved such a Convergence theorem As long as a solution series given by the homotopy analysis method converges, it must be one of solutions of the original nonlinear problem.

23. 3. Basic ideas Function of Control and adjust the convergence region of approximation series Function of Provides freedom to choose the base functions of approximation series

24. 3. Basic ideas Three Rules • Rule of Solution Expression • Rule of Ergodicity • Rule of Solution Existence are established to choose • the initial guess • the auxiliary linear operator • the auxiliary function

25. 3. Basic ideas Unification of non-perturbation methods It can be proved that : • Lyapunov’s artificial small parameter method • Delta-expxnaion method • Adomian’s decomposition method are only special cases of the homotopy analysis method. So, the homotopy analysis method is more general.

26. 3. Basic ideas Advantages: • independent upon small/large parameters; • valid for highly nonlinear problems; • with a convenient way to control and adjust the convergence region of approximation series; • with freedom to choose basis functions so as to more efficiently approximate solutions.

27. 3. Basic ideas In this way, nearly all restrictions of the above-mentioned analytic techniques are overcome.

28. 4. Some applications 4.1 Blasius viscous flow Governing equation: Boundary conditions:

29. 4.1 Blasius viscous flow Inner solution: Blasius (1908) gave a power seires where and must be numerically given

30. 4.1 Blasius viscous flow Outer solution: Restriction of Blasius’ solution: • the approximation is notuniformly valid; • it is a semi-analytic solution, because must be given by numerical methods;

31. 4.1 Blasius viscous flow homotopy analysis solution • when, we have valid in the region

32. 4.1 Blasius viscous flow Homotopy analysis solution • when , it is the same as Blasius power series. Thus, our solution is more general. • when , it converges in an infinite region so thatouter solution is unnecessary

34. 4.1 Blasius viscous flow This example shows that the auxiliary parameter provides us with a convenient way to control and adjust the convergent region of solution series.

35. 4.1 Blasius viscous flow Homotopy analysis method (B) When We have an explicit solution: where the coefficient is given by recurrence formulas

36. 4.1 Blasius viscous flow Advantage of our solution • Uniformly valid in theinfiniteregion • It is unnecessary to give numerical value of. Thus, it is a purely analytic solution.

37. 4.1 Blasius viscous flow To the best of our knowledge, it is the firsttime such a kind of explicit, purely analytic solution of Blasius’ flow is published.

38. 4.1 Blasius viscous flow • The 1st case: power series • The 2nd case: exponential functions This example indicates that the homotopy analysis method provides us with freedom to choose basis function.

39. 4.2 Volterra’s population model Consider a nonlinear integro-differential equation subject to the initial condition

40. 4.2 Volterra’s population model Symbols: numerical result Solid line: Homotopy analysis approximation The HAM is Valid for nonlinear integro-differential problems

41. 4.3 Soliton of Vakhnenko equation Consider the propagation of high-frequency waves in a relaxing medium, governed by Vakhnenko equation We search for its loop soliton solution

42. 4.3 Soliton of Vakhnenko equation Symbols: exact solution Solid line: Our 10th-order approximation The HAM is Valid for soliton solutions with loops

43. 4.4 Soliton of Camassa-Holm equation Soliton waves in shallow water, governed by Camassa-Holm equation:

44. 4.4 Soliton of Camassa-Holm equation Symbols: exact solution Solid line: Our 10th-order Approximation The HAM is valid for soliton solution with discontinuity

45. 4.5 Viscous flow past a sphere

47. 4.5 Viscous flow past a sphere Homotopy analysis solution • For the first time, a uniformly valid solution is given for both near and far field of the flow; • Our drag formula agrees with experimental data in a considerably larger region of Reynolds number than all previous published theoretical results

48. 4.5 Viscous flow past a sphere This example indicates that homotopy analysis method can be applied to attack some unsolved nonlinear problems.

49. 4.6 New solutions of Cheng-Minkowycz boundary layer flow Cheng and Minkowycz (1977) equation: Solutions decaying exponentially were found by means of numerical techniques.

50. Solutions decaying exponentially when By means of the HAM, Liao and Pop (2004) gave explicit analytic solutions decaying exponentially.