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A new approach to get series solution of nonlinear PDEs

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### 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

Outline

- Motivation
- Purpose
- Basic ideas
- Some applications
- Chance and challenge

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

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

1. Motivation

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

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

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)

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.

2. Purpose

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

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)

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.

3. Basic ideas

Concept of generalized homotopy

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.

3. Basic ideas

Consider a nonlinear equation

Where is a nonlinear operator.

( the boundary/initial conditions can

be treated in the similar way )

3. Basic ideas

Construct a family of equations

3. Basic ideas

Expand in Taylor series with respect to the embedding parameter q, i.e.

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!

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.

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

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

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.

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.

3. Basic ideas

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

4.1 Blasius viscous flow

Inner solution:

Blasius (1908) gave a power seires

where

and

must be numerically given

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;

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

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.

4.1 Blasius viscous flow

Homotopy analysis method

(B) When

We have an explicit solution:

where the coefficient is given by recurrence formulas

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.

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.

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.

4.2 Volterra’s population model

Consider a nonlinear integro-differential equation

subject to the initial condition

4.2 Volterra’s population model

Symbols:

numerical result

Solid line:

Homotopy analysis

approximation

The HAM is

Valid for nonlinear

integro-differential

problems

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

4.3 Soliton of Vakhnenko equation

Symbols:

exact

solution

Solid line:

Our 10th-order

approximation

The HAM is

Valid for soliton

solutions with

loops

4.4 Soliton of Camassa-Holm equation

Soliton waves in shallow water, governed by Camassa-Holm equation:

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

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

4.5 Viscous flow past a sphere

This example indicates that

homotopy analysis method can be applied to attack some unsolved nonlinear problems.

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.

Solutions decaying exponentially when

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

Solutions decaying algebraically

When

there are an infinite number of solutions decaying algebraically

(Magyari, 2004)

By means of the HAM, Liao and Magyari found that, when

,

there are an infinite number of solutions decaying algebraically.

Domain of existence of

algebraically decaying boundary layers,

The potential of the HAM to find new solutions

This example illustrates that the HAM can be applied to findnew solutions of nonlinear problems, which aredifficultto find even by means ofnumerical techniques or other analytic methods.

Application of the HAM in finance

- Zhu SP, An exact and explicit solution for the valuation of American put options, Quantitative Finance 6 (3): 229-242 JUN (2006)
- Zhu SP, A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield, ANZIAM JOURNAL 47: 477-494 Part 4 APR (2006)

4.8 Gelfand equation

In case of N = 2 (two dimension)

By means of the auxiliary operator

the original 2nd-order nonlinear PDE is replaced by an infinite number of 4th-order linear ODEs

4.8 Gelfand equation

In case of N = 3 (three dimension)

By means of the auxiliary operator

the original 2nd-order nonlinear PDE is replaced by an infinite number of 6th-order linear ODEs

4.8 Gelfand equation

Comparison between the HAM series and numerical results

4.8 Gelfand equation

It illustrates that one nth-order nonlinear differential equation can be replaced by an infinite number of kth-order linear ODEs, where the order n is unnecessary to be equal to the order k.

4.8 Gelfand equation

- This example illustrates that we have much larger freedom to solve nonlinear problems than we traditionally thought.
- Using such kind of freedom, nonlinear problems might be solved in a much easier way.

Advantages of the HAM

- Generality: independent of small/large physical parameters;
- Validity: provides a simple way to ensure the convergence of solution series;
- Flexibility: provides large freedom to choose the basis functions and related auxiliary linear operators;
- Unification: logically contains other non-perturbation techniques;

5. Chances and challenge

Chances

Independent upon small parameters, the homotopy analysis method is more general, and provides us with a new way to solve highly nonlinear problems, and even to attack some classical unsolved nonlinear problems

5. Chances and challenge

Challenge

Nonlinear phenomena are very complicated and thus it is nearly impossible to develop a general method valid for all kinds of nonlinear problems, especially those with chaotic solutions. The homotopy analysis method should be further improved so as to be valid for asmany nonlinear problems as possible.

Question:How to find the best auxiliary linear operator and best basis function ?

Acknowledgement

Thanks to

- Prof. A. Campo (Idaho State University)
- Prof. K.F. Cheung (University of Hawaii)
- Prof. A.T. Chwang (Hong Kong University)
- Prof. E. Magyari, (Swiss Federal Institute of

Technology Züric, Switzerland)

- Prof. I. Pop (University of Cluj, Romania)
- all of my students

for their co-operation and value discussions.

谢谢！

Thank You!

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