Analytical and Numerical methods for a Mean curvature flow equation with applications to financial Mathematics and image processing

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University of

Michigan

Analytical and Numerical methods

for a Mean curvature flow equation with applications to

financial Mathematics and image processing

Alireza Zavareh

Thesis for the Master of Science degree (two years)

In Mathematical Modeling and Simulation

30 credits point (30 ECTS credits)

February 2012

Blekinge Tekniska Hogskola (BTH), Sweden

University of Michigan (UofM), USA

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Contact Information

Alireza Zavareh:

alirezazavareh10@gmail.com

Prof. Nail Ibragimov:

nailhib@gmail.com

Assis. Prof. Arash Fahim:

fahimara@umich.edu

Assoc. Prof. Claes Jogréus:

claes.jogreus@bth.se

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Dedicated to my dear family

and

my lovely son, Arsalan

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Abstract

This thesis provides an analytical and two numerical methods for solving a parabolic equation of two-‐dimensional mean curvature flow with some applications. In analytical method, this equation is solved by Lie group analysis method, and in numerical method, two algorithms are implemented in MATLAB for solving this equation. A geometric algorithm and a step-‐wise algorithm; both are based on a deterministic game theoretic representation for parabolic partial differential equations, originally proposed in the genial work of Kohn-‐Serfaty [1].

Keywords:

Mean curvature flow, Lie group analysis, Parabolic equation, Deterministic game theoretic

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Acknowledgements

It gives me great pleasure to express my sincere and heartfelt thanks to all of those who helped me during my research to complete this thesis.

Firstly, I would like to thank to Professor Nail Ibragimov, for his motivational advice during my studies of his five courses on Lie group analysis and his valuable help on the analytical part of this thesis. I also owe my deepest gratitude to Assis. Professor Arash Fahim for his guidance, patience and profound knowledge, which enabled me to successfully complete my work at the university of Michigan.

Secondly, I would like to show special thanks to Professor Peter J. Olver for his useful help during discussions about Lie methods and his suggestion to check the obtained symmetries by Maple.

Furthermore, I am grateful to Assoc. Professor Claes Jogréus, Professor Elisabeth Rakus-‐Andersson, Assis. Professor Raisa Khamitova, Dr. Mattias Eriksson and all other teachers and managers in Blekinge Tekniska Hogskola for their help, during my studies in Karlskrona, Sweden.

Last but not least, I would like to express my sincere thanks with heartfelt feeling to Zahra Zavareh, Afrouz Shirian, Alireza Niki, Sharif Zahiri, Amir Eskandari, Damoon Rastegar, Hossein Keshavarz, Salli Baker and all of my friends for their support and encouragement during my studies.

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List of figure

Figure 1. Motion with constant velocity Figure 2. Motion by curvature

Figure 3. Paul’s quandary – if he tries to go north, Carol will send him south.

Figure 4. Left. Paul can exit from a well-‐chosen concentric circle in just one step. Right. The construction can be repeated

Figure 5. Motion by curvature for Rectangle (𝜀 = 2.5) Figure 6. Motion by curvature for Rectangle (𝜀 = 0.4) Figure 7. Motion by curvature for convex polygon (𝜀 = 2.5) Figure 8. Motion by curvature for convex polygon (𝜀 = 0.4) Figure 9. Motion by curvature for nonconvex initial boundary

Figure 10. Motion by curvature for nonconvex initial Boundary With a Montacarlo method

Figure 11. Numerical solution vs. Analytical solution

Figure 12. Processing of the 2-‐D image by mean curvature flow

Figure 13. The heart ventricle in open phase – visualized from the result of acquis-‐ ition (left) and after the 3-‐D image processing by mean curvature flow (right)

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Chapter 1 Deterministic control interpretation

1.1 Introduction

When the velocity of the moving surface depends only on the normal direction, the level-‐set description of the motion is a first-‐order PDE (a Hamilton-‐Jacobi equation), but when the velocity depends on curvature, the level-‐set description is a second-‐order parabolic or elliptic PDE. From [1] we see that for geometric evolutions, the first and second-‐order cases are actually quite similar.

I write sections 1.2, 1.3 and 1.4 from reference [1], and in order to have a clear explanation, I have not changed these explanations.

1.2 Motion with constant velocity

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Figure 1. Motion with constant velocity [1]

This evolution is completely characterized by the arrival time

𝑢 𝑥 = 𝑡𝑖𝑚𝑒 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑚𝑜𝑣𝑖𝑛𝑔 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑝𝑎𝑠𝑠𝑒𝑠 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑥.

This function solves the stationary Hamilton-‐Jacobi equation

| 𝛻𝑢 | = 1 in Ω

(1) with 𝑢 = 0 at the boundary. Equation (1) is characterized by the optimization

𝑢 𝑥 = min

_!∈!!

𝑑𝑖𝑠𝑡(𝑥, 𝑧).

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1.3 Motion by curvature

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Figure 2. Motion by curvature [1]

−𝑑𝑖𝑣 (𝛻𝑢/|𝛻𝑢|) is the curvature of a level set of 𝑢, and the velocity of the moving front is 1/|𝛻𝑢|, so the arrival time of motion by curvature solves

|𝛻𝑢|𝑑𝑖𝑣 (𝛻𝑢/|𝛻𝑢| ) + 1 = 0 in Ω

(3) with 𝑢 = 0 at the boundary. This PDE is to motion by curvature as eikonal equation (1) is to motion with constant velocity.

1.4 Paul-‐Carol game interpretation

Both evolutions are similar in the sense that motion by curvature also has a deterministic control interpretation, similar to (2). It involves a game with two

players, Paul and Carol, and a small parameter ℇ. Paul is initially at some point

𝑥 ∈ Ω; he wants to exit as soon as possible, and Carol tries to delay his exit as long as possible. This game has the following rules:

.

Paul chooses a direction, i.e. a unit vector 𝑣 = 1.

.

Carol can either accept or reverse Paul’s choice, i.e. she chooses 𝑏 = ±1.

.

Paul then moves distance 2𝜀 in the possibly-‐reversed direction, i.e. from 𝑥

to 𝑥 + 2𝜀bv.

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For example, if Paul is near the top of the rectangle, one might think he should choose 𝑣 pointing north. But that’s a bad idea, because if he does so, Carol will reverse him and he’ll have to go south (Figure 3).

Figure 3. Paul’s quandary – if he tries to go north, Carol will send him south. [1]

Can Paul exit? Yes indeed. This is easiest to see when 𝜕Ω is a circle of radius R. The midpoints of secants of length 2 2𝜀 trace a concentric circle, whose radius is smaller by approximately 𝜀!_{/𝑅. Paul can exit in one step if and only if he starts}

on or outside this concentric circle (Figure 4, left). This construction can be repeated of course, producing a sequence of circles from which he can exit in a fixed number of steps (Figure 4, right).

Figure 4. Left. Paul can exit from a well-‐chosen concentric circle

in just one step. Right. The construction can be repeated [1]

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1/𝑅 = curvature. We have determined Paul’s optimal strategy: if Ω = 𝐵_!(0) and his present position is 𝑥 then his optimal 𝑣 is perpendicular to 𝑥. And we have linked his minimum exit time to motion by curvature. This calculation is fundamentally local, so it is not really limited to balls. It suggests that Paul’s scaled arrival time,

𝑢

_!

𝑥 =

𝜀

!

_.

minimum number of steps Paul needs to exit starting

from 𝑥, assuming Carol behaves optimally.

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converges as 𝜀 → 0 to the arrival-‐time function of motion by curvature.

The circle was too easy. To analyze more general domains a key tool, from [1], is the dynamic programming principle:

𝑢

_!

𝑥 = min

_{! !!}

max

_!!±!

𝑢

_!

(𝑥 + 2𝜀𝑏𝑣+𝜀

!

_.

₍₅₎

1.5 Implementation of the geometric algorithm

In this thesis I have implemented the geometric algorithm described above, in MATLAB for general region Ω, in both convex and nonconvex initial boundary condition. All codes are available. Contact me if needed.

As a result of running this program for some different initial boundaries, we get the figures 5-‐9, and figure 10 is from [2] with a Montecarlo method.

As we see in figures 5-‐8, in motion by curvature, when the initial boundary is convex, it stays convex, and also, in figures 9 and 10, we see that even if the initial boundary is not convex, after some time steps, the boundary becomes convex and after some more time steps it will become circle.

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Figure 5. Motion by curvature for Rectangle (𝜀 = 2.5)

Figure 6. Motion by curvature for Rectangle (𝜀 = 0.4)

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Figure 7. Motion by curvature for convex polygon (𝜀 = 2.5)

Figure 8.

Motion by curvature for convex polygon (𝜀 = 0.4)

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Figure 9. Motion by curvature for

nonconvex initial boundary

Figure 10. Motion by curvature for nonconvex initial

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1.6 Deterministic control interpretations of fully

nonlinear PDEs

The preceding discussion seems strongly linked to the geometric character of the problem. In particular Paul’s value function 𝑢! converged to the level-‐set

description of a geometric motion. Now we consider the following deterministic game, from [1], approach to a large class of fully-‐nonlinear parabolic and elliptic equations. To explain the main idea consider a final-‐value problem of the form

𝑣

_!

+ 𝑓 𝐷

_!

, 𝐷

!

= 0 for 𝑡 < 𝑇, with 𝑣 = 𝜙 at 𝑡 = 𝑇

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on all ℝ!_{. We assume the PDE is parabolic, in the sense that}

𝑓 𝑝, Γ ≤ f(p, Γ

!

₎

_if

_{Γ ≤ Γ}

!_{as symmetric matrices. (7)}

The game still has two players, whom we call Helen and Marks (for a reason to

be explained later). There’s a marker, that’s initially at position 𝑥 ∈ ℝ!_{at time T.}

At each time step

1. Helen chooses a vector 𝑝 ∈ ℝ!_{and a symmetric 𝑛×𝑛 matrix Γ; then (after}

hearing Helen’s choice) Mark chooses a vector 𝑤 ∈ ℝ!_.

2. Helen pays penalty 𝜀𝑝. 𝑤 +!_!!

Γ𝑤, 𝑤

− 𝜀!_{𝑓 𝑝, Γ .}

3. The marker moves from 𝑥 to 𝑥 + 𝜀𝑤 and the clock steps from 𝑡 to 𝑡 + 𝜀!_.

The game continues this way until the final time 𝑇, when Helen collects a bonus 𝜙 𝑥 𝑇 . Her goal is maximize her bonus minus accumulated penalties. Mark

does all he can against her. Helen’s value function 𝑣_! 𝑥, 𝑡 now satisfies the

dynamic programming principle

𝑣

_!

𝑥, 𝑡 = max

_!,!

min

_!

𝑣

!

𝑥 + 𝜀𝑤, 𝑡 + 𝜀

!

_{− 𝜀𝑝. 𝑤}

−

!_!!

Γ𝑤, 𝑤 + 𝜀

!

_{𝑓(𝑝, Γ)}

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1.7 Mean curvature flow equation

In this part of the thesis I solve the following Mean curvature flow equation from [2], by applying the algorithm explained above.

𝑢

_!

− ∆𝑢 +

!".!_!"!!"!_!

= 0

(9)

with 𝑢(0, 𝑥) = 𝑔(𝑥)

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If we rewrite the equation (9) for 2-‐D , when 𝑥 = 𝑥, 𝑦 𝜖ℝ!_and

𝑔 𝑥 = 𝑥! _{+ 𝑦}!_{− 1, which means the initial boundary is a circle of radius 1, we}

get the following equation

𝑢

_!

− 𝑢

_!!

− 𝑢

_!!

+

𝑢

𝑥2

𝑢

𝑥𝑥

+2𝑢

𝑥

𝑢

𝑦

𝑢

𝑥𝑦

+𝑢

𝑦2

𝑢

𝑦𝑦

𝑢

_𝑥2

+𝑢

_𝑦2

= 0

(11)

with 𝑢 0, 𝑥, 𝑦 = 𝑥

!

_{+ 𝑦}

!

_{− 1}

₍₁₂₎

By comparing equations (6) and (11) we have

𝑓 𝐷

_!

, 𝐷

!_!

= −𝑢

_!!

− 𝑢

_!!

+

𝑢

𝑥

2

_𝑢

_𝑥𝑥

_+2𝑢

_𝑥

_𝑢

_𝑦

_𝑢

_𝑥𝑦

_+𝑢

_𝑦2

_𝑢

_𝑦𝑦

𝑢

_𝑥2

+𝑢

_𝑦2

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I have applied the algorithm described above, in MATLAB, to solve equation (11) with the initial boundary (12). In this program, I have calculated the first and second derivatives, in (13), by finite difference method. In figure 11, I have compared the Numerical solution, which I got by this algorithm, versus the Analytical solution, which I got by Lie Group Analysis method, to be described in chapter 2, both with the same initial boundary (12), in some different time steps.

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Figure 11. Numerical solution vs. Analytical solution

Circles: Analytical solution *: Numerical solution

1.8 Application in financial Mathematics

Mean curvature flow equations have some applications in Portfolio optimization, Hedging and Super hedging in financial Mathematics. In reference

[2] we can find some models of Portfolio optimization, Vasicek model, CEV-‐SV

model and Heston model, which are related to the mean curvature problem.

Also the game interpretation described in section 1.6, which has two players, is closely related to the well-‐known fact that European options can be perfectly hedged in a binomial tree market. In this financial interpretation, with 𝜀 > 0,

 𝑥 = the stock price

 −𝑝 = the amount of stocks in Helen’s hedge portfolio

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portfolio

 𝑣_! 𝑥, 𝑡 = time-‐t value of the option with payoff 𝜙 at time 𝑇 We have:

𝑣

_!

𝑥

_!

, 𝑡

_!

+

(𝜀𝑝. 𝑤 +

𝜀

!

2 Γ𝑤, 𝑤 − 𝜀

!

𝑓 𝑝, Γ ) = 𝜙(𝑥 𝑇 )

1.9 Application in image processing

Numerical approximation of a nonlinear equation of mean curvature

flow has some applications in image processing. Using the level set equation to initial image yields the silhouettes smoothing. In figures 12-‐ 14, from reference [5], we see some examples related to image processing.

Figure 12. Processing of the 2-‐D image by mean curvature flow [5]

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Figure 13. The heart ventricle in open phase – visualized from the result of acquisition (left) and after the 3-‐D image processing

by mean curvature flow (right) [5]

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Figure 14. Extraction of the chromosomes from the 3-‐D image by mean curvature flow [5]

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Chapter 2 Lie Group Analysis

2.1 Introduction

In this chapter, in section 2.2, I will calculate Lie point symmetries of the mean curvature equation (11). By using the symmetries of the equation, I will find invariant solutions of this equation in section 2.3. I follow the notations used in reference [3] and apply the method described in references [3] and [4].

Let 𝐺 be a one-‐parameter group of transformations of independent variables

𝑥 = 𝑥!_{, 𝑥}!_{, … , 𝑥}! _{and dependent variable 𝑢, of the form:}

𝑥

!

= 𝑓

!

_{(𝑥, 𝑢, 𝑎),}

_{𝑢 = 𝑔(𝑥, 𝑢, 𝑎).}

₍₁₄₎

By expanding functions 𝑓!_{and 𝑔 into Taylor’s series near 𝑎 = 0, we obtain the}

infinitesimal transformations

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𝑓

!

_|

!!!

= 𝑥

!

, 𝑔|

!!!

= 𝑢.

(16)

Now, the generator of the group 𝐺 is

𝑋 = 𝜉

!

_{𝑥, 𝑢}

! !!!

+ 𝜂 𝑥, 𝑢

! !!

(17)

that

𝜉

!

₌

!!! !!

|

!!!

, 𝜂 =

!! !!

|

!!!

.

(18) Transformations (14) can be obtained from infinitesimal transformation (15) and also from the generator (17), by deriving the following system of ordinary differential equations, which are called the Lie equations:

𝑑_𝑑!_!!

= 𝜉

!

_(𝑥

!

_{, 𝑢}

!

_{), 𝑥}

!

_|

!!!

= 𝑥

!

,

(19)

that

𝑑_𝑑!_!

= 𝜂(𝑥

!

, 𝑢), 𝑢|

_!!!

= 𝑢.

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Now, consider the equation

𝐹 𝑥, 𝑢 = 0.

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Let G be a Lie transformation group, generated by the operator (17). After

prolongation, it is written as following formula

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𝜁

_!

= 𝐷

_!

𝜂 − 𝑢

_!

𝐷

_!

𝜉

!

_,

₍₂₃₎

and the higher-‐order prolongation formula is written as

𝜁

_!_!_…!_!

= 𝐷

_!_!

𝜁

_!_!_…!_!!!

− 𝑢

_!!_!_…!_!!!

𝐷

_!_!

(𝜉

!

_).

₍₂₄₎

Group 𝐺 is called a group of the symmetries of the differential equation

2.2 Lie symmetries of the mean curvature flow

equation

Consider the mean curvature flow equation (11) as follows

𝑢

_!

− 𝑢

_!!

− 𝑢

_!!

+

𝑢

! !

_𝑢

!!

+ 2𝑢

!

𝑢

!

𝑢

!"

+ 𝑢

!!

𝑢

!!

𝑢

_!!

_{+ 𝑢}

! !

= 0,

then

!!!! !_!! !!!!!!!!!!!!!!!!!!!!!!!!!!" !_!!!!_!!

= 0.

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We know

𝑢

_!!

+ 𝑢

_!!

≠ 0,

therefore we can write our mean curvature flow

equation as following

𝑢

_!

𝑢

_!!

_{+ 𝑢}

!

𝑢

!!

− 𝑢

!!

𝑢

!!

− 𝑢

!!

𝑢

!!

+ 2𝑢

!

𝑢

!

𝑢

!"

= 0,

(26)

with the initial boundary

𝑢 0, 𝑥, 𝑦 = 𝑥

!

_{+ 𝑦}

!

_{− 1.}

₍₂₇₎ Let

𝐹 ≡ 𝑢

_!

𝑢

_!!

_{+ 𝑢}

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For equation (15), we find the maximal Lie algebra of symmetries

𝑋 = 𝜉

!

_{𝑡, 𝑥, 𝑦, 𝑢}

𝜕

𝜕𝑡

+ 𝜉

!

𝑡, 𝑥, 𝑦, 𝑢

𝜕

𝜕𝑥

+𝜉

!

_{𝑡, 𝑥, 𝑦, 𝑢}

𝜕 𝜕𝑦

+ 𝜂 𝑡, 𝑥, 𝑦, 𝑢

𝜕𝑢𝜕

.

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According to the prolongation formula (22), prolongation of 𝑋 is

𝑋 = 𝜉

!

𝜕

𝜕𝑡

+ 𝜉

!

𝜕

𝜕𝑥

+ 𝜉

!

𝜕

𝜕𝑦

+ 𝜂

𝜕

𝜕𝑢

+𝜁

_!_!!! !

+ 𝜁

! ! !!_!

+ 𝜁

! ! !!_!

+ 𝜁

!! ! !!_!!

+ 𝜁

!! ! !!_!!

+ 𝜁

!" ! !!_!"

,

(29) which by applying formulas (23) and (24) we have

(26)

𝑋 𝐹 |

_!≡!

= 0,

so the determining equation is written as

[𝜁_!

𝑢

𝑥2

+ 𝑢

𝑦2

+ 𝜁

₂

2𝑢

𝑡

𝑢

𝑥

− 2𝑢

𝑦𝑦

𝑢

𝑥

+ 2𝑢

𝑦

𝑢

𝑥𝑦

+ 𝜁

₃

2𝑢

𝑡

𝑢

𝑦

− 2𝑢

𝑥𝑥

𝑢

𝑦

+

2𝑢

𝑥

𝑢

𝑥𝑦

− 𝜁

₂₂

𝑢

𝑦2

− 𝜁

₃₃

𝑢

𝑥2

+ 2𝜁

₂₃

𝑢

𝑥

𝑢

𝑦

]

_𝐹≡0

= 0.

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By substituting 𝜁_!, 𝜁_!, 𝜁_!, 𝜁_!!, 𝜁_!! and 𝜁_!", after simplification, it is clear that the left-‐hand side of the equation (19) is a polynomial in the variables

𝑢

_!

, 𝑢

_!

,

𝑢

_!

, 𝑢

_!!

, 𝑢

_!!

, 𝑢

_!"

, 𝑢

_!"

, 𝑢

_!"

, …,

which all coefficients should be equal to zero. Therefore, we have the following linear system of partial differential equations:

𝜂

_!

= 0, 𝜂

_!

= 0, 𝜂

_!

= 0

𝜉

! !

−

1

2 𝜉

!!

= 0, 𝜉

!!

+ 𝜉

!!

= 0, 𝜉

!!

−

1

2 𝜉

!!

= 0

𝜉

! !

= 0, 𝜉

!!

= 0, 𝜉

!!

= 0, 𝜉

!!

= 0, 𝜉

!!!

= 0, 𝜉

!!!

= 0

By solving the previous system of equations, we obtain

𝜉

!

_{𝑡, 𝑥, 𝑦, 𝑢 = −𝑓}

!

𝑢 𝑦 +

!_!

𝑓

!

𝑢 𝑥 + 𝑓

!

(𝑢),

𝜉

!

_{𝑡, 𝑥, 𝑦, 𝑢 = 𝑓}

!

𝑢 𝑥 +

!_!

𝑓

!

𝑢 𝑦 + 𝑓

!

(𝑢),

𝜉

!

_{𝑡, 𝑥, 𝑦, 𝑢 = 𝑓}

!

𝑢 𝑡 + 𝑓

!

(𝑢),

𝜂 𝑡, 𝑥, 𝑦, 𝑢 = 𝑓

_!

(𝑢),

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2.3 Invariant solution

Linear combination of the symmetries 𝑋_!and 𝑋_!, in section 2.1, is written as

𝑋

₁

+ (0.5)𝑋

₂

=

𝜕 𝜕𝑢

+ (0.5)

𝜕 𝜕𝑡

(31)

We write the characteristic equation of the equation (31) as following

𝑑𝑢 = 2𝑑𝑡,

(32)

by solving the characteristic equation, we obtain

𝑢 − 2𝑡 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡,

(33)

and solving the characteristic equation of the symmetry 𝑋_!

is

(28)

!" !

= −

!" !

(34)

which gives

𝑥

!

_{+ 𝑦}

!

_{= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.}

₍₃₅₎

According to (19) and (20), 𝑢 − 2𝑡 is a function of 𝑥!_{+ 𝑦}!_{, so}

𝑢 − 2𝑡 = 𝜑 𝑥

!

_{+ 𝑦}

!

_,

₃₆ or

𝑢 = 2𝑡 + 𝜙 𝑟 ,

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where

𝑟 = 𝑥

!

_{+ 𝑦}

!

_.

₍₃₈₎

By rewriting equation (15) in variables 𝑡, 𝑟 and 𝜙, as a function of 𝑟, we obtain the following first-‐order ordinary differential equation

𝜙

!

𝑟 − 2𝑟 = 0,

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which gives

𝜙 𝑟 = 𝑟

!

_{+ 𝑐.}

₍₄₀₎ After substitution (23) and (25) in (22), we obtain the invariant solution to the equation (15) as follows

𝑢 𝑡, 𝑥, 𝑦 = 2𝑡 + 𝑥

!

_{+ 𝑦}

!

_{+ 𝑐.}

₍₄₁₎ Due to the initial boundary (15), we have

𝑥

!

+ 𝑦

!

+ 𝑐 = 𝑥

!

+ 𝑦

!

− 1,

hence

𝑐 = −1

Therefore, we obtain the following particular solution to the 2-‐D mean curvature flow equation (11)

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As we see, the solution to the motion by curvature, if the initial boundary be a circle, is a circle of the radius that is depended on the time 𝑡. In figure 11, in section 1.7, I have compared this solution versus the Numerical solution, which I got by implementation of the deterministic control interpretation algorithm.

Conclusion and further work

In conclusion, in motion by curvature, when the initial boundary is a circle, it remains circle, that in each time step it’s radius is depended on the time 𝑡. As a result of the numerical method described in section 1.5, figures 5-‐9, which I got by implementation of the geometric algorithm, in motion by curvature, even if the initial boundary is not convex, after some time steps, the boundary will be convex and after some more time steps it will become circle.

In this thesis I have worked on 2-‐D mean curvature flow equations, further work can be working in higher-‐dimension mean curvature flow equations. In Lie method also, I have worked on 2-‐D problem which initial boundary is circle; so further work can be 2-‐D problems with non-‐circle initial boundary, and also working on higher-‐dimension problems.

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