CALIFORNIA STATE UNIVERSITY, NORTHRIDGE SIMILARITY ANALYSIS OF THE ONE-DIMENSIONAL FOKKER-PLANCK EQUATION A thesis submitted in partial fulfillment of the requirements For the degree of Master of Science in Mathematics By Areej Alsafri May 2014 The thesis of Areej Alsafri is approved: Dr. Rabia Djellouli Date Dr. Emmanuel Yomba Date Dr. Ali Zakeri, Chair Date California State University, Northridge ii Acknowledgments I would like to express my special appreciation and thanks to my advisor Professor Ali Zaker, you have been a tremendous mentor for me. I would like to thank you for encouraging my research and for allowing me to grow as a research scientist. Your advice on both research as well as on my career have been priceless. I would also like to thank my committee members, professor Rabia Djellouli and professor Emmanuel Yomba, for serving as my committee members even at hardship. I also want to thank you for letting my defense be an enjoyable moment, thanks to you. A special thanks to my family and to my family-in-law. Words cannot express how grateful I am to my mother, father, mother-in-law, fater-in-law, and uncl Hamed for all of the sacrifices that you've made on my behalf. Your prayer for me was what sustained me thus far. I would also like to thank all of my brothers and sisters who supported and motivated me to strive towards my goal. At the end I would like express appreciation to my husband Essam who spent sleepless nights with and was always my support in the moments when there was no one to answer my queries. iii Table of Contents Signature Page ii Acknowledgments iii Abstract v Introduction 1 1 Local Lie Group 3 2 Stretching Group 5 3 The Lie Variables 19 4 The General Similarity Method 28 5 The Fokker-Planck Equation 43 6 Similarity Solution Under The Stretching Transformation 44 7 A General Similarity Solution For The Fokker-Planck Equation 54 8 The Clarkson-Kruskal Reduction Method 63 Conclusion 73 Bibliography 74 iv ABSTRACT SIMILARITY ANALYSIS OF THE ONE-DIMENSIONAL FOKKER-PLANCK EQUATION By Areej Alsafri Master of Science in Mathematics The goal of this study is to investigate the conditions under which the Fokker- Planck equation assumes closed form solutions. The Fokker-Planck equation arises in the study of fluctuations in physical and biological systems. This equation has been subject of several recent studies. We have analyzed the existence of solutions of this equation analytically and have established re- sults using various invariant approaches. For example, we have employed a group of stretching transformations that keeps the original equation invariant under this group actions. This resulted in obtaining a class of closed form solutions. Then we analyzed the conditions under which a Clarkson-Kruskal type similarity functional form would result in closed form solutions. The importance of this type of solutions relies on the fact that they can be used as a test functional tool in checking the accuracy of numerical algorithms. General form of such class of solutions is obtained and several specific exam- ples of this class is discussed. In addition, a general group invariant approach is investigated. v Introduction There are many important nonlinear partial differential equations that we are unable to solve analytically. However, under certain conditions, for some of these nonlinear partial differential equations we can transform them into other equations that are invariant or unchanged due the symmetries proper- ties in the partial differential equations. Basically, the method of similarity is determining the set of transformations under which the partial differen- tial equations is invariant and then using the transformations to find special conditions where the partial differential equations can be simplified. These conditions drive a new single independent variable that is a function of the independent variables in the partial differential equations; where the new in- dependent variable reduces the partial differential equation into an ordinary differential equation. The solution to the ordinary differential equation pro- vides a class of self-similar solutions to the partial differential equation. Such solutions are very impotent closed form solutions that can be used to study the physics of the problem, and even if the obtained solutions have no physical values, they can be used as the check solutions for complex numerical method for accuracy of the method. In this thesis, we give an elementary brief review of the general Lie group method with ample examples and explain key defini- tions and main results that are applicable in the problem that we are going to study.We like to point out that G.W. Bluman has already obtained a classs of similarity solutions for the one-dimensional Fokker-Planck equation in the @2P @ @P form of @x2 + @x (f(x)P ) = @t for the initial value P (x; 0=x0) = δ(x − x0). α He assumed that f(x) is an odd function, given by f(x) = x + βx where β > 0 and −∞ < α < 1. In this thesis, we consider a general case of Fokker- @W @ (1) @2 (2) Planck equation in the form of @t = − @x [D (x; t)W ] + @x2 [D (x; t)W ]. So, in Bluman assumed D(1)(x; t) = −f(x) for that particular form and D(2)(x; t) = 1. The main goal of this thesis is to analyze and study the 1 closed form solution of one-dimensional Fokker-Plank equation that is used to study fluctuations in physical and biological models. To this end, we first study a group of stretching infinitesimal transformations. We obtain such solution and the required specific condition on the drift and diffusion coeffi- cients for which such solutions exist. Then, we consider the general Lie group of transformations. Finally, we use the Clarkson-Kruskal type solutions by selecting a specific closed form expression and then we obtained the neces- sary requirements for the existence of this class of similarity solutions. A few special cases are analyzed and closed form solutions are given. 2 1 Local Lie Group We consider a transformation T of the plane that is mapping from R2 to R2, i.e. T (x; t) = (¯x; t¯). This means that T : R2 ! R2 which is defined as T : t¯= φ(x; t); x¯ = (x; t) where φ and are given functions. Geometrically, this means T transforms the point (x; t) to another point (¯x; t¯) in the plane R2. We could obtain the inverse transformation T −1 defined by equations: T −1 : t = φ(¯x; t¯); x = (¯x; t¯) Clearly, TT −1 and T −1T are identity transformation I given by the equations t¯= t; x¯ = x. The types of symmetry transformations under the invariance properties of an equation are transformations of the plane with the additional stipulation that the transformation also depends on a real parameter which we denote by , where varies over an open interval jj < 0 and we consider the collection T of all transformations included in the family defined by T : t¯= φ(x; t; ); x¯ = (x; t; ) (1) 2 where φ and are given functions on R × (−0; 0), and (1) is said to be a one parameter family of transformation on R2. A one parameter family of transformations T is called a local Lie group of transformation if it satisfies the local closer property, contains the identity, and inverses exist for small . Therefore, we want (1) to satisfy the local closure property, namely if 1 and 2 are sufficiently small, then there exists 3 in the interval jj < 0 for which T1 T2 = T3 . 3 Also, we assume that the family T contains the identity transformation I when = 0 t¯= φ(x; t; 0) = t; x¯ = (x; t; 0) = x Finally, for each sufficiently small we want the transformation T1 to have an inverse T2 . Examples of local Lie groups: Translation Group t¯= t + , x¯ = x + Rotation Group t¯= t cos − x sin ,x ¯ = t sin + x cos Stretching Group t¯= eat,x ¯ = ebx where a; b are constants. 4 2 Stretching Group Similarity solutions establish by the invariance properties of the partial dif- ferential equations under local Lie groups of transformations. In this section, we discuss an example of local Lie groups of transformations which is the Stretching Group. The idea is to find new independent and dependent vari- ables that are multiples of the old ones where the partial differential equation in the new coordinate system is the same up to some multiple as in the old coordinate system. Example 1 Consider the heat equation ut − uxx = 0 (2) first we make change of variables of the form x¯ = ax t¯= bt u¯ = cu where is a real parameter in the open interval containing 1, and a,b, and c are fixed real numbers. so, we have c−b c−2a u¯t¯ − u¯x¯x¯ = ut − uxx (3) if we choose a, b and c such that c − b = c − 2a so b = 2a then we get 5 c−b u¯t¯ − u¯x¯x¯ = (ut − uxx) (4) therefore, we can say the original equation is stretched by a multiple constant c−b. Consider a single second order partial differential equation with one depen- dent variable u, and independent variables x and t, u = u(x; t) F (x; t; u; ux:ut; uxx; uxt; utt) = 0 (5) 3 3 define the family of transformation T : R ! R such that x¯ = ax t¯= bt (6) u¯ = cu where a, b and c are fixed real constants and is real parameter that varies over an open interval I that contains 1. Then we can say that T contains the identity transformation T1.