Application of Symmetry Methods to Partial Differential Equations

Transcription

1 Application of Symmetry Methods to Partial Differential Equations by Omar Mayez Hijazi B. Mathematics and Information Technology (University of Western Sydney) B.Sc. (Hons) Mathematics (University of Western Sydney) A thesis submitted to the College of Health and Science in fulfillment of the requirements for the degree of Doctor of Philosophy School of Computing and Mathematics University of Western Sydney, March 2013

2 Acknowledgements I would like to express my deepest thanks to my supervisors, Associate Professor Richard Ollerton, Dr Alec Lee and Dr Stephen Weissenhofer who pointed me in the right direction and contributed in every possible way to this thesis. Finally, I gratefully acknowledge the support of my wife Iman, my son Mayez and my two daughters Souraya and Sarah for their support to write and finish this thesis. ii

3 Declaration This thesis contains no materials which have been accepted for the award of any other higher degree or graduate diploma in any tertiary institution and to the best of my knowledge and belief, the thesis contains no materials previously published or written by another person, except where due reference is made in the text of the thesis. Omar M. Hijazi March, 2013 iii

4 Contents Acknowledgements ii Declaration iii List of Figures viii Abstract ix 1 Introduction Overview Structure of the Thesis Literature Review Introduction Mathematical Models of Predator-Prey Dynamical Systems Single Species Models The Lotka-Volterra Equations A PDE Model of Predator-Prey Dynamics A Second Order PDE Predator-Prey Model iv

5 2.2.5 Coupled Non-Linear Burgers Equations Qualitative Analysis of a Diffusive Predator-Prey Model Summary PDE Solution Methods: An Overview The Classical Approach The Nonclassical Approach Potential Symmetries Initial and Boundary Value Problems Definitions of Some of the Main Terminology of Lie Symmetry Conclusion PDE Solution Methods: Worked Examples Introduction Classical Approach Burgers Equation Nonclassical Approach Burgers Equation Potential Symmetries Approach Burgers Equation Conclusion Review of Hasimoto s System Introduction Determination of the Infinitesimals Solution Process Conclusion Modification of Hasimoto s System 55 v

6 5.1 Introduction Classical Approach Determination of the Infinitesimals Solution Process: Special Cases Comments on a More General Approach Solution Process Conclusion Further Solutions for Modifications of Hasimoto s System Introduction Modification of the Interaction Terms Introduction Classical Approach Solution Process Conclusion Further Solutions for Modifications of Hasimoto s System Introduction Classical Approach Determination of the Infinitesimals Solution Process Special Case Conclusion Non-Linear Predator-Prey System 118 vi

7 8.1 Introduction Modification of the Velocity Terms Introduction Classical Approach Modification of the Interaction Terms Introduction Classical Approach Potential Symmetry Coupled Non-Linear Burgers Equations Introduction Classical Approach Nonclassical Approach Potential Symmetries Approach Graph of Solution Conclusion Conclusion and Suggestions for Further Research Summary Summary of New Solutions Suggestions for Further Research Conclusion Bibliography 169 vii

8 List of Figures 1 Graph of the general solution (4.18) of Hasimoto s system (1.1) at various times with initial and boundary conditions (4.19) Graph of the exact solution (6.81) for system (6.80) at various times with initial and boundary conditions (6.1) Graph of the trajectories of the implicit solution (7.85) for system (7.84) near the critical points (0,0) and (B,A), where A = 5 and B = 10. The trajectories start at the boundary (G = 5), leading downwards and curving to finish at the right hand boundary (F = 10). This occurs because (0,0) is an unstable saddle point Graph of the exact soliton solution (7.92) for system (7.78) at various times Graph of solution(8.114) for system(8.3) at various times with initial and boundary conditions (8.84) and (8.85) viii

9 Abstract In this thesis, solutions of variations of the following system of semi-linear coupled equations by classical (Lie [52]), nonclassical (Bluman and Yan [24]) and potential symmetry methods (Anco and Bluman [5]) are considered: u t +a u x = λ 1uv, v t +b v x = λ 2uv, (1) where a, b, λ 1 and λ 2 are constants, and where u and v are functions of x and t and may represent prey and predator densities respectively. The general solution of this system was first presented by Hasimoto[39] by means of a transformation analogous to that used by Hopf [41] and Cole [29] in their derivation of the solution of the Burgers equation. Later, Chow [28] obtained the general solution by the classical symmetry technique. Chow s solution procedure is reviewed before solutions are found by use of the classical approach for the following modification of Hasimoto s system: u t +a u = m(x, t, u, v), x v t +b v = n(x, t, u, v), (2) x where m and n are functions of (x, t, u, v) and may model the interaction between prey and predator species along the x-axis at time t with respect to the densities of the species u and v. While general and exact solutions are not found for system (2) with general forms of m and n, exact solutions are obtained for special cases of m and n by use of the classical method. The first special case is u t +a u x = m(t)uv, v t +b v = n(t)uv, (3) x ix

10 where the functions m and n may model time varying strengths of inter-species interaction. The second special case of (2) is u t +a u x = λ 1F(u)G(v), v t +b v x = λ 2F(u)G(v), (4) which may model more general time independent interactions between prey and predator. The solution process gave rise to special cases of system (4), one such system involving soliton solutions and another involving u and v to powers of constants on the right hand sides. Exact solutions are obtained by the classical approach and presented graphically. Some limitations of the classical approach are also considered. Two further modifications of Hasimoto s system are then explored: and u t +u u x = λ 1uv, v t +v v x = λ 2uv, (5) u t +u u x = λ 1 x (uv), v t +v v x = λ 2 (uv). (6) x Exact solutions are found by the classical method, while nonclassical and potential symmetry methods gave no new solutions. Finally, system (6) is modified to become a system of coupled Burgers equations: u t 2 u x +u u 2 x = λ 1 x (uv), v t 2 v x +v v 2 x = λ 2 (uv). (7) x x

11 Classical, nonclassical and potential symmetry approaches are applied and new exact solutions obtained and presented graphically. xi

12 Chapter 1 Introduction 1.1 Overview Dynamical systems (for example, predator-prey relationships, traffic flow, the flow of water in a pipe, the human body) have been a dominant theme in mathematics, physics, engineering and biology. In particular, interaction between species has been modelled by the use of difference equations, ordinary differential equations (ODEs) and partial differential equations (PDEs). PDEs arise in mathematics, physics, engineering and many other disciplines. Symmetry methods are important as they can be used to obtain special reduced solutions of PDEs. Symmetry methods include the classical approach, the nonclassical approach and the potential symmetries method. In this thesis they will be applied to selected PDE systems over time and a single space dimension in order to find new solutions. Some of the solutions will be presented graphically. As a unifying theme, these PDE models will be described in terms of predator-prey interactions. 1

13 1.2 Structure of the Thesis A literature review of the mathematical models of predator-prey systems is presented in the first part of chapter 2. The review provides a broad historical overview of the field from the Lotka-Volterra model onwards for ODE and PDE models. It also contains a description of all terms, their physical meanings, assumptions made, the methods used to solve the equations and their solutions. Predator-prey systems will provide a verifying theme for the systems of PDEs considered in this thesis. A historical overview of symmetry methods (classical, nonclassical and potential symmetries) and worked examples to illustrate the methods are provided in the second part of chapter 2 and in chapter 3. A semi-linear coupled predator-prey system of equations by Hasimoto [39], first solved in 1974 by means of a transformation analogous to that used by Hopf [41] and Cole [29] in their derivation of the solution of the Burgers equation, is given by: u t +a u x = λ 1uv, v t +b v x = λ 2uv. (1.1) System (1.1) is considered in chapter 4. Chow [28] applied the classical approach to this system showing that symmetry methods are powerful tools to solve some PDE systems. She deduced the same results obtained by Hasimoto. A summary of Chow s solution method is given and the general solution is presented graphically for some initial conditions. In this research modifications are made to Hasimoto s system (1.1) and solved by the use of the different symmetry techniques. The first modification described in 2

14 chapter 5, is the introduction of two new functions of (x,t,u,v) on the right hand side of (1.1) which alter the interaction terms to a more general form giving u t +a u x = m(x,t,u,v), v t +b v x = n(x,t,u,v). (1.2) The functions m and n may model the interaction between prey and predator species along the x-axis at time t with respect to the densities of the species u and v. Chapter 5 also describes the solution process for this modification. While general and exact solutions for system (9.2) using the classical technique are not found, special cases of system (9.2) are derived and solved. The final section of chapter 5 considers a special case of the system (9.2): u t +a u x = 2(x+at)u 1 v, v t +b v x = 2(x+bt)v 1 u. (1.3) The aim of exploring the above system is to highlight some limitations of the classical approach. The second special case of (9.2) involves replacing m(x,t,u,v) and n(x,t,u,v) by functions of time multiplied by the product uv: u t +a u x = m(t)uv, v t +b v x = n(t)uv. (1.4) The modified functions m and n in (1.4) may model time varying strengths of interspecies interaction. The solution process for (1.4) is described in chapter 6. 3

15 The third special case of (9.2) is obtained by changing the right hand sides to λ 1 F(u)G(v) and λ 2 F(u)G(v) respectively, giving u t +a u x = λ 1F(u)G(v), v t +b v x = λ 2F(u)G(v), (1.5) where λ 1 and λ 2 are constants and may model the rate of nonlinear growth, decay and interaction of the two populations expressed by the functions F and G. After the classical procedure is applied to (1.5) in chapter 7, a transformation is applied in order to consider a special case: u t +a u x = c(α+βu)j (γ +δv) k, v t +b v x = d(α+βu)j (γ +δv) k. (1.6) where α, β, γ, δ, c, d, j and k are constants. The special case is solved by use of the classical method. In addition, the following special case of (9.2) is also explored by using the classical approach: u t +a u x = λ 1u A v, v t +b v x = λ 2v B u, (1.7) where the constants A and B are strictly greater than v and u respectively. The right hand side of the modified system (1.7) may model the interaction between the two species limited by external resources. A further modification to (1.1), presented in chapter 8 involves the construction of coupled Burgers equations (Kaya [47]). This is done in three steps. The first step 4

16 is the replacement of the constants a and b on the left hand sides of (1.1) by u and v respectively. The motivation behind this modification is to make the velocity of the species dependent on the density. The second step introduces the spatial gradient of the product of the population densities to the interaction terms on the right hand sides. Finally, diffusion terms 2 u x and 2 v modelling the effect of transportation in 2 x2 the habitat while propagating along the x-axis, are introduced to the left hand side giving u t 2 u x +u u 2 x = λ 1 x (uv), v t 2 v x +v v 2 x = λ 2 (uv). (1.8) x The solution processes for the new modified systems are presented in chapter 8. Some of the solutions are presented graphically. Finally, chapter 9 presents conclusions and suggestions for further research. 5

17 Chapter 2 Literature Review 2.1 Introduction Understanding and predicting the population dynamics of species have been of significant interest to both mathematicians and ecologists, especially since 1925 when Lotka and Volterra introduced their ODE model (A.J. Lotka [56], V. Volterra [65] and Edelstein-Keshet [48]). Discrete and continuous time models have been used by mathematicians to model population dynamics. Discrete time models are described by difference equations, discrete dynamical systems or iterative maps (Liao et al [50] and [51], Loladze [54] and [55] and Skider [63]). Continuous time models may be described by differential equations or dynamical systems. In this thesis, only continuous models are considered. Section 2.2 of the literature review will consider single species systems modelled by continuous (ODE and PDE) systems. This is intended to provide a wide ranging background to the practical application of these models. Section 2.3 will then describe 6

18 the main symmetry methods for solution of PDEs. 2.2 Mathematical Models of Predator-Prey Dynamical Systems Single Species Models For a single species, the simplest models are given by the exponential model ẋ = r 0 x, and the logistic model ẋ = r 0 x(1 x ), where x(t) denotes the population of the k species at time t, k is the carrying capacity of the environment and r 0 is the essential growth rate (Liu and Xiao [53]). The corresponding discrete models are given by x n+1 = x n +r 0 x n and x n+1 = x n +r 0 x n (1 x n k ), where x n is the population at stage n The Lotka-Volterra Equations Lotka and Volterra constructed a predator-prey system which models interaction and competition between two species. Volterra was an Italian mathematician who became interested in the area of population biology after a discussion with a colleague U. d Ancona (Edelstein-Keshet [48]). During World War I, commercial fishing in the Adriatic Sea fell to very low levels. The population of the commercially valuable fish declined on average while the number of sharks increased. In 1925, Volterra, with the help of the American biophysicist Lotka proposed a continuous model to describe predator-prey interactions in the fish populations. Lotka and Volterra used some simplifying assumptions to produce their system. 7

19 The following are some of the assumptions made: 1. Prey grow in an unlimited way when predators do not keep them under control. 2. Predators depend on the presence of their prey to survive (i.e predator population decays in absence of prey). 3. The rate of predation depends on the likelihood that prey are encountered by predators. 4. The growth rate of the predator population is proportional to food intake. These assumptions lead to the following system of equations: dx dt dy dt = ax bxy, = cy +dxy, (2.1) where a, b, c and d are positive constants and the functions x and y represent the prey and predator populations respectively at time t. Other physical systems such as competitive hunters can be modelled by taking different signs on the right hand side of (2.1). Interpretation of the Lotka-Volterra Terms In system (2.1) the constant a represents the net growth rate of the prey population when predators are absent and ax represents the growth term. The constant c represents the net death rate of the predator population in the absence of prey and cx represents the decay term. The xy terms appearing in both equations represent the interaction between the two populations, given that both species move about randomly and are uniformly distributed over their habitat. Finally, the derivative terms 8

20 represent the rates of change in both populations with respect to time t. From (2.1), a relationship between x and y may be found in the form where k is an arbitrary positive constant. Modifications of the Lotka-Volterra Equations y a x c = k, (2.2) eby+dx A number of modifications have been made over the years to the Lotka-Volterra equations. Some of these are listed below. 1. Density dependence: More realistic prey growth-rate assumptions in which a in (2.1) is replaced by a density-dependent function f(x): (a) f(x) = r(1 x)modelslogisticgrowthlimitedbyexternalresources(pielou k [59]), (b) f(x) = r( k 1) models a hyperbolic relationship between population density and the instantaneous per-individual birth rate (Schoener x [62]). 2. Attack rate: More realistic predation interaction rates where the term bx in (2.1) is replaced by h(x) in which the attack capacity of predators is limited. (a) If h(x) = k(1 e cx ), then k(1 e cx ) models behaviour based on observations that a lone exploiter will attack k(1 e cx ) victims in a fixed amount of time (Ivlev [46]). (b) If h(x) = kx g, where 1 g > 0, then kx g represents the predator kill rate and also a relatively low tendency for exploiters to become hungry or satiated and to modify their behavior accordingly (Rosenzweig [60]). 9

21 2.2.3 A PDE Model of Predator-Prey Dynamics System Overview A semi-linear system of first order partial differential equations has been proposed by Hasimoto [39]. The system models the interaction between two species, predator and prey, and is given by u t +a u x = λ 1uv, v t +b v x = λ 2uv. (2.3) System(2.3) is the simplest hyperbolic system which describes the non-linear coupling between two waves propagating along the x-axis and may be used to express a time history of the distribution of prey u(x, t) and predator v(x, t) runing on a straight line in the same or the opposite direction with constant velocities. Interpretation of Hasimoto s Model The terms u t and v t in (2.3) model the rate of change of the prey densities u(x,t) and predator densities v(x, t) respectively with respect to the time t. The terms u x and v model the change of the prey and predator densities according to the change x in position on the x-axis. The constants a and b in(2.3) represent, respectively, the velocity of the two waves while propagating along the x-axis. The signs of a and b determine the directions of propagation of the two waves. The interaction between the two species is modelled by the terms λ 1 uv and λ 2 uv, where λ 1 and λ 2 indicate the strength of the interaction. In other words, the number of interactions is proportional to the product of the population sizes. 10

22 System Analysis System (2.3) has been solved previously by Hasimoto [39], Yoshikawa [67] and Chow [28]. The latter used classical symmetry methods to solve the system and obtained u(x, t) = v(x, t) = (a b)f (x at) λ 2 [F(x at)+g(x bt)], (b a)g (x bt) λ 1 [F(x at)+g(x bt)], (2.4) where F and G are arbitrary functions of x at and x bt respectively. Note that, if the parameters λ 1 and λ 2 are zero then there is no interaction between the two species and (2.4) would not be valid. In that case the two species are isolated from each other. The solutions are then in the form u = h(x at) and v = k(x bt), where h and k are arbitrary functions of values x at and x bt respectively A Second Order PDE Predator-Prey Model Introduction Kanel and Zhou [49] investigated the existence of wavefront solutions and estimated the wave speed for the following competition diffusion system: u t d 2 u 1 x = u(a 2 1 b 1 u c 1 v), v t d 2 v 2 x = v(a 2 2 b 2 u c 2 v). (2.5) Model Analysis System (2.5) is a modification of the Lotka-Volterra system where more terms are added to the right hand sides of the system and second spatial derivatives are introduced to the left hand sides. The dependent variables u(x,t) and v(x,t) represent 11

23 the space and time dependent densities of the prey and predator populations respectively. The terms u v and model the rate of change for the prey and predator t t 2 u population with respect to time t. The terms d 1 x and d 2 v 2 2 model the effect of x2 transportation in the habitat where constants d 1 and d 2 represent the diffusivity of each species while propagating along the x-axis. For suitably signed constants, the constant a 1 represents the growth factor of the prey in the absence of the predator while a 2 represents the death factor of the predator in the absence of the prey. The constants a 1 b 1 and a 2 c 2 are the carrying capacities of the species while c 1 and b 2 are the strength of the interaction for the two species. The main purpose of the paper by Kanel and Zhou [49] was to investigate the existence of travelling wave solutions of the form u = u(ξ), v = v(ξ) where ξ = x+ct, where c is a constant representing wave speed and u = u(ξ) and v = v(ξ) are positive and monotone about ξ in the interval (, + ) and satisfy the boundary conditions u( ) = u, u(+ ) = u +, v( ) = v and v(+ ) = v + where (u,v ) and (u +,v + ) are the rest points of (2.5). Such solutions were found. Similar systems were examined by different authors. For example, Ahmed and Lazer [2] investigated the same system but for N-equations Coupled Non-Linear Burgers Equations Introduction Esipov [33] derived the following coupled non-linear Burgers equations: u t 2 u x +u u 2 x +λ 1 (uv) = F(x,t), x v t 2 v x +v v 2 x +λ 2 (uv) = G(x,t), (2.6) x 12

24 where F and G are given functions and λ 1 and λ 2 are constants. System (2.6) arose as a simple model of sedimentation of scaled volume concentrations of two kinds of particles in fluid suspensions under the effect of gravity. Model Analysis A number of mathematicians have examined (2.6) for travelling wave-front solutions using different methods. As an example, Kaya [47] considered the system with appropriate initial values using the decomposition method. The author calculated the solution in the form of a convergent power series with easily computable components. Also, Inan et al [45] applied Bäcklund transformation and similarity reduction to (2.6) and obtained solitary wave solutions and travelling wave solutions. In addition, Arrigo et al [9] considered the nonclassical symmetries of a class of Burgers systems. System (2.6) may also be interpreted in terms of predator-prey interactions. This is due to the similarity of the terms involved in (2.3), (2.5) and (2.6), where u(x,t) and v(x,t) may model the densities of prey and predator respectively. The terms u t, v t, d 2 u 1 x and d 2 v 2 2 may have the same interpretation as those in (2.3) and (2.5). x2 The interaction between the two species is modelled by the spatial gradient of the product of the population densities. In this research, the classical, nonclassical and potential symmetry methods are applied to search for different types of solutions. To the best of my knowledge some of the solutions which are obtained have not been previously reported. 13

25 2.2.6 Qualitative Analysis of a Diffusive Predator-Prey Model Introduction Chen and Wang [25] studied the qualitative properties of a diffusive predator-prey model subject to a homogeneous Neumann boundary condition using a comparison argument and iteration technique. Under some hypotheses they proved that the positive steady-state solution is globally asymptotically stable. They also established the existence and non-existence of non-constant positive steady-states (stationary patterns) by use of degree theory. Model Analysis Assuming predator and prey densities are spatially inhomogeneous in a bounded domain Ω R N with smooth boundary, Chen and Wang [25] considered the following predator-prey model with diffusion: ( u t d 1 u = u r 1 b 1 u a 1v v t d 2 v = v 1+c 1 u ), x Ω, t > 0, ( r 2 a ) 2v, x Ω, t > 0, c 2 +u v u = v v = 0, x Ω, t > 0, u(x, 0) = u 0 (x) > 0, v(x, 0) = v 0 (x) > 0, x Ω, (2.7) where the densities of the prey and predator are represented by u and v respectively and are required to be non-negative. In system (2.7), v is the outward unit normal vector on Ω and v = / v, and where = x 2 1 x 2 2 x 2 n is the Laplacian. It should be noted that the Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. The homogeneous 14

26 Neumann boundary condition means that (2.7) is self-contained and there is no population flux across the boundary Ω. All the parameters appearing in the model are assumed to be positive constants. The constants d 1 and d 2 are the diffusion coefficients corresponding to u and v, and the initial data u 0 (x) and v 0 (x) are continuous functions. Note that (2.7) has a unique global solution (u, v) (Chen and Wang [25]). In addition, by virtue of u 0 > 0, v 0 > 0, the solution is positive. For simplicity, using the non-dimensional variables t t b 1 c 1, u c 1 u, v v a 1c 1 b 1, d 1 d 1 c 1 b 1, d 2 d 2 a 1 a 2 b 1, Chen and Wang [25] obtained the following non-dimensionalized form of (2.7): ( u t d 1 u = u a u v τv t d 2 v = v 1+u ), x Ω, t > 0, ( b v ), x Ω, t > 0, c+u v u = v v = 0, x Ω, t > 0, u(x, 0) = u 0 (x) > 0, v(x, 0) = v 0 (x) > 0, x Ω. (2.8) Note that when u is small it will be like uv and when it is large it will be like v, where a = c 1r 1 b 1, b = a 1r 2 b 1 a 2, c = c 1 c 2, and τ = a 1 a 2 b 1. The main aim of Chen and Wang s [25] paper was to study the large-time behaviour of solutions to (2.8), and the existence or non-existence of non-constant positive steady states of the model. The steady state problem of (2.8) is the following elliptic system: ( d 1 u = u a u v d 2 v = v 1+u ), x Ω, ( b v ), x Ω, c+u v u = v v = 0, x Ω. (2.9) 15

27 By direct computation, Chen and Wang[25] noted that the positive constant solutions of (2.9) have the following cases: Case A: a = bc > 1+b. System (2.9) has a unique positive constant solution (ǔ 0, ˇv 0 ): ǔ 0 = a 1 b, and ˇv 0 = b(c+ǔ). Case B: a > bc. (2.9) has a unique positive constant solution (ǔ 1, ˇv 1 ): ǔ 1 = 1 2 a 1 b+ (a 1 b) 2 +4(a bc), and ˇv 1 = b(c+ǔ 1 ). Case C: 1+b < a < bc and (a 1 b) 2 +4(a bc) 0. The two possible positive constant solutions of (2.9) are (ǔ 2, ˇv 2 ) and (ǔ 3, ˇv 3 ): ǔ 2 = 1 2 a 1 b+ (a 1 b) 2 +4(a bc), ˇv 2 = b(c+ǔ 2 ) and ǔ 3 = 1 2 a 1 b+ (a 1 b) 2 +4(a bc), ˇv 3 = b(c+ǔ 3 ). Firstly, by a comparison argument and an iteration technique, they proved that the positive constant steady state (ǔ 1,ˇv 1 ) is globally asymptotically stable. Secondly, by an energy method, they gave some non-existence results of non-constant positive classical solutions of (2.9) for a certain range of the parameters. Finally, based on degree theory, they considered the existence and bifurcation of non-constant positive solutions for (2.9) Summary Different types of predator-prey systems modelling various types of species interaction were presented in this section. This was done to give a good understanding about the field and to show the methods used and some of the obtained solutions. Some 16

28 of these models will be modified and examined in this research. Some others may be used in future research. An overview of the mathematical methods that are used to solve the PDE systems is given next. 2.3 PDE Solution Methods: An Overview Symmetry methods for PDEs were first developed in the late 19th century by Marius Sophus Lie [52]. They are important in the study of PDEs arising in mathematics, physics, engineering and many other disciplines since they can be used to obtain special reduced solutions of the PDEs. In general, a symmetry for a system of PDEs is any transformation that maps its solutions to other solutions. The following are the main symmetry methods which will be utilised in this thesis: Lie s work on invariance of PDEs (the classical approach) includes: Lie symmetries, infinitesimals, mapping of solutions to solutions and invariant (similarity) solutions (Hill [43], Lie [52] and Olver [58]). The nonclassical approach which represents an extension of Lie s classical approach (Bluman [15], Bluman and Cole [17], Bluman, Yan [24], Hill [43], and Clarkson and Mansfield [31]); The potential symmetries involving conservation laws and nonlocal extensions (Bluman and Kumei [19], Bluman et al [20], Anco and Bluman [4], [5], [6], Bluman and Cheviakov [16] and Bluman, Cheviakov and Anco [8]). One of the main difficulties that arises when using symmetry methods is the huge amount of algebraic work necessary to derive solutions for a given PDE. Previously, 17

29 this has limited the use of Lie s theory, but more recently the availability of powerful computer algebra systems such as Mathematica and Maple has reduced the problem of huge algebraic manipulations. In this research a Mathematica package called MathLie, written by Gerd Baumann [13], will be used to apply the different methods of symmetry and a Maple package called GeM, written by Alex F. Cheviakov [26] and [27] is used to compute conservation laws (CLs) that lead to potential systems The Classical Approach For a single PDE involving a dependent variable u and two independent variables x and t the following transformation is considered (Hill [43], Bluman and Kumei [21], Lie [52] and Olver [58]): x 1 = x 1 (x, t, u, ε) = x+εξ(x, t, u)+o(ε 2 ) = e εx x, t 1 = t 1 (x, t, u, ε) = t+εη(x, t, u)+o(ε 2 ) = e εx t, (2.10) u 1 = u 1 (x, t, u, ε) = u+εζ(x, t, u)+o(ε 2 ) = e εx u = U(x, t, u, ε), where ξ and η are the infinitesimals of the transformation for the independent variables, ζ is the infinitesimal for the dependent variable and X is the differential operator defined by X = ξ(x,t,u) x +η(x,t,u) t +ζ(x,t,u) u. (2.11) The invariance condition is then used to determine the infinitesimal functions ξ, η and ζ. If the transformation (2.32) leaves a given PDE invariant and if u = α(x, t), then from u 1 = α(x 1, t 1 ) on equating terms of order ε, the following is obtained: ξ(x, t, u) u +η(x, t, u) u x t 18 = ζ(x, t, u). (2.12)

30 For known functions ξ, η and ζ, equation (2.12) is a first order PDE which may be solved for u. This yields the functional form of the similarity solution in terms of an arbitrary function. This arbitrary function is determined by substitution of the functional form of the solution into the given PDE. It should be noted that equation (2.12) only yields a special class of solutions for a PDE invariant under the transformation (2.32). In the case of a system of PDEs with two dependent variables u and v, the transformation (2.32) is extended to: x 1 = x 1 (x, t, u, v, ε) = x+εξ(x, t, u, v)+o(ε 2 ) = e εx x, t 1 = t 1 (x, t, u, v, ε) = t+εη(x, t, u, v)+o(ε 2 ) = e εx t, (2.13) u 1 = u 1 (x, t, u, v, ε) = u+εζ(x, t, u, v)+o(ε 2 ) = e εx u = U(x, t, u, v, ε), v 1 = v 1 (x, t, u, v, ε) = v +εχ(x, t, u, v)+o(ε 2 ) = e εx v = V(x, t, u, v, ε). Instead of (2.12) the following is used: ξ(x, t, u, v) u +η(x, t, u, v) u x t ξ(x, t, u, v) v +η(x, t, u, v) v x t = ζ(x, t, u, v), which, in principle, will yield functional forms of u and v. = χ(x, t, u, v), (2.14) The Nonclassical Approach The nonclassical method of reduction of PDEs was introduced by Bluman and Cole [17] as an extension of Lie s classical approach. They applied their new method to the one dimensional linear heat equation u xx u t = 0, 19

31 in order to derive solutions which are different to those obtained by the classical method. The nonclassical approach, which is considerably more complicated to apply, as the determining equations are no longer linear, makes use of (2.12) before the functions ξ, η and ζ are determined and includes the infinitesimals from the classical method as special cases (Hill [43]). The direct method due to Clarkson and Kruskal (Clarkson, Mansfield [30] and Clarkson [31]) is also a special case of the nonclassical method. The basic idea of the direct method is to seek a solution for a PDE in the following form: ( ( u(x,t) = F x,t,w z(x,t)) ), and require that w(z) satisfies an ODE. This imposes conditions upon F(x,t,w), z(x, t) and their derivatives in the form of an overdetermined system of equations whose solution yields the desired reductions. Many new solutions to nonlinear PDEs have been found using the nonclassical approach Potential Symmetries The potential symmetries method involves the derivation of new systems of PDEs (potential systems) from the original system whose solutions are also solutions of the original system. Classical or nonclassical methods can then be applied to the potential systems. The infinitesimals obtained from the classical and the nonclassical methods will not only depend on the local dependent variables of the original equation but on the new potential variables of the affiliated systems. Potential systems are constructed by the use of conservation laws (CLs). Finding CLs is important in the study of physical systems (Anco and Bluman 20

32 [3]). They have many significant uses, particularly with regard to integrability and linearisation, constants of motion, analysis of solutions and numerical solution methods (Wolf [66] and Cheviakov [27]). CLs represent multipliers used with the original PDE system and do not change its specifications. Different approaches are available in the literature to find CLs (Bluman and Kumei ([19], [20]), Anco and Bluman ([4], [5], [6]), Bluman and Cheviakov [16], Bluman, Cheviakov and Anco [8] and Wolf [66]). In this research the direct method by Anco and Bluman ([5], [6]) and Bluman, Cheviakov and Anco [8] is used. They show how CLs can be found systematically. Given the PDE G(x,t,u, u,, k u) = 0, (2.15) a CL is an expression D x X(x,t,u, u,..., r u)+d t T(x,t,u, u,..., r u) = 0 (2.16) which holds for any solution of (2.15), where D x and D t are the operators of total differentiation with respect to x and t respectively. The differential functions X and T are called the flux and density of the CL. The search for nontrivial CLs of (2.15) is equivalent to finding all local multipliers µ(x,t, U,..., q U) so that for all U µ(x,t, U,..., q U)G(x,t,U, U,..., k U) D x X(x,t,U, U,..., r U)+D t T(x,t,U, U,..., r U) for some {X(x,t,U, U,..., r U), T(x,t,U, U,..., r U)}. 21

33 Anco and Bluman ([4], [5], [6]) and Bluman, Cheviakov and Anco [8] showed that µ(x,t, U,..., q U) is a multiplier for a local CL of (2.15) if and only if E U ( µ(x,t, U,..., q U)G(x,t,U, U,..., k U) ) 0 where the Euler operator is E U = ( ) U D x +D t U x U ( t ) + (D x ) 2 +(D t ) 2 +D x D t U xx U tt U xt. Definition: Euleroperator[13]: Letf = f(x,u,u x, )bethedensityofafunctional F[u]. Then we call the functional derivative of F and F u := ( ( 1) n dn f dx n ϵ := n=0 n=0 u (n) ( 1) n D n u (n) an Euler operator. D n = dn dx n denotes the nth-order total derivative. then and Example: As an example of the Euler operator let f = u 2 +uv v 2 +u 2 t +v 2 t, E u f = 2u+v +2u tt E v f = u 2v +2v tt. If a multiplier is known, there is an integral formula to obtain the conserved densities. A multiplier µ(x,t,u, u,..., n 1 u) for a CL of PDE G = 0 is a higher order ) symmetry U = µ u 22

34 of G = 0 if and only if the PDE G = 0 admits a variational principle. Note that U is a symmetry of G = 0 if and only if µ(x,t,u, u,..., n 1 u) satisfies its linearised system (Frechet derivative) G u µ+g ux D x µ+g ut D t µ+ G=0 = 0. More generally, a multiplier µ(x,t,u, u,..., n 1 u) for a CL of PDE G = 0 is a solution of the adjoint of its linearised system G u ν +G ux D x ν +G ut D t ν + G=0 = 0, that is, solves µ(x,t,u, u,..., n 1 u) G u µ D x (G ux µ) D t (G ut µ)+ G=0 = 0. After CL (2.16) is found the following formula is used to form the potential system P of (2.15) (Bluman and Kumei [19]): v t = X(x,t,u, u,..., r u), v x = T(x,t,u, u,..., r u), G(x,t,u, u,..., k u) = 0. (2.17) If (u, v) solves potential system P (2.17) then u clearly solves G = 0. Conversely, if u solves G = 0 then there exists a solution (u, v) and (u, v +C) for any constant C of the potential system P. Suppose the differential operator X = ξ(x, t, u, v) x +η(x, t, u, v) t +ζ(x, t, u, v) u +χ(x, t, u, v) v (2.18) 23

35 is admitted by system (2.17), then solutions not obtained by the classical approach to the given PDE (2.15) may be found if and only if (ξ v ) 2 +(η v ) 2 +(ζ v ) 2 0. (2.19) If (2.19) is identical to zero then no new solutions are obtained. Many new solutions to PDEs were obtained by the use of the potential symmetry approach. Finally, the following four steps are followed when using the method of potential symmetries [13]: Determine the integrating multipliers. Extract these integrating multipliers which allow an equivalent conservation law. Create for each integrating factor from the second step the potential representation. Examine the potential system either with the classical or the nonclassical method. It should be noted that if a given PDE system has n local conservation laws which, in turns respectively yield n potential variables, then one can obtain up to 2 n 1 nonlocally related PDE systems by considering the obtained potential systems oneby-one(n singlets, each with one potential variable), in pairs(n(n 1)/2 couplets, each with two potential variables) and all together (one n-plet containing all n potential variables). Furthermore, a given PDE system may admit more potential systems by considering subsystems by excluding one of the dependent variables from the given PDE system through some means that connects each solution of the given PDE system and subsystem. 24

36 As an example of the method, Bluman, Cheviakov and Anco [8] and Bluman and Temuerchaolu [23] computed conservation laws (CLs) for the nonlinear telegraph (NLT) systems H 1 [u,v] = v t F(u)u x G(u) = 0, H 2 [u,v] = u t v x = 0 (2.20) by using multipliers that are functions of independent and dependent variables. CLs are used to find potential systems for the NLT. Potential systems are used to find new invariant solutions for the NLT. Details about the interpretation of the terms in(2.20) are given in Bluman, Cheviakov and Anco [8] and Bluman and Temuerchaolu [23]. Two functions ξ = ξ(x,t,u,v) and ϕ = ϕ(x,t,u,v) are multipliers of a conservation law of system (2.20) if they satisfy ξh 1 [U,V]+ϕH 2 [U,V] = D x X +D t T (2.21) for all differentiable functions U(x, t) and V(x, t) and some differentiable functions X = X(x,t,U,V) and T = T(x,t,U,V). Consequently, the conservation law D x X +D t T = 0 (2.22) holds for all solutions U = u(x,t) and V = v(x,t) of system (2.20) with flux X(x,t,u,v) and density T(x,t,u,v). The necessary and sufficient conditions for ξ(x,t,u,v) and ϕ(x,t,u,v) to yield multipliers for a conservation law of (2.20) are that the Euler operators E U and E V with respect to U and V, respectively, annihilate the left-hand side of (2.21), thus E U [ϕ(x,t,u,v)(u t V x )+ξ(x,t,u,v)(v t F(U)U x G(U))] = 0, E V [ϕ(x,t,u,v)(u t V x )+ξ(x,t,u,v)(v t F(U)U x G(U))] = 0, (2.23) 25

37 for all differentiable functions U(x,t) and V(x,t), where E U = U D x D t, U x U t E V = V D x D t, (2.24) V x V t and where D x and D t are the total derivative operators with respect to the independent variables x and t such as D x = x +U x U +V x V +U xx, U x + U xt U t +V xx V x +V xt V t +, etc, D t = t +U t U +V t V +U tt, U t + U xt U x +V tt V t +V xt V x +, etc. (2.25) Using (2.24) implies that E U = ξ(x,t,u,v)f (U)U x ξ(x,t,u,v)g (U)+D x (ξ(x,t,u,v)f(u)) D t (ϕ(x,t,u,v)), E V = D x (ϕ(x,t,u,v)) D t (ξ(x,t,u,v)). (2.26) The following system is obtained by using (2.25) in system (2.26) then substituting the results into system (2.23), consider ξ(x,t,u,v)f (U)U x ξ(x,t,u,v)g (U)+ξ(x,t,U,V) U t F (U) +F(U)(ξ t (x,t,u,v)+ U t ξ U(x,y,U,V))+ V t ξ V(x,t,U,V) +U t ξ(x,t,u,v)f (U)+U t ξ U (x,t,u,v)f(u)+v t ξ V (x,t,u,v)f(u) = 0, 26

38 ϕ x (x,t,u,v)+ U x ϕ U(x,t,U,V)+U x ϕ U (x,t,u,v)+v x ϕ V (x,t,u,v) ξ t (x,t,u,v) U t ξ U(x,t,U,V) V t ξ V(x,t,U,V) U t ξ U (x,t,u,v) V t ξ V (x,t,u,v) = 0. (2.27) It can be easily shown from system (2.20) that U t = V x and V t = F(U)U x +G(U). By eliminating the dependent variables U t and V t from (2.27) the following is obtained: ξ U U x F ξg +V x Fξ V +Fξ x ϕ t V x ϕ U Gϕ V Fϕ V U x = 0, ξ t +Gξ V +U x Fξ V U x ϕ U +ξ U V x ϕ V V x ϕ x = 0. (2.28) Finally, the following determining equations are obtained by letting the coefficients of U x, V x and the remaining terms equal to zero: ϕ V ξ U = 0, ϕ U F(U)ξ V = 0, ϕ x ξ t G(U)ξ V = 0, F(U)ξ x ϕ t G(U)ξ U G (U)ξ = 0, (2.29) with x, t, U, V as independent variables and ξ, ϕ dependent variables. System (2.29) can be solved for the multipliers ξ and ϕ where different cases may rise, Bluman, Cheviakov and Anco [8] and Bluman and Temuerchaolu [23]. After finding the multipliers the fluxes X and densities T are computed by using the following formula U V x X = ξ(x,t,s,b)f(s)ds ϕ(x,t,u,s)ds G(a) ξ(x,t,a,b)ds, a b T = U a ϕ(x,t,s,b)ds+ V b ξ(x, t, U, s)ds, (2.30) 27

39 where the constants a and b are chosen so that the integrals are not singular. After the fluxes and densities are found potential systems are constructed by using (2.17) Initial and Boundary Value Problems Most of the biological, and physical problems require the PDE to be solved subject to suitable initial and boundary conditions (IBVPs). The general procedure of applying the Lie symmetry method to study IBVPs of PDEs requires (Anco and Bluman [7]) the determination of a one parameter Lie group of transformations that leaves the problem invariant, and to use these transformations to construct the invariant solution or to obtain similarity reduction. This may be summarised by the following: 1: Determining the infinitesimals of the governing PDE. 2: Using the most general form of the infinitesimals found in the first step and finding the conditions under which it leaves the boundaries invariant. 3: Finding the restrictions on the infinitesimals that are imposed due to invariance of boundary conditions on the boundary. 4: All of the above steps will determine the symmetry that leaves the IBVP invariant. The obtained symmetry is then used to find the invariant solution of the IBVP Definitions of Some of the Main Terminology of Lie Symmetry Lie groups: Is a group which, in addition to the group properties, carries the structure of a differentiable manifold. More precisely, it is required that a Lie group 28

40 G be C manifold endowed with a group structure in which multiplication and the inversion are C operations, (Anco and Bluman [7] and Baumann [13]). Lie algebra, Baumann [13]: Consider a finite dimensional vector space V over a field K of real or complex numbers. The vector spave V is called a Lie algebra over K if there is a rule of composition ( v, w) [ v, w] in V which satisfies the following axioms: 1. Antisymmetry: [ v, w] = [ w, v] for all v, w V. 2. Linearity: [α v+β w, u] = α[ v, u]+β[ w, u]forallα,β K and u, v, w V. 3. Jacobi identity: [ v,[ w, u]+[ w,[ u, v]+[ u,[ v, w] = 0 for all v, u, w V. The operator [, ] is the multiplication relation of the algebra and is known as Lie product or Lie bracket. Group of transformations: Anco and Bluman [7] used the following definition of a group of transformations: Let x = (x 1,x 2,...,x n ) lie in region D R n. The set of transformations x = X(x;ε) (2.31) defined for each x in D and parameter ε in set S R, with ϕ(ε,δ) defining a law of composition of parameters ε and δ in S, forms a one-parameter group of transformations on D if the following holds: (i) For each ε in S the transformations are one-to-one onto D. Hence, x lies in D. (ii) S with the law of composition ϕ forms a group G. 29

41 (iii) For each x in D, x = x when ε = ε 0 corresponds to the identity e, i.e., X(x;ε 0 ) = x. (iv) If x = X(x;ε), x = X(x ;δ) then x = X(x;ϕ(ε,δ)). One-parameter Lie group of transformations, Anco and Bluman [7]: A oneparameter group of transformations defines a one-parameter Lie group of transformations if, in addition to satisfying axioms (i)-(iv) of the above definition, the following hold: (v) ε is a continuous parameter, i.e., S is an interval in R. Without loss of generality, ε = 0 corresponds to the identity element e. (vi)xisinfinitelydifferentiablewithrespecttoxind andananalyticfunction of ε in S. (vii) ϕ(ε,δ) is an analytic function of ε and δ, ε S, δ S. Prolongation: For a single PDE involving a dependent variable u and two independent variables x and t the prolongation is defined by, (Anco and Bluman [7], Bluman and Kumei [21], Hill [43], Lie [52] and Olver [58]) x 1 = x 1 (x, t, u, ε) = x+εξ(x, t, u)+o(ε 2 ) = e εx x, t 1 = t 1 (x, t, u, ε) = t+εη(x, t, u)+o(ε 2 ) = e εx t, u 1 = u 1 (x, t, u, ε) = u+εζ(x, t, u)+o(ε 2 ) = e εx u = U(x, t, u, ε), where ξ and η are the infinitesimals of the transformation for the independent variables, ζ is the infinitesimal for the dependent variable and X is the differential operator defined by X = ξ(x,t,u) x +η(x,t,u) t +ζ(x,t,u) u. 30

42 Infinitesimals generators, Anco and Bluman [7]: The infinitesimal generator of the one-parameter Lie group of transformations X = X(x,ε) is the operator X = X(x) = ξ(x). = n i=1 ξ i (x) x i, where is the gradient operator ( ) =,,..., x 1 x 2 x n For any differentiable function F(x) = F(x 1,x 2,...,x n ), one has XF(x) = ξ(x). F(x) = n i=1 ξ i (x) F(x) x i. Symmetry: A symmetry of a system of differential equations is a transformation that maps any solution to another solution of the system, Anco and Bluman [7] Invariant solutions, (Anco and Bluman [7], Bluman and Kumei [21], Hill [43], Baumann [13], Lie [52] and Olver [58]): u = Θ(x) is an invariant solution of PDE F(x,u, u, u 2,..., u k ) = 0 resulting from its admitted point symmetry with the infinitesimal generator if and only if: X = ξ i (x,u) x i +η(x,u) u (i) u = Θ(x) is an invariant surface of infinitesimal generator; and (ii) u = Θ(x) solves the PDE. 31

43 2.4 Conclusion A broad historical overview of the predator-prey models of interaction from Lotka- Volterra onwards was given in the first section of this chapter. This was done to give a good understanding of the field and to present some of the models that will be used in this research. In section 2.3, the main symmetry methods including the classical, nonclassical and potential symmetry approaches were defined. Before the symmetry methods are applied to some predator-prey models from the literature review, they will be applied in the next chapter to the simple Burgers equation in order to develop a good grasp of the processes involved in each methods. 32

44 Chapter 3 PDE Solution Methods: Worked Examples 3.1 Introduction In this chapter symmetry methods (classical, nonclassical and potential symmetries) are applied in detail to the following basic form of Burgers equation (Arrigo and Hickling [11] and Biler et al [14]): u t +2uu x u xx = 0 (3.1) to provide an overview of the methods used later in this thesis. The Mathematica package MathLie by Baumann [13] is used in this chapter to produce the determining equations for the infinitesimals ξ, η and ζ. It is also used to solve the obtained linear (for the case of the classical approach) and non-linear (for the nonclassical approach) determining equations for ξ, η and ζ. 33

45 3.2 Classical Approach Burgers Equation Equation (3.1) has u as the dependent variable and x and t as the independent variables. Using (2.32) with (3.1) then eliminating u t coefficients of ϵ to zero the following prolongation of (3.1) is obtained: by (3.1) and equating the 2ζu x +u 3 x( 2uη uu +ξ uu ) u xx η t +ζ t +2u xt η x 2uu xx η x +2u xx ξ x +2uζ x +u 2 x(u xx η uu ζ uu 4uη xu +2ξ xu ) +u xx η xx +u x (2u xt η u ξ t +2u xx ξ u +2u xx η xu 2ζ xu +2u(η t +2uη x ξ x η xx )+ξ xx ) ζ xx = 0. (3.2) In order to obtain the determining equations for ξ, η and ζ the coefficients of the derivatives of u are equated to zero. After simplification and elimination of repeated coefficients, the following system is obtained: ξ u = 0, η u = 0, ζ uu = 0, η x = 0, 2ζ ξ t 2uξ x +2uη t +ξ xx 2ζ xu = 0, ζ t +2uζ x ζ xx = 0, 2ξ x η t = 0. (3.3) The infinitesimals ξ, η and ζ need to satisfy (3.3). In order to solve (3.3) for the infinitesimals ξ, η and ζ the package MathLie may be used. The following command: Infinitesimals[{D[u[x,t],t]+2*u[x,t]*D[u[x,t],x] -D[u[x,t],x,x]},{u},{x,t}] gives 34