The importance of the nonlinear systems due to their common occurrence in the study of physical phenomena and also the various limitations posed by the linear systems theory have been the prime reasons for the studies of nonlinear models as such. The physical situations in which nonlinear equations arise tend to be highly idealized with the assumption of constant coefficients. Due to this, much attention has been paid on study of nonlinear equations with variable coefficients. This book deals with nonlinear partial differential equations with variable coefficients representing some interesting physical systems viz. coupled KdV system, generalized Hirota-Satsuma KdV, variant Boussinesq, modified Boussinesq and a family of non-evolution equations, from the view point of their underlying Lie point symmetries and to obtain their exact solutions. Since the nonlinear systems with variable coefficients are very difficult to handle, hence this book shall provide freedom for the researchers to obtain the symmetries and exact solutions for variable coefficients nonlinear systems and further to simulate the desired physical situations.

This book deals with applications of symmetry groups to solve physically important Einstein field equations, which are-the nondiagonal Einstein-Rosen metrics, the Einstein-Maxwell equations, Einstein-Maxwell equations for the magnetostatic fields, the Einstein-Maxwell equations for non-static Einstein and Rosen metrics, Einstein Vacuum equations for axially symmetric gravitational fields. To solve these highly nonlinear systems of partial differential equations (PDEs), Lie Classical method, symmetry reduction method and (G'/G)-expansion method are utilized. Symmetries are derived to reduce these systems of PDEs to ODEs and some exact solutions are obtained. Some of these solutions are also represented graphically.

This book deals with wide range of applications of continuous symmetry groups to some physically important systems which are: the coupled Klein-Gordon-Schrodinger equation with its generalized form, the Dullin-Gottwald-Holm Equation, the Generalized Bretherton equation with variable coefficients. For all the three different equations, we have derived some special type of solutions including traveling wave solutions, periodic solutions, kink wave solutions, solitons etc. and we have plotted some figures also to see the propogation and asymptotic behaviour of all types of waves.

General relativity is a physical theory which nowadays plays a key role in astrophysics and in physics and in this way it is important for a number of ambitious experiments and space missions. Einstein equations are central piece of general relativity. Einstein equations are expressed in terms of coupled system of highly nonlinear partial differential equations describing the matter content of space-time. The present work is to give an exposition of parts of the theory of partial differential equations that are needed in this subject and to represent exact solutions to Einstein equations. This book deals with various system of non linear partial differential equations corresponding to the Einstein equations for non diagonal Einstein-Rosen Metrics, Cylindrically Symmetric Null Fields, Vacuum Field Equations etc. from the view point of underlying symmetries and then to obtain their some new explicit exact solutions by using symmetry techniques like Lie symmetry analysis, symmetry reduction etc. These exact solutions play a significant for understanding of various phenomenons and are utilized for checking validity of numerical and approximation techniques and programs.

Keeping in view the rich treasure and wide applicability of nonlinear equations in almost every field, we have in this book carried out the application of Lie group analysis for obtaining exact solutions to nonlinear partial differential equations. In particular, this book is devoted to a wide range of applications of continuous symmetry groups to two physically important systems i.e. the (2+1)-dimensional Calogero Degasperis equation with its variable coefficients form and the (2+1)-dimensional potential Kadomstev Petviashvili equation along its generalized form. In recent years, much attention has also been paid to equations with variable coefficients as the physical situations in which nonlinear systems arise tend to be highly idealized due to assumption of constant coefficients. This has led us to undertake the study of equations with variable coefficients and to derive the admissible forms of the coefficients along with their exact solutions. The efforts are thus concentrated on finding the symmetries, reductions and exact solutions of certain nonlinear equations by using various methods.

This book is an outgrowth of our results on the existence and stability of solutions to nonlinear dynamical systems, stochastic systems, and impulsive systems over the last five years. In particular, we present the Razumikhin-type exponential stability criteria for impulsive stochastic functional differential systems, the stability analysis of neutral stochastic delay differential equations by a generalization of Banachs contraction principle and the globally asymptotical stability in the mean square for stochastic neural networks with time-varying delays and fixed moments of impulsive effect. Also, we discuss oscillation criteria based on a new weighted function for linear matrix Hamiltonian systems and the existences of the positive solutions or nontrivial solutions of nonlinear differential equations.

Vibrations and dynamic chaos are undesired phenomenon in structures as they cause the 4D. They are: disturbance, discomfort, damage and destruction of the system or the structure. For these reasons, money, time and effort are spent to eliminate or control vibrations,noise and chaos or to minimize them. The main object of this thesis is the investigation of the behavior two systems. They are simple and spring pendulums. A tuned absorber (passive control), in the transverse direction and/or the longitudinal one is connected to both systems to reduce the oscillations. Negative velocity feedback or its quadratic or cubic value is applied to the systems (active control). Also active control is applied to the systems via negative acceleration feedback or negative angular displacement or its quadratic or cubic value. Multiple scale is applied to determine approximate closed form solutions for the differential equations describing the systems. Both frequency response equations and the phase plane technique are applied to study systems stability. Optimum working conditions of both systems are extracted when applying both passive and active control, to be used in the design of such systems.

Exact solutions to nonlinear evolution equations (NEEs) play an important role in nonlinear physical science, since the characteristics of these solutions may well simulate real-life physical phenomena. One of the benefits of finding new exact solutions to such nonlinear partial differential equations (PDEs) is to give a better understanding on the various characteristics of the solutions. The main task of this work is to show that our proposed methods, improved tanh and sech methods, are very efficient in solving various types of NEEs and PDEs including special equations than using classical tanh and sech methods. This efficiency is because of their rich with the multiple traveling wave solutions than classical tanh and sech methods. From the obtained results, we can not only recover the previous solutions obtained by some authors but also obtain some new and more general solitary wave, singular solitary wave and periodic solutions. Illustrating the theory of nonlinear transmission lines (NLTLs), showing the ability of NLTL to generate solitons and solving the model equation of NLTL in presence of loss are other tasks of this book.

The Boundary value/periodic problems for the nonlinear equation (or, more generally, second order nonlinear ODEs) have been the focus of nonlinear analysis study for a long time. The goal of this book is to show how the equivariant degree theory can be used for the systematic study of multiple solutions to several (symmetric) generalizations of BVP and for the classification of symmetric properties of these solutions. There are several classical methods of nonlinear analysis used to solve the BVP. However, their application encounters serious difficulties if: the group of symmetries is large, the dimension of the problem is high, and multiplicities of eigenvalues of linearizations are large, etc. In this book, we: (i) set up the abstract functional analysis framework for studying symmetric properties of multiple solutions to symmetric generalizations of the BV problem via the equivariant degree approach; (ii) describe wide classes of second order BVPs admitting dihedral symmetries to which the abstract theory can be effectively applied; (iii) and apply the obtained results to several classes of implicit second order symmetric differential equations.

In this book, we make an extension to the unified method that unifies all the known methods in the literature for finding the exact solutions of scalar or vector nonlinear PDE's with constant coefficients in the nonlinear sciences. The extended unified method unable us to investigate the effects of the inhomogeneity of the diffusion, diffraction dispersion super-diffusion of the medium trough considering the coefficients space-dependent. On the other hand, some problems have been studied when these coefficients are taken as time-dependent. The main objectives of the extended unified method are; (a) Constructing the necessary conditions for the existence of solutions to evolution equations. (b) Whenever the solutions exist, this method suggests a new classification to the solution structures namely; the polynomial solutions, the rational solutions and the polynomial-rational solutions. In each type, we mean that the obtained equations are accomplished by a set of auxiliary equation whose solution gives rise to an auxiliary function.

The study of nonlinear problems is of crucial in the areas of Applied Mathematics, Physics and Engineering, as well as other disciplines. The differential equations are linear or nonlinear, autonomous or non-autonomous. Practically, numerous differential equations involving physical phenomena are nonlinear. Methods of solutions of linear differential equations are comparatively easy and highly developed. Whereas, very little of a general character is known about nonlinear equations. An important approach to the study of such nonlinear oscillations is the small parameter expansion of Krylov-Bogoliubov-Mitropolski (KBM). This book is concerned the critically damped nonlinear systems by use of the KBM method. The presentation of this book is easy and intelligible by the beginners. Researchers who thoroughly cover the book will be well prepared to make important contributions to analyze nonlinear systems. It will be helpful in the area of mechanics, physics, engineering etc. The book contains a wide bibliography.

Difference equations are very useful in daily life. There are lot of applications of difference equations in business, statistics, economics, computer programming and numerical solutions of differential equations. In mathematics, there are two reasons for using the difference equations. Firstly, difference equations play an important role in the designing of mathematical models which are used in mechanics and mathematical physics. Such kind of models relay on symmetries. The existence of exact analytical solution of the difference equation and their conservation laws are related to their continuous symmetries. Secondly, in the theory of differential equation (D.E), system of D.E. can be replaced by using difference equations and meshes. In this book, a complete symmetry analysis for the multidimensional discrete heat equation is presented. For this, generalized prolongations are reported for the considered equation. Furthermore, Lie point generators are computed for n=2, 3 and then generalized for the arbitrary value of n. A relationship between the number of the symmetries and the value of n is given at last.

In this book, we study the validity of the maximum principle (one of the most useful and best known tools employed in the study of partial differential equations) for some nonlinear elliptic systems, with variable coefficients, involving the weighted p-Laplcaian operators on bounded and unbounded domains. Also, using a different methods like the nonlinear theory of monotone operators, subsuper solutions method and an approximation (perturbation) method, we study the existence of positive weak solutions for some nonlinear elliptic systems, with variable coefficients, involving the weighted p-Laplcaian operators on bounded and unbounded domains.

Many mathematical models of natural phenomena admit symmetries that are related to some physical or geometric regularities. In this way, we arrive at the following paradigmatic question: what is the link between symmetries of a differential system and symmetries of its solutions? In this book, we introduce an abstract functional analysis framework and equivariant degree methodology for studying implicit boundary value/Hopf bifurcation problems in the presence of group symmetries via the transition to the equivariant multivalued differential inclusions. The methodology is applied to three types of problems related to symmetric implicit differential systems and the abstract theoretical results are supported by concrete examples for which full equivariant analysis is performed. Since the symmetry groups are allowed to be of a considerable size, the conducted analysis is only possible with the usage of special computer programs created by us. These programs, based on the existing GAP routines, provide a platform for effective algebraic computations of Burnside ring and Euler ring. Thus this book should be especially useful to researchers in symmetric systems and symbolic computing.

In this book, solutions of nonlinear parabolic differential equations are investigated. More specifically, existence and non-existence theorems of solutions are presented. The first part involves the related results for the Cauchy problem for a set of nonlinear parabolic differential equations with critical exponent. Then the necessary and sufficient conditions are determined for the non-existence results of solutions for the corresponding Dirichlet and mixed boundary value problems. The proofs make use of the dilation and comparison arguments. Moreover, to generalize these results, proofs are given in n-dimension and for general type of domains. Finally, numerical simulations are also provided to verify the theoretical analysis

## symmetries and exact solutions for nonlinear systems в наличии / купить интернет-магазине

## Symmetries and Exact Solutions for Nonlinear Systems

The importance of the nonlinear systems due to their common occurrence in the study of physical phenomena and also the various limitations posed by the linear systems theory have been the prime reasons for the studies of nonlinear models as such. The physical situations in which nonlinear equations arise tend to be highly idealized with the assumption of constant coefficients. Due to this, much attention has been paid on study of nonlinear equations with variable coefficients. This book deals with nonlinear partial differential equations with variable coefficients representing some interesting physical systems viz. coupled KdV system, generalized Hirota-Satsuma KdV, variant Boussinesq, modified Boussinesq and a family of non-evolution equations, from the view point of their underlying Lie point symmetries and to obtain their exact solutions. Since the nonlinear systems with variable coefficients are very difficult to handle, hence this book shall provide freedom for the researchers to obtain the symmetries and exact solutions for variable coefficients nonlinear systems and further to simulate the desired physical situations.

## Symmetries and Exact Solutions OF Einstein Field Equations

This book deals with applications of symmetry groups to solve physically important Einstein field equations, which are-the nondiagonal Einstein-Rosen metrics, the Einstein-Maxwell equations, Einstein-Maxwell equations for the magnetostatic fields, the Einstein-Maxwell equations for non-static Einstein and Rosen metrics, Einstein Vacuum equations for axially symmetric gravitational fields. To solve these highly nonlinear systems of partial differential equations (PDEs), Lie Classical method, symmetry reduction method and (G'/G)-expansion method are utilized. Symmetries are derived to reduce these systems of PDEs to ODEs and some exact solutions are obtained. Some of these solutions are also represented graphically.

## Symmetries and Invariant Solutions to Some Nonlinear Systems

This book deals with wide range of applications of continuous symmetry groups to some physically important systems which are: the coupled Klein-Gordon-Schrodinger equation with its generalized form, the Dullin-Gottwald-Holm Equation, the Generalized Bretherton equation with variable coefficients. For all the three different equations, we have derived some special type of solutions including traveling wave solutions, periodic solutions, kink wave solutions, solitons etc. and we have plotted some figures also to see the propogation and asymptotic behaviour of all types of waves.

## Applications of Symmetries for Solutions of Einstein Equations

General relativity is a physical theory which nowadays plays a key role in astrophysics and in physics and in this way it is important for a number of ambitious experiments and space missions. Einstein equations are central piece of general relativity. Einstein equations are expressed in terms of coupled system of highly nonlinear partial differential equations describing the matter content of space-time. The present work is to give an exposition of parts of the theory of partial differential equations that are needed in this subject and to represent exact solutions to Einstein equations. This book deals with various system of non linear partial differential equations corresponding to the Einstein equations for non diagonal Einstein-Rosen Metrics, Cylindrically Symmetric Null Fields, Vacuum Field Equations etc. from the view point of underlying symmetries and then to obtain their some new explicit exact solutions by using symmetry techniques like Lie symmetry analysis, symmetry reduction etc. These exact solutions play a significant for understanding of various phenomenons and are utilized for checking validity of numerical and approximation techniques and programs.

## Applications of Lie Group to Some Nonlinear Equations

Keeping in view the rich treasure and wide applicability of nonlinear equations in almost every field, we have in this book carried out the application of Lie group analysis for obtaining exact solutions to nonlinear partial differential equations. In particular, this book is devoted to a wide range of applications of continuous symmetry groups to two physically important systems i.e. the (2+1)-dimensional Calogero Degasperis equation with its variable coefficients form and the (2+1)-dimensional potential Kadomstev Petviashvili equation along its generalized form. In recent years, much attention has also been paid to equations with variable coefficients as the physical situations in which nonlinear systems arise tend to be highly idealized due to assumption of constant coefficients. This has led us to undertake the study of equations with variable coefficients and to derive the admissible forms of the coefficients along with their exact solutions. The efforts are thus concentrated on finding the symmetries, reductions and exact solutions of certain nonlinear equations by using various methods.

## Existence and stability of solutions to nonlinear dynamical systems

This book is an outgrowth of our results on the existence and stability of solutions to nonlinear dynamical systems, stochastic systems, and impulsive systems over the last five years. In particular, we present the Razumikhin-type exponential stability criteria for impulsive stochastic functional differential systems, the stability analysis of neutral stochastic delay differential equations by a generalization of Banachs contraction principle and the globally asymptotical stability in the mean square for stochastic neural networks with time-varying delays and fixed moments of impulsive effect. Also, we discuss oscillation criteria based on a new weighted function for linear matrix Hamiltonian systems and the existences of the positive solutions or nontrivial solutions of nonlinear differential equations.

## Improving Solutions of Nonlinear Differential Equations Using Control

Vibrations and dynamic chaos are undesired phenomenon in structures as they cause the 4D. They are: disturbance, discomfort, damage and destruction of the system or the structure. For these reasons, money, time and effort are spent to eliminate or control vibrations,noise and chaos or to minimize them. The main object of this thesis is the investigation of the behavior two systems. They are simple and spring pendulums. A tuned absorber (passive control), in the transverse direction and/or the longitudinal one is connected to both systems to reduce the oscillations. Negative velocity feedback or its quadratic or cubic value is applied to the systems (active control). Also active control is applied to the systems via negative acceleration feedback or negative angular displacement or its quadratic or cubic value. Multiple scale is applied to determine approximate closed form solutions for the differential equations describing the systems. Both frequency response equations and the phase plane technique are applied to study systems stability. Optimum working conditions of both systems are extracted when applying both passive and active control, to be used in the design of such systems.

## Improved Tanh and Sech Methods for Obtaining New Exact Solutions

Exact solutions to nonlinear evolution equations (NEEs) play an important role in nonlinear physical science, since the characteristics of these solutions may well simulate real-life physical phenomena. One of the benefits of finding new exact solutions to such nonlinear partial differential equations (PDEs) is to give a better understanding on the various characteristics of the solutions. The main task of this work is to show that our proposed methods, improved tanh and sech methods, are very efficient in solving various types of NEEs and PDEs including special equations than using classical tanh and sech methods. This efficiency is because of their rich with the multiple traveling wave solutions than classical tanh and sech methods. From the obtained results, we can not only recover the previous solutions obtained by some authors but also obtain some new and more general solitary wave, singular solitary wave and periodic solutions. Illustrating the theory of nonlinear transmission lines (NLTLs), showing the ability of NLTL to generate solitons and solving the model equation of NLTL in presence of loss are other tasks of this book.

## Equivariant Degree with Symmetric Nonlinear Boundary Value Problems

The Boundary value/periodic problems for the nonlinear equation (or, more generally, second order nonlinear ODEs) have been the focus of nonlinear analysis study for a long time. The goal of this book is to show how the equivariant degree theory can be used for the systematic study of multiple solutions to several (symmetric) generalizations of BVP and for the classification of symmetric properties of these solutions. There are several classical methods of nonlinear analysis used to solve the BVP. However, their application encounters serious difficulties if: the group of symmetries is large, the dimension of the problem is high, and multiplicities of eigenvalues of linearizations are large, etc. In this book, we: (i) set up the abstract functional analysis framework for studying symmetric properties of multiple solutions to symmetric generalizations of the BV problem via the equivariant degree approach; (ii) describe wide classes of second order BVPs admitting dihedral symmetries to which the abstract theory can be effectively applied; (iii) and apply the obtained results to several classes of implicit second order symmetric differential equations.

## Exact Solutions of Evolution Equations with Variable Coefficients

In this book, we make an extension to the unified method that unifies all the known methods in the literature for finding the exact solutions of scalar or vector nonlinear PDE's with constant coefficients in the nonlinear sciences. The extended unified method unable us to investigate the effects of the inhomogeneity of the diffusion, diffraction dispersion super-diffusion of the medium trough considering the coefficients space-dependent. On the other hand, some problems have been studied when these coefficients are taken as time-dependent. The main objectives of the extended unified method are; (a) Constructing the necessary conditions for the existence of solutions to evolution equations. (b) Whenever the solutions exist, this method suggests a new classification to the solution structures namely; the polynomial solutions, the rational solutions and the polynomial-rational solutions. In each type, we mean that the obtained equations are accomplished by a set of auxiliary equation whose solution gives rise to an auxiliary function.

## On Some Asymptotic Solutions of Critically Damped Nonlinear Systems

The study of nonlinear problems is of crucial in the areas of Applied Mathematics, Physics and Engineering, as well as other disciplines. The differential equations are linear or nonlinear, autonomous or non-autonomous. Practically, numerous differential equations involving physical phenomena are nonlinear. Methods of solutions of linear differential equations are comparatively easy and highly developed. Whereas, very little of a general character is known about nonlinear equations. An important approach to the study of such nonlinear oscillations is the small parameter expansion of Krylov-Bogoliubov-Mitropolski (KBM). This book is concerned the critically damped nonlinear systems by use of the KBM method. The presentation of this book is easy and intelligible by the beginners. Researchers who thoroughly cover the book will be well prepared to make important contributions to analyze nonlinear systems. It will be helpful in the area of mechanics, physics, engineering etc. The book contains a wide bibliography.

## Discrete Symmetries for the higher dimensional heat equation

Difference equations are very useful in daily life. There are lot of applications of difference equations in business, statistics, economics, computer programming and numerical solutions of differential equations. In mathematics, there are two reasons for using the difference equations. Firstly, difference equations play an important role in the designing of mathematical models which are used in mechanics and mathematical physics. Such kind of models relay on symmetries. The existence of exact analytical solution of the difference equation and their conservation laws are related to their continuous symmetries. Secondly, in the theory of differential equation (D.E), system of D.E. can be replaced by using difference equations and meshes. In this book, a complete symmetry analysis for the multidimensional discrete heat equation is presented. For this, generalized prolongations are reported for the considered equation. Furthermore, Lie point generators are computed for n=2, 3 and then generalized for the arbitrary value of n. A relationship between the number of the symmetries and the value of n is given at last.

## Maximum Principle and Existence of Weak Solutions

In this book, we study the validity of the maximum principle (one of the most useful and best known tools employed in the study of partial differential equations) for some nonlinear elliptic systems, with variable coefficients, involving the weighted p-Laplcaian operators on bounded and unbounded domains. Also, using a different methods like the nonlinear theory of monotone operators, subsuper solutions method and an approximation (perturbation) method, we study the existence of positive weak solutions for some nonlinear elliptic systems, with variable coefficients, involving the weighted p-Laplcaian operators on bounded and unbounded domains.

## Symmetric Systems of Implicit Functional Differential Equations

Many mathematical models of natural phenomena admit symmetries that are related to some physical or geometric regularities. In this way, we arrive at the following paradigmatic question: what is the link between symmetries of a differential system and symmetries of its solutions? In this book, we introduce an abstract functional analysis framework and equivariant degree methodology for studying implicit boundary value/Hopf bifurcation problems in the presence of group symmetries via the transition to the equivariant multivalued differential inclusions. The methodology is applied to three types of problems related to symmetric implicit differential systems and the abstract theoretical results are supported by concrete examples for which full equivariant analysis is performed. Since the symmetry groups are allowed to be of a considerable size, the conducted analysis is only possible with the usage of special computer programs created by us. These programs, based on the existing GAP routines, provide a platform for effective algebraic computations of Burnside ring and Euler ring. Thus this book should be especially useful to researchers in symmetric systems and symbolic computing.

## Solutions of Nonlinear Parabolic Differential Equations

In this book, solutions of nonlinear parabolic differential equations are investigated. More specifically, existence and non-existence theorems of solutions are presented. The first part involves the related results for the Cauchy problem for a set of nonlinear parabolic differential equations with critical exponent. Then the necessary and sufficient conditions are determined for the non-existence results of solutions for the corresponding Dirichlet and mixed boundary value problems. The proofs make use of the dilation and comparison arguments. Moreover, to generalize these results, proofs are given in n-dimension and for general type of domains. Finally, numerical simulations are also provided to verify the theoretical analysis