Method For Finding Exact Solutions Research Articles

Solving nonlinear partial differential equations using a novel Cham method

Nonlinear partial differential equations (NLPDEs) have been of great interest in recent years due to their numerous applications. While there are several methods for finding exact solutions to various NLPDEs, more solutions are still required. This paper first proposes the Cham method, a new method for solving NLPDEs that can generate eight families of solutions. The method is then successfully employed to solve the (2+1)-dimensional Bogoyavlenskii's breaking soliton equations. The dynamic behaviour of these equations and the bifurcation of traveling waves are also discussed. Finally, we graphically depict some solutions corresponding to some discovered solutions with different coefficient values. The Cham method is general, effective, and adaptable to many NLPDEs.

Solitary waves of the complex Ginzburg-Landau equation with anti-cubic nonlinearity

The complex Ginzburg-Landau equation with anti-cubic nonlinearity is considered. Using a modification of the simplest equation method for finding exact solutions of nonlinear partial differential equations we get new solitary waves with arbitrary amplitude and speed of this special case of the complex Ginzburg -Landau equation. The first integral of nonlinear ordinary differential equation is found. The general solution of equation is presented taking into account the first integral of equation. Conserved density of the solitary wave solution is calculated.

An analytical method for finding exact solutions of a nonlinear partial differential equation arising in electrical engineering

In this investigation, we aim to investigate the novel exact solutions of nonlinear partial differential equations (NLPDEs) arising in electrical engineering via the -expansion method. New acquired solutions are kink, particular kink, bright, dark, periodic combined-dark bright and combined-dark singular solitons, and hyperbolic functions solutions. The achieved distinct types of solitons solutions contain critical applications in engineering and physics. Numerous novel structures (3D, contour, and density plots) are also designed by taking the appropriate values of involved parameters. These solutions express the wave show of the governing models, actually.

Hidden Time-Reversal Symmetry, Quantum Detailed Balance and Exact Solutions of Driven-Dissipative Quantum Systems

Driven-dissipative quantum systems generically do not satisfy simple notions of detailed balance based on the time symmetry of correlation functions. We show that such systems can nonetheless exhibit a hidden time-reversal symmetry which most directly manifests itself in a doubled version of the original system prepared in an appropriate entangled thermofield double state. This hidden time-reversal symmetry has a direct operational utility: it provides a general method for finding exact solutions of non-trivial steady states. Special cases of this approach include the coherent quantum absorber and complex-$P$ function methods from quantum optics. We also show that hidden TRS has observable consequences even in single-system experiments, and can be broken by the non-trivial combination of nonlinearity, thermal fluctuations, and driving. To illustrate our ideas, we analyze concrete examples of driven qubits and nonlinear cavities. These systems exhibit hidden time-reversal symmetry but not conventional detailed balance.

Traveling wave solutions of Benny Luke equation via the enhanced (G'/G)-expansion method

In this article, we execute the enhanced (G'/G)-expansion method to search for new and further general closed-form wave solutions to the nonlinear partial differential equation, namely the Benny Luke equation. This method is one of the powerful techniques that come into view in recent time for establishing more exact wave solutions to nonlinear partial differential equations. We have achieved some new exact solutions including soliton and periodic wave solutions with arbitrary parameters and the solutions are expressed in terms of hyperbolic and trigonometric functions. The efficiency of this method for finding exact solutions has been established. It is shown that the enhanced (G'/G)-expansion method is direct, effective and can be used for many other nonlinear partial differential equations in mathematical physics and engineering.

Construction of Dirac spinors for electron vortex beams in background electromagnetic fields

Exact solutions of the Dirac equation, a system of four partial differential equations, are rare. The vast majority of them are for highly symmetric stationary systems. Moreover, only a handful of solutions for time dependent dynamics exists. Given the growing number of applications of high energy electron beams interacting with a variety of quantum systems in laser fields, novel methods for finding exact solutions to the Dirac equation are called for. We present a method for building up solutions to the Dirac equation employing a recently introduced approach for the description of spinorial fields and their driving electromagnetic fields in terms of geometric algebras. We illustrate the method by developing several stationary as well as non-stationary solutions of the Dirac equation with well defined orbital angular momentum along the electron's propagation direction. The first set of solutions describe free electron beams in terms of Bessel functions as well as stationary solutions for both a homogeneous and an inhomogeneous magnetic field. The second set of solutions are new and involve a plane electromagnetic wave combined with a generally inhomogeneous longitudinal magnetic field. Moreover, the developed technique allows us to derive general physical properties of the dynamics in such field configurations, as well as provides physical predictions on the self-consistent electromagnetic fields induced by the dynamics.

Solitary waves of the generalized Sasa–Satsuma equation with arbitrary refractive index

Traveling wave reduction of the generalized Sasa–Satsuma equation with arbitrary refractive index is considered. The system of equations corresponding to the real and imaginary parts is presented. The two compatibility conditions to this system equations are given. Methods for finding exact solutions of nonlinear differential equations with arbitrary powers of nonlinearity are presented and discussed. Periodic and solitary wave solutions of the generalized Sasa–Satsuma equation with arbitrary refractive index are found.

Exact solutions of space-time fractional (2+1)-dimensional breaking soliton equation

This paper suggests a direct algebraic method for finding exact solutions of the space-time fractional (2+1)-dimensional breaking soliton equation. The solution procedure is reduced to solve a large system of algebraic equations, which is then solved by Wu?s method.

On the Derivation of Nonclassical Symmetries of the Black–Scholes Equation via an Equivalence Transformation

The nonclassical symmetries method is a powerful extension of the classical symmetries method for finding exact solutions of differential equations. Through this method, one is able to arrive at new exact solutions of a given differential equation, i.e., solutions that are not obtainable directly as invariant solutions from classical symmetries of the equation. The challenge with the nonclassical symmetries method, however, is that governing equations for the admitted nonclassical symmetries are typically coupled and nonlinear and therefore difficult to solve. In instances where a given equation is related to a simpler one via an equivalent transformation, we propose that nonclassical symmetries of the given equation may be obtained by transforming nonclassical symmetries of the simpler equation using the equivalence transformation. This is what we illustrate in this paper. We construct four nontrivial nonclassical symmetries of the Black–Scholes equation by transforming nonclassical symmetries of the heat equation. For completeness, we also construct invariant solutions of the Black–Scholes equation associated with the determined nonclassical symmetries.

Exact solutions of a quantum system placed in a Kratzer potential and under a uniform magnetic field

We propose Whittaker function approach as a theoretical method for finding exact solutions of a quantum mechanical system placed in the Kratzer potential. We then show that the effect of an external uniform magnetic field on this system can be satisfactorily determined using variational method. By following the one-step treatment suggested in this study, we increase the reliability and the accuracy of the solutions of Schrodinger equation for a quantum mechanical system placed in potential energy and perturbed by a uniform magnetic field that proves to be useful in modelling physical phenomena. We find that the achieved numerical and analytical results agree very well with those already published and those calculated using the Numerov method.

Exactly Solvable Models for the First Vlasov Equation

Construction of the method for finding exact solutions of the first equation from the chain of Vlasov equations, formally similar to the continuity equation, is considered. The equation under investigation is written for the scalar function $$f$$ and the vector field $$\left\langle {\vec {v}} \right\rangle $$ . Depending on the formulation of the problem, the function $$f$$ can correspond to the density of probabilities, charge, mass, or the magnetic permeability of a magnetic material. The vector field $$\left\langle {\vec {v}} \right\rangle $$ can correspond to the probability flow, velocity field of a continuous medium, or magnetic field strength. Mathematically, the same equation is applicable for describing statistical, quantum, and classical systems. The exact solution obtained for one physical system can be mapped onto the exact solution for another system. Availability of exact solutions of model nonlinear systems is important for designing complex physical facilities, such as the SPD detector for the NICA project. These solutions are used as tests for writing a program code and can be encapsulated into finite-difference schemes to numerically solve boundary-value problems for nonlinear differential equations.

Construction of exact solutions of the problem of the motion of a fluid with a free boundary using infinite systems of differential equations

We consider the well-known hydrodynamics problem for the planar potential motion of an ideal incompressible fluid with a free boundary without capillarity and propose a method for finding exact solutions based on reducing the boundary-value problems to systems of ordinary differential equations. We obtain two examples of flows: a stationary motion of a heavy fluid over a smooth bottom and a nonstationary motion of a fluid that originally filled a wedge.

Exponential operator method for finding exact solutions of the propagator equation in the presence of a magnetic field

A general method of obtaining the exact analytical solutions of the propagator equations for charged particles in the presence of a constant magnetic field is developed. The method is applied to calculate the exact electron propagator in a strong magnetic field.

Hardening networks against strategic attackers using attack graph games

We consider the problem faced by a network administrator (defender) when deploying limited security resources to protect a network against a strategic attacker. To evaluate the effectiveness of a defense strategy, one must consider possible counterattacks that an attacker can choose. We use game theory to model the interaction between the defender and the attacker. Game theory provides relevant concepts and algorithms for computing optimal strategies in environments with multiple decision makers. To model the space of attacker’s possible actions, we use attack graphs, that compactly represent all known sequences of attacker’s action that may lead to successful attack for a given network. We demonstrate our approach on a specific type of defense actions, where the defender deploys deceptive hosts and services (honeypots) to detect and mitigate attacks.We assume the worst-case attacker who has a complete knowledge of the (typically randomized) defense strategy. We seek the optimal defense strategy against this attacker in the form of a Stackelberg equilibrium. Computing this solution exactly using standard techniques has limited scalability, so we investigate several approaches for increasing scalability to realistic problems. We introduce optimization methods for finding exact solutions for these games and then propose a variety of polynomial heuristic algorithms that scale to significantly larger games. We analyze the scalability and the quality of these heuristic solutions on realistic network topologies. We show that the strategies found by the heuristics are often near-optimal and that they outperform non-game-theoretic baselines. Finally, we show how attack graph games can be used to answer various research questions relevant to network security administrators.

The first integrals and exact solutions of a two-component Belousov–Zhabotinskii reaction system

The popular Belousov–Zhabotinskii (BZ) system of equations for description of a two component reaction is considered. The Painlevé test is applied to determine integrability of this system. It is shown that the system of equations is nonintegrable in general case. The parameter values of the mathematical model are found for the case when the system of equations passes the Painlevé test. Simplest solutions of the system of equations are presented. Additional conditions are specified when the general solutions of the system can be found. These general solutions are found using the new generalized method for finding exact solutions and the first integrals. Two first integrals of the Belousov–Zhabotinskii reaction system are given at additional conditions on parameter values for the mathematical model.

The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations

The approach by Painlevé to construct the general solutions of some nonlinear second-order ordinary differential equations is generalized for finding exact solutions of nonintegrable differential equations. We demonstrate that this approach generalizes some other well-known algorithms of finding exact solutions: the tanh-function method, the G′/G-expansion method, the Exp-function method and some variants of the simplest equation method. The method allows us to search for both solution in the form of solitary waves and periodic solution expressed through elliptic functions. We demonstrate possibilities of the method for finding exact solutions of nonintegrable ordinary differential equations which are obtained as reductions of the modified Korteweg–de Vries equation with source, the generalized Burgers equation, the generalized Fisher equation, the generalized Gardner equation and the generalized Kuramoto–Sivashinsky equation.

Symmetry analysis and some new exact solutions of Born–Infeld equation

This paper propounds the Lie group analysis method for finding exact solutions of Born–Infeld (BI) equation arising in nonlinear electrodynamics. We obtain generators of infinitesimal transformations, commutator table of Lie algebra, the complete geometric vector field, group symmetries and reduction equations. For the set of geometric vector field, we find an optimal system of the vector fields. Each element in this system helps to reduce the main equation into an ordinary differential equation, which provides analytical solution to the BI equation. We perform numerical simulation to obtain an appropriate visual appearance and dynamic behavior of the traced solutions. The nature of the solutions is investigated both analytically and physically through their evolutionary profile by considering appropriate choices of arbitrary constants.

Exact solutions of the equation for surface waves in a convecting fluid

A method for finding exact solutions and the first integrals is presented. The basic idea of the method is to use the value of the Fuchs index that appears in the Painlevé test to construct the auxiliary equation for finding the first integrals and exact solutions of nonlinear differential equations. It allows us to obtain the first integrals and new exact solutions of some nonlinear ordinary differential equations. The main feature of the method is that we do not assign a solution function at the beginning, we find this function during calculations. This approach is conceptually equivalent to the third step of the Painlevé test and sometimes allows us to change this step. Our approach generalizes a number of other methods for finding exact solutions of nonlinear differential equations. We demonstrate a method for finding the traveling wave solutions and the first integrals of the well-known nonlinear evolution equation for description of surface waves in a convecting liquid. The general solution of this equation at some conditions on parameters and new traveling wave solutions of the fourth-order equation are found.

On Exact Solutions and Perturbative Schemes in Higher Spin Theory

We review various methods for finding exact solutions of higher spin theory in four dimensions, and survey the known exact solutions of (non)minimal Vasiliev’s equations. These include instanton-like and black hole-like solutions in (A)dS and Kleinian spacetimes. A perturbative construction of solutions with the symmetries of a domain wall is also described. Furthermore, we review two proposed perturbative schemes: one based on perturbative treatment of the twistor space field equations followed by inverting Fronsdal kinetic terms using standard Green’s functions; and an alternative scheme based on solving the twistor space field equations exactly followed by introducing the spacetime dependence using perturbatively defined gauge functions. Motivated by the need to provide a higher spin invariant characterization of the exact solutions, aspects of a proposal for a geometric description of Vasiliev’s equation involving an infinite dimensional generalization of anti de Sitter space are revisited and improved.

Using biologically inspired computing to effectively improve prediction models

The complexity of certain problems causes that classical methods for finding exact solutions have some limitations. In this paper we propose an incremental heterogeneous ensemble model for time series prediction where biologically inspired algorithms offer a suitable alternative. Ensemble learning techniques are advantageously used for improving performance of various prediction methods. The quality of this kind of machine learning approaches depends on proper combination of used methods. The influence of each of the used method can change on the fly and is determined by proper choice of its weights. Finding optimal weights in prediction methods represents typical optimization problem with objective function reflecting error minimization, where biologically inspired algorithms can be used. In the proposed paper, we study several biologically inspired algorithms in the process of weights optimization. We investigate and compare ensembles using base models and ensembles op- timized by biologically inspired algorithms. We demonstrate that the ensemble learning prediction models optimized by biolog- ically inspired algorithms outperformed the base prediction methods. We present performance and accuracy results of proposed ensemble models that were evaluated on power load datasets with concept drifts.