Form Of Rational Function Research Articles

Novel exact solutions, bifurcation of nonlinear and supernonlinear traveling waves for M-fractional generalized reaction Duffing model and the density dependent M-fractional diffusion reaction equation

In the present work, we study the dynamical behavior of nonlinear traveling waves for the M-fractional generalized reaction Duffing model and density dependent M-fractional diffusion reaction equation. Novel exact solutions in the form of rational, hyperbolic and trigonometric functions are obtained using the new extended direct algebraic method. The obtained solutions are also verified for the aforesaid equations with the help of symbolic soft computations. Furthermore, some produced results are graphically illustrated using 3D surface graphs and 2D line plots, which provide useful information about the dynamical behavior. Moreover, we assurance that all the obtained solutions are accurate, efficient and an excellent contribution in the literature of solitary wave theory. Bifurcation behavior of some nonlinear traveling waves of the proposed equations is also studied with the help of bifurcation theory of planar dynamical systems through numerical simulation. It is observed that proposed equations supports nonlinear solitary waves, periodic waves and most important supernonlinear periodic waves.

On the Dynamic Solution of the Volterra Integral Equation in the Form of Rational Spline Functions

On the Dynamic Solution of the Volterra Integral Equation in the Form of Rational Spline Functions

New abundant analytic solutions for generalized KdV6 equation with time-dependent variable coefficients using Painlevé analysis and auto-Bäcklund transformation

This paper considers the generalized KdV6 equation with time-dependent variable coefficients. The integrability of the considered equation is being examined by the Painlevé analysis method. Further, an auto-Bäcklund transformation method has been adopted to obtain the analytic solutions. Using this technique, five novel families of analytic solutions in the form of rational, exponential, hyperbolic and trigonometric functions have been successfully found for the considered equation. New kink-antikink and periodic soliton solutions have been discovered using this method. The solutions are graphically depicted to show the physical significance of the problem under consideration.

Lyapunov function computation for autonomous systems with complex dynamic behavior

A computational approach is presented in this paper to construct local Lyapunov functions for autonomous dynamical systems with multiple isolated locally asymptotically stable (such as point-like, periodic, or strange) attractors. We consider systems of nonlinear ODEs, where the right-hand-side of the dynamic equations is given in the form of rational functions (i.e., fraction of polynomials). The Lyapunov function is searched in a parameterized quadratic form of rational terms of the state variables. The quadratic decomposition of the rational state-dependent inequalities is performed using the linear fractional transformation (LFT) and further algebraic/numeric simplification steps. Unlike the sum of squares (SOS) approach, the sufficient linear matrix inequality (LMI) conditions for the Lyapunov function are formulated only locally on a compact polytopic subset of the state space, which allows indefinite matrix solutions for the quadratic decomposition. The local solution is enforced using affine annihilators with matrix Lagrange multipliers. Alongside the typical Lyapunov conditions, further boundary LMI constraints are prescribed using Finsler’s lemma to ensure the required geometric properties of the Lyapunov function. The results are illustrated on four planar benchmark models having either multiple locally stable equlibria or a limit cycle, and on the Lorenz system.

The dynamical study of Biswas–Arshed equation via modified auxiliary equation method

The presented work deals with the investigation of Biswas–Arshed model which has great importance in the theory of solitons. The proficient approach, namely, modified auxiliary equation (MAE) method is practiced. This method provides results in the form of hyperbolic, trigonometric and rational functions. The dynamical behavior of obtained results is observed through graphs by assigning particular values to free parameters.

Painlevé analysis, auto-Bäcklund transformation and new exact solutions of [formula omitted] and [formula omitted]-dimensional extended Sakovich equation with time dependent variable coefficients in ocean physics

This article considers time-dependent variable coefficients (2+1) and (3+1)-dimensional extended Sakovich equation. Painlevé analysis and auto-Bäcklund transformation methods are used to examine both the considered equations. Painlevé analysis is appeared to test the integrability while an auto-Bäcklund transformation method is being presented to derive new analytic soliton solution families for both the considered equations. Two new family of exact analytical solutions are being obtained successfully for each of the considered equations. The soliton solutions in the form of rational and exponential functions are being depicted. The results are also expressed graphically to illustrate the potential and physical behaviour of both equations. Both the considered equations have applications in ocean wave theory as they depict new solitary wave soliton solutions by 3D and 2D graphs.

Explore Optical Solitary Wave Solutions of the kp Equation by Recent Approaches

The study of nonlinear evolution equations is a subject of active interest in different fields including physics, chemistry, and engineering. The exact solutions to nonlinear evolution equations provide insightful details and physical descriptions into many problems of interest that govern the real world. The Kadomtsev–Petviashvili (kp) equation, which has been widely used as a model to describe the nonlinear wave and the dynamics of soliton in the field of plasma physics and fluid dynamics, is discussed in this article in order to obtain solitary solutions and explore their physical properties. We obtain several new optical traveling wave solutions in the form of trigonometric, hyperbolic, and rational functions using two separate direct methods: the (w/g)-expansion approach and the Addendum to Kudryashov method (akm). The nonlinear partial differential equation (nlpde) is reduced into an ordinary differential equation (ode) via a wave transformation. The derived optical solutions are graphically illustrated using Maple 15 software for specific parameter values. The traveling wave solutions discovered in this work can be viewed as an example of solutions that can empower us with great flexibility in the systematic analysis and explanation of complex phenomena that arise in a variety of problems, including protein chemistry, fluid mechanics, plasma physics, optical fibers, and shallow water wave propagation.

On the Dynamic Solution of the Volterra Integral Equation in the Form of Rational Spline Functions

Приближенное решение интегрального уравнения Вольтерры второго рода представлено в виде коллокационных рациональных сплайн-функций на последовательных отрезках, исчерпывающих всю область решения. Получены также оценки скорости сходимости приближенных решений к точному в равномерной метрике через модуль непрерывности решения и его производных первого и второго порядков. Библиография: 11 названий.

Elastic diffusion vibrations of an isotropic Kirchhoff-Love plate under an unsteady distributed transverse load

We investigated an unsteady elastic diffusion vibration of a simply supported rectangular isotropic Kirchhoff-Love plate. The plate is under the action of a distributed transverse load. A model that describes coupled elastic diffusion processes in a multicomponent continuum is used for the mathematical problem formulation. The model is taking into account the diffusion fluxes relaxation. The transverse vibration equations of a rectangular isotropic Kirchhoff-Love plate with diffusion were obtained from the model using the d'Alembert variational principle. The initial-boundary value problem of a freely supported isotropic rectangular plate bending is formulated on the basis of the obtained equations. The plate is under the action of elastic diffusion perturbations distributed over the surface. The problem solution of an unsteady elastic diffusion plate vibration is sought in an integral form. The surface Green's functions are the kernels of the integral representations. To find the Green's functions, we used the Laplace transform in time and the expansion into double trigonometric Fourier series in spatial coordinates. Green's functions in the image domain are represented in the form of rational functions and depend on the Laplace transform parameter. The transition to the original domain is done analytically through residues and tables of operational calculus. The surface Green's function analytical expressions are obtained. As a calculation example, we considered a freely supported elastodiffusive plate under the action of suddenly applied unsteady bending moments distributed over the plate surface. By using a three-component continuum, a numerical study of interactions between unsteady mechanical and diffusion fieldsis done for an isotropic plate. The influence of relaxation effects on the kinetics of mass transfer is investigated. The solution is presented in the analytical form, as well as in the graphs of the displacement fields and concentration increments on time and coordinates. At the end of the publication, the main conclusions are given about the fields coupling effect and the relaxation of diffusion fluxes on the stress-strain state and mass transfer in the plate.

Mixed soliton solutions for the (2+1)-dimensional generalized breaking soliton system via new analytical mathematical method

In this research, we study the dispersive phenomena for the (2+1)-dimensional generalized breaking soliton system (GBSS), which describe the interaction phenomena between Riemann wave and long wave via two space variable in nonlinear media. The constructed solitary wave results for the (2+1)-dimensional GBSS via extended modified rational expansion (EMRE) method in the form of rational, elliptic, and trigonometric functions including mixed solitons, single bright and dark solitions, combined bright and dark solitons, kink and anti-kink wave solitions, traveling waves, and periodic waves. The physical phenomena of demonstrated results represented by higher dimensional and contour plots graphically with the help of symbolic computation. The obtained results are very helpful in the study of interaction phenomena in fiber optics, mathematical physics, fluid dynamics, geophysics, engineering and many other various areas of scientific fields.

Abundant optical solitons for Lakshmanan–Porsezian–Daniel model by the modified auxiliary equation method

In this paper, the Lakshmanan–Porsezian–Daniel (LPD) model is investigated for optical solitons. Abundant exact solitary wave solutions of the LPD model are extracted using the modified auxiliary equation method (MAE). The proposed model is investigated for Kerr law of nonlinearity. The MAE method provides the solutions in the form of hyperbolic, trigonometric and rational functions. The 3D plots and 2D contours are presented for physical demonstration of retrieved the solutions.

New Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean

Let Ipx be the modified Bessel function of the first kind of order p. The upper and lower bounds in the form of simple rational functions about cosht and (sinht)/t for the function I0x are obtained. The corresponding inequalities for the Toader-Qi mean do not match those in the existing literature.

Multiloop Multirate Continuous-Discrete Drone Stabilization System: An Equivalent Single-Rate Model

The article discusses the UAV lateral motion stabilization system, as a MIMO multiloop multirate continuous-discrete system, specified in the form of an input–output model in the domain of discrete Laplace transform or in the form of a structural diagram. Approaches to the construction of equivalent T and NT single-rate models for MIMO multiloop multirate continuous-discrete systems are considered. Here, T is the largest common divisor of the sampling periods of the system, N is a natural number that is the smallest common multiple of the numbers characterizing the sampling periods of the system. The resulting impulse representations of the outputs of equivalent models are in the form of rational functions. The basis for the construction of these models is a matrix of sampling densities—a structural invariant of sampling chains. An example of the construction of the indicated matrix and an equivalent single-rate model are given. Obtaining equivalent single-rate models for MIMO multiloop multirate systems allows us to extend the methods of research and synthesis of MIMO continuous and continuous-discrete systems to a common theoretical base—the theory of polynomials and rational functions, which are typical elements of the description of these classes of systems.

Some New Solutions of Non Linear Evolution Equations With Mutable Coefficients

We construct the traveling wave solutions of some NonLinear Evolution Equations (NLEEs) with mutable coefficients arising in different branches of physics and mathematics. we apply a novel (G′G)-formalism to construct more general solitary traveling wave solutions of NLEEs such as Sharma-Tasso-Olver with mutable coefficients and Zakharov Kuznetsov equation. Interesting solutions of NLEEs are investigated by traveling wave solutions which are in form of trigonometric, rational, and hyperbolic functions. This may build more unified new solutions for different kinds of such NLEEs with mutable coefficients arising in mathematics and physics. Wolfram Mathematica 11 is used to perform the computation work and their corresponding plots and counter graphs are plotted. This method is found to be more useful and efficient for searching the exact solutions of NLEEs.

Dispersion analysis and soliton solution of space–time fractional Bi-Hamiltonian Boussinesq system

In this paper, the physical phenomena in fractional calculus of the space–time fractional Bi-Hamiltonian Boussinesq system involving the Riemann–Liouville derivatives is studied. The wave dispersion relationship in the linear form of the governing system is analyzed and then that relationship is used to obtain the velocity of the phase and group. Soliton solutions are developed using the fractional sub-equation method. The soliton solutions obtained in the form of trigonometry, hyperbolic and rational function illustrate that the approach of fractional sub-equation is very effective in evaluating the physical phenomena.

Abundant closed-form solutions and solitonic structures to an integrable fifth-order generalized nonlinear evolution equation in plasma physics

This paper investigates the evolution dynamics of closed-form solutions for a new integrable nonlinear fifth-order equation with spatial and temporal dispersion which describes shallow water waves moving in two directions. Some novel computational soliton solutions are obtained in the form of exponential rational functions, trigonometric and hyperbolic functions, and complex-soliton solutions. Some dynamical wave structures of soliton solutions are achieved in evolutionary dynamical structures of multi-wave solitons, double-solitons, triple-solitons, multiple solitons, breather-type solitons, Lump-type solitons, singular solitons, and Kink-wave solitons using the generalized exponential rational function (GERF) technique. All newly established solutions are verified by back substituting into the considered fifth-order nonlinear evolution equation using computerized symbolic computational work via Wolfram Mathematica. These newly formed results demonstrate that the considered fifth-order equation theoretically possesses very rich computational wave structures of closed-form solutions, which are also useful in obtaining a better understanding of the internal mechanism of other complex nonlinear physical models arising in the field of plasma physics and nonlinear sciences. The physical characteristics of some constructed solutions are also graphically displayed via three-dimensional plots by selecting the best appropriate constant parameter values to easily understand the complex physical phenomena of the nonlinear equations. Eventually, the results validate the effectiveness and trustworthiness of the used technique.

Global Well-Posedness of the Ocean Primitive Equations with Nonlinear Thermodynamics

We consider the hydrostatic Boussinesq equations of global ocean dynamics, also known as the “primitive equations”, coupled to advection–diffusion equations for temperature and salt. The system of equations is closed by an equation of state that expresses density as a function of temperature, salinity and pressure. The equation of state TEOS-10, the official description of seawater and ice properties in marine science of the Intergovernmental Oceanographic Commission, is the most accurate equations of state with respect to ocean observation and rests on the firm theoretical foundation of the Gibbs formalism of thermodynamics. We study several specifications of the TEOS-10 equation of state that comply with the assumption underlying the primitive equations. These equations of state take the form of high-order polynomials or rational functions of temperature, salinity and pressure. The ocean primitive equations with a nonlinear equation of state describe richer dynamical phenomena than the system with a linear equation of state. We prove well-posedness for the ocean primitive equations with nonlinear thermodynamics in the Sobolev space {{mathcal {H}}^{1}}. The proof rests upon the fundamental work of Cao and Titi (Ann. Math. 166:245–267, 2007) and also on the results of Kukavica and Ziane (Nonlinearity 20:2739–2753, 2007). Alternative and older nonlinear equations of state are also considered. Our results narrow the gap between the mathematical analysis of the ocean primitive equations and the equations underlying numerical ocean models used in ocean and climate science.

Exact and solitary wave solutions of conformable time fractional Clannish Random Walker’s Parabolic and Ablowitz-Kaup-Newell-Segur equations via modified mathematical methods

The nonlinear fractional differential equations (FDEs) composed by mathematical modeling through nonlinear corporeal structures. The study of these kind models has an energetic position in different field of applied sciences. The dominant intension of this article is to extract novel analytical wave solutions of the conformable time fractional Clannish Random Walker’s Parabolic (CRWP) and (2 + 1)-dimensional fractional Ablowitz-Kaup-Newell-Segur (AKNS) equations in the sense of conformable derivative with the aid of modified mathematical methods, called modified extended auxiliary equation mapping and modified F-expansion schemes. By the virtue of employed techniques, different types of solutions are obtained in the form of trigonometric, hyperbolic, exponential and rational functions respectively. To prompt the essential propagated features, some investigated solutions are exhibited in form of 3D and 2D is planned by passing on the precise values to the parameters under the constrain conditions. The accomplished solutions show that these presented schemes are reliable, applicable and efficient.

Adequate wide-ranging closed-form wave solutions to a nonlinear biological model

The auxiliary equation method and the extended tanh function method are put in use to establish wide-ranging, further general and standard closed-form traveling wave solutions to the time fractional nonlinear biological model. The fractional derivative is considered in the sense of beta-derivative. The ascertained solutions are in the form of exponential, rational, hyperbolic and trigonometric functions with significant physical meaning. The achieved solutions demonstrate that the methods are effective, compatible and efficient mathematical tools. We analyze the attained solutions through sketching graphs for the defined values of the associated parameters to perceive the internal structure by making use the Mathematica software. The developed traveling wave solutions might be helpful to illustrate the internal infrastructure of tangible phenomena associated with the consider model.

Explicit Travelling Wave Solutions to Nonlinear Partial Differential Equations Arise in Mathematical Physics and Engineering

To describe the interior phenomena of the mysterious problems around the real world, non-linear partial differential equations (NLPDEs) plays a substantial role, for which construction of analytic solutions of those is most important. This paper stands for a goal to find fresh and wide-ranging solutions to some familiar NLPDEs namely the non-linear cubic Klein-Gordon (cKG) equation and the non-linear Benjamin-Ono (BO) equation. A wave variable transformation is made use to convert the mentioned equations into ordinary differential equations. To acquire the desired precise exact travelling wave solutions to the above-stated equations, the rational -expansion method is employed. Consequently, three types of equipped solutions are successfully come out in the forms of hyperbolic, trigonometric and rational functions in a compatible way. To analyse the physical problems arisen relating to nonlinear complex dynamical systems, our obtained solutions might be most helpful. So far we know, these achieved solutions are different than those in the literature. The applied method is efficient and reliable which might further be used to find different and novel solutions to many other NLPDEs successfully in research field.