Explicit Exact Solutions Research Articles

Discovering asymptotic expansions for problems in mechanics using symbolic regression

Recently, symbolic regression (SR) has demonstrated its efficiency for discovering basic governing relations in physical systems. A major impact can be potentially achieved by coupling symbolic regression with asymptotic methodology. The main advantage of asymptotic approach involves the robust approximation to the sought for solution bringing a clear idea of the effect of problem parameters. However, the analytic derivation of the asymptotic series is often highly nontrivial especially, when the exact solution is not available.In this paper, we adapt SR methodology to discover asymptotic series. As an illustration we consider three problem in mechanics, including two-mass collision, viscoelastic behavior of a Kelvin–Voigt solid and propagation of Rayleigh–Lamb waves. The training data is generated from the explicit exact solutions of these problems. The obtained SR results are compared to the benchmark asymptotic expansions of the above mentioned exact solutions. Both convergent and divergent asymptotic series are considered. A good agreement between SR expansions and exact analytical results is observed. It is demonstrated that the proposed approach can be used to identify material parameters, e.g. Poisson’s ratio, and has high prospects for utilizing experimental and numerical data.

Exact Solutions and Dynamics in Schrödinger–Hirota Model Having Multiplicative White Noise via Itô Calculus

For the Schrödinger–Hirota model having multiplicative white noise described via Itô calculus, to find exact explicit solutions, the corresponding differential system of the amplitude component is formulated, which is a planar dynamical system with a singular straight line. In this paper, by using the techniques from dynamical systems to analyze the parameter conditions of the associated system and to find the corresponding phase portraits, the dynamical behavior of the amplitude component can be derived. Under a special parameter condition, exact explicit homoclinic solutions, periodic solution families as well as compacton solutions can be found.

Explicit Exact Solutions of Nonlinear Transient Thermal Models of a Porous Moving Fin using Laplace transform - Exp-function method

The present paper develops explicit and non-power series exact solutions to the nonlinear heat transfer models of conductive-radiative-convective moving-porous fin using Laplace transform - Exp-function method, is presented. The developed solutions are employed for investigation of the included parameters on the transient and steady states studies of the moving-porous fin. The results submitted that the fin temperature is augmented with time increase due to increase heat transfer rate as time progresses. However, thermal parameter of the fin reduces from the its base to its tip. As the porosity, moving, convective-conductive-radiative parameters are increased, the fin temperature are decreased due to increased heat transfer rate. The opposite trend is displayed for the conductive-radiative number. It can be stated that present work will be useful in the analysis of the device.

An exact closed-form explicit solution of free transverse vibration for non-uniform multi-cracked beam

Non-uniformity and damage are the two primary subjects in studying the vibrations of the beam-type elements. An exact closed-form explicit solution for the transverse displacement of a non-uniform multi-cracked beam with any type of boundary conditions is introduced. The generalized functions and the distributional derivative concepts are adopted. Four fundamental functions are introduced. These functions make the boundary conditions' process and compute the frequency equation more convenient. By introducing the non-dimensional parameters, the non-dimensional motion equation of the damaged beam with an arbitrary count of cracks is derived, and its exact closed-form explicit solution is obtained based on the four introduced fundamental functions. The standard method of computing these functions is presented, and the closed-form of these functions is determined for eight cases like uniform and conical beams. The closed-form of the frequency equation and mode shapes of the non-uniform multi-cracked beam are derived for several boundary conditions. The influence of the count of cracks, their location and intensity, and the boundary conditions on the natural frequency and mode shape are assessed by running a numerical study. The first and second frequencies of a conical beam are computed to verify the obtained results by applying this newly presented closed-form solution and the Differential Quadrature Element Method. A good agreement is evident when the obtained results are compared.

A novel numerical scheme for fractional differential equations using extreme learning machine

In this paper, we propose a neural network-based approach with an Extreme Learning Machine (ELM) for solving fractional differential equations. The solution procedure for the linear and nonlinear fractional differential equations has been derived. Also the convergence and stability of the proposed method is provided. Then we examine the numerical solution of several fractional-order ordinary and partial differential equations. As a last example the Burgers equation without an explicit exact solution. The effect of changing the number of neurons on the accuracy of the solution is obtained graphically.

Modeling of micropolar nanofluid flow over flat surface with slip velocity and heat transfer: Exact multiple solutions

This study presents new exact solutions for the flow of a non–Newtonian micropolar fluid in a water–based nanofluid, which is initiated by a bidirectional moving permeable surface. The surface is either stretching or shrinking linearly along its length, and experiences velocity slip under constant wall temperature or linearly growing wall temperature. By using a similarity reduction of the micropolar nanofluid equations, a coupled system of ordinary differential equations is obtained, which is governed by various parameters such as the moving parameter, mass transpirations, velocity slip, material parameter, and nanofluid concentration parameter. Explicit exact solutions are derived for both the general flow and the gyration velocity fields. The effects of these parameters on the fluid flow and temperature distributions are demonstrated through graphical analysis. The study finds that single solution exist for the stretching sheet case, and dual solutions appear in the region s⩾sc, where sc=1.4943 for the shrinking sheet. One of the two components of the dual solutions exhibits algebraic decay for large war flux values. It is also found that when either a=0, or velocity slip is very large, their thermal solutions coincide and are entirely dependent on mass transpiration.

Novel Exact Traveling Wave Solutions for Nonlinear Wave Equations with Beta-Derivatives via the sine-Gordon Expansion Method

The main objectives of this research are to use the sine-Gordon expansion method (SGEM) along with the use of appropriate traveling transformations to extract new exact solitary wave solutions of the (2 + 1)- dimensional breaking soliton equation and the generalized Hirota-Satsuma coupled Korteweg de Vries (KdV) system equipped with beta partial derivatives. Using the chain rule, we convert the proposed nonlinear problems into nonlinear ordinary differential equations with integer orders. There is then no further demand for any normalization or discretization in the calculation process. The exact explicit solutions to the problems obtained with the SGEM are written in terms of hyperbolic functions. The exact solutions are new and published here for the first time. The effects of varying the fractional order of the beta-derivatives are studied through numerical simulations. 3D, 2D, and contour plots of solutions are shown for a range of values of fractional orders. As parameter values are changed, we can identify a kink-type solution, a bell-shaped solitary wave solution, and an anti-bell shaped soliton solution. All of the solutions have been carefully checked for correctness and could be very important for understanding nonlinear phenomena in beta partial differential equation models for systems involving the interaction of a Riemann wave with a long wave and interactions of two long waves with distinct dispersion relations.

Rogue waves and breathers of the derivative Yajima-Oikawa long wave-short wave equations on theta-function backgrounds

Based on the Baker-Akhiezer functions on Riemann surfaces, a general method is proposed to construct rogue-wave solutions and breather solutions of integrable nonlinear evolution equations associated with higher-order matrix spectral problems on theta-function periodic backgrounds. The derivative Yajima-Oikawa long wave-short wave equation associated with a 3×3 matrix Lax pair is chosen as an illustrative example of our method. First, a theta-function seed solution for the derivative Yajima-Oikawa long wave-short wave equation is derived by using the Weierstrass functions. Then, the general solution of the 3×3 matrix spectral problem and its derivatives with respect to the spectral parameter are obtained with the help of the Baker-Akhiezer functions. Finally, by means of Darboux transformations, explicit exact solutions such as rogue waves and breathers on theta-function backgrounds for the derivative Yajima-Oikawa long wave-short wave equation are constructed. Taking the advantage of the Jacobi θ-functions, these new solutions are numerically calculated and illustrated. In addition, by observing the explicit solutions, it is found that two rogue waves of different shapes may appear in one solution, which is a new phenomenon different from the solutions on constant backgrounds.

On the Construction of Exact Numerical Schemes for Linear Delay Models

Exact numerical schemes have previously been obtained for some linear retarded delay differential equations and systems. Those schemes were derived from explicit expressions of the exact solutions, and were expressed in the form of perturbed difference systems, involving the values at previous delay intervals. In this work, we propose to directly obtain expressions of the same type for the fundamental solutions of linear delay differential equations, by considering vector equations with vector components corresponding to delay-lagged values at previous intervals. From these expressions for the fundamental solutions, exact numerical schemes for arbitrary initial functions can be proposed, and they may also facilitate obtaining explicit exact solutions. We apply this approach to obtain an exact numerical scheme for the first order linear neutral equation x′(t)−γx′(t−τ)=αx(t)+βx(t−τ), with the general initial condition x(t)=φ(t) for −τ≤t≤0. The resulting expression reduces to those previously published for the corresponding retarded equations when γ=0.

Explicit exact solutions and conservation laws in a medium with competing weakly nonlocal nonlinearity and parabolic law nonlinearity

Explicit exact solutions and conservation laws in a medium with competing weakly nonlocal nonlinearity and parabolic law nonlinearity

Time-dependent non-Abelian waves and their stochastic regimes for gauge fields coupled to external sources

In this paper we explore explicit exact solutions of the $SU(2)$ Yang-Mills (YM) and Yang-Mill-Higgs (YMH) equations with homogeneous and inhomogeneous external sources. Whereas in the case of YM we have confirmed our analytical findings with the numerical simulations, the numerical corroborations in the YMH case yielded the stochastic character of motion for the ensuing fields.

An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions

Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the discovery of integrability of these equations has greatly promoted the understanding of their generality, and thus promoted their rapid development. A key feature of an integrable nonlinear evolution equation is the fact that it can be expressed as the compatibility condition of two linear spectral problems, i.e., a Lax pair, which plays a crucial roles in the Darboux transformation. A major difficulty, however, is the problem of associating nonlinear evolution equations with appropriate spectral problems. Therefore, it is interesting for us to search for the new spectral problem and corresponding nonlinear evolution equations. In this paper, a new integrable nonlinear wave model and its integrable nonlinear reduction are presented by using the introduced 2 × 2 matrix spectral problem. Based on the resulting gauge transforms between the 2 × 2 matrix Lax pairs, Darboux transforms are derived for the integrable nonlinear wave model and its integrable nonlinear reduction, from which an algebraic algorithm for solving this integrable nonlinear wave model and its integrable nonlinear reduction is given. As an application of the Darboux transform, explicit exact solutions of the integrable nonlinear reduction are obtained, including solitons, breathers, and rogue waves.

Soliton structures and dynamical behaviors for the integrable system of Drinfel’d–Sokolov–Wilson equations in dispersive media

This study is used to investigate the exact explicit solutions and dynamical behaviors of the (1+1)-dimensional integrable system of Drinfel’d–Sokolov–Wilson equations in dispersive media. Firstly, the unified method is implemented to find explicit solutions in polynomial and rational forms. These solutions include rational, dark and bright soliton structures. After that, the planar dynamical system of the considered model is obtained using the Galilean transformation. The phase portraits of the bifurcations are drawn from the planar dynamical system for different physical parametric values using the fourth-order Runge–Kutta method. Further, an external force is imposed on the dynamical system to observe the phase portraits of chaotic trajectories. These tracks are outlined for different values of the strength and frequency of the external force acting on the dynamical system. The outcomes show that a given model has chaotic behvior as its solution becomes disordered by taking small variation in the strength and frequency of the external force. The results are new and can be helpful in investigating the dynamical behaviors of the other nonlinear physical models arising in optics, bio-mathematics and so many other fields of science.

Exact solutions for flow through porous media with the Klinkenberg effect

The effect of gas compressibility during flow through a porous medium modifies the governing equation for pressure, and for low pressures, this leads to the so-called Klinkenberg effect. Darcy’s law remains unchanged, and accounting for the equation of state leads to a parabolic governing equation for pressure. The increased flow rate at low pressure modifies the nonlinearity of this governing equation. We present two broad approaches for constructing exact solutions to the steady state problem. First, geometric reduction is employed to construct exact solutions in Cartesian and polar coordinates. Next, the governing equation is interpreted so that a stream function exists for the flow, and this is used to demonstrate that solutions can only be found when the flow is irrotational. A second, rather broad, class of exact solutions is thus constructed from potential flows and their generalization for variable permeability cases. The latter leads to a non-constant coefficient problem, and we provide both an algorithm illustrating how to use an existing linear numerical solver to solve the nonlinear problem and an explicit exact solution for an annular domain.

Perturbed traveling wave solutions of the CDGKS equation and its dynamics characteristics

Based on the traveling wave reduction method with a perturbed initial solution and the F-expansion method, a class of explicit exact solutions of the (2+1)-dimensional CDGKS equation are obtained through the symbolic computation. Moreover, both the interaction behavior between parameters and the perturbation degree of periodic wave and Gauss wave to rational pulse wave, and the correlation of parameters to the superposition degree of the interaction energy between solitary wave and rational pulse wave are discussed. Finally, numerical simulations are shown to demonstrate the mechanism of the above solutions.

Exact linearization and exact solutions of a generalized Thomas equation with general variable coefficients

In this paper, we investigate a generalized Thomas equation with general variable coefficients, which describes the processes in chemical kinetics with ion exchange under sorption in a reagent flow. It is shown that the generalized Thomas equation with variable coefficients can be exactly linearized as long as the variable coefficient functions satisfy some constraint conditions. Firstly, we establish a nonlinear transformation between the generalized Thomas equation with variable coefficients and the linear hyperbolic equation with variable coefficients. Then we obtain a series of explicit exact solutions of the generalized Thomas equation with variable coefficients by solving reduced linear hyperbolic equation with variable coefficients under some coefficient conditions. In some cases, these exact solutions contain an arbitrary function.

Dynamics of Nonautonomous Matter Waves in a Waveguide With an External Source

We explore explicit exact Lorentzian-type matter wave solutions of nonautonoumous nonlinear Schrödinger equation (NLSE) with an external source and a harmonic trap. We explicate the mechanism to control the dynamical behaviour of these waves of Bose-Einstein condensate (BEC) for different sets of nonlinearity and trapping modulations. Our results raise the possibility of some experiments and potential applications related to BEC in the presence of an inhomogeneous external source.

On the multiple explicit exact solutions to the double‐chain DNA dynamical system

The nonlinear dynamics of deoxyribonucleic acid (DNA) is the subject of this research. For an organism to flourish and reproduce, it needs the genetic information that is contained in its DNA. In the DNA molecule, the hydrogen bonds between the base pairs of the two polynucleotide chains are represented by an elastic membrane (or a series of linear springs) that connects the model's two long elastic homogeneous strands (or rods). Using two integration norms, namely, the new auxiliary equation method (NAEM) and the extended sinh‐Gordon equation expansion method (ShGEEM), we obtain a variety of nonlinear dynamical solitary wave structures in various shapes, including hyperbolic, trigonometric, and exponential function solutions, as well as some unique solitary wave solutions, including bell‐shaped, kink, singular, combined, and multiple solitons. Using the logarithmic transformation assistant, we examine a variety of wave structures and recover singular periodic wave solutions with unknown parameters, along with the validity constraints. According to the results, the governing model contains an unusually large number of theoretically exact solitary wave solutions. The physical characterization of some of the presented results can be visualized in 3D and 2D profiles by selecting the appropriate parameter values.

Assorted exact explicit solutions for the generalized Atangana’s fractional BBM–Burgers equation with the dissipative term

In this study, the generalized Atangana’s fractional BBM–Burgers equation (GBBM-B) with the dissipative term is investigated by utilizing the modified sub-equation method and the new G'/(bG' + G + a)-expansion method; with the aid of symbolic computations, many types of new exact explicit solutions including solitary wave solutions, trigonometric function periodic solutions, and the rational function solutions are obtained. Some 3D and 2D plots of these solutions are simulated, which show the novelty and visibility of the propagation behavior and dynamical structure of the corresponding equation. Moreover, with the selection of different values on the parameters and orders, we can deduce many types of exact solutions in special cases. We also discussed the changes and characteristics of these solutions, which can help us further understand the inner structure of this equation. The obtained solutions indicate that the approach is easy and effective for nonlinear models with high-order dispersion terms.

BÄCKLUND TRANSFORMATION AND DIVERSE EXACT EXPLICIT SOLUTIONS OF THE FRACTAL COMBINED KdV–mKdV EQUATION

A fractal modification of the combined KdV–mKdV equation which plays a key role in various fields of physics is presented in this work for the first time. Aided by the fractal two-scale transform, the homogeneous balance method is employed to construct the fractal Bäcklund transformation. By means of the Bäcklund transformation, some new exact explicit solutions such as the algebraic solitary wave solution of rational function, single-soliton solution, double-soliton solutions, N-soliton solutions, singular traveling solutions and the periodic wave solutions of trigonometric function are obtained. Finally, some solutions are illustrated with different fractal orders in the form of the 3D plot, 3D density and 2D curves by assigning reasonable parameters with the help of Mathematica. The findings in this paper are expected to present some new insights into the fractal theory of the fractal PDEs.