Narrow Results

In this paper, two algebraic methods are applied for solving a class of conformable fractional partial differential equations (FPDEs). We use these methods for the time-fractional Radhakrishnan–Kundu–Lakshmanan equation. With these methods, further solutions can be obtained compared with other approaches and techniques. The exact particular solutions include the exponential solution, trigonometric function solution, rational solution and hyperbolic function solution. These methods are very effective to obtain exact solutions of many fractional differential equations.

This paper proposes a new method for the development of the Caputo time fractional model. The method relies on generalized Fourier’s and Fick’ laws to describe the flow behavior of Brinkman-type fluids. An analysis of the free convection flow through a channel is carried out using a new transformation method. This transformation affects fluid energy and concentration equations. The specific governing equations are solved using a Laplace transform and Fourier sine transform. We obtain the solutions of the governing partial differential equations (PDEs) in terms of the Mittag–Leffler function. Mathematical software has been used for both graphical and numerical computation in order to examine the effects of embedded parameters. From graphical and tabular analysis, fractional-order solution provides more than one layer for fluid behavior, thermal, and concentration distribution in the channel. Experimentalists and engineers can choose from many best-fitted layers to compare their data and results. A deviation in the velocity profile’s behavior is also seen for larger values of the Brinkman parameter.

In this paper, we investigate the exact solutions of a modified generalized multidimensional fractional Kadomtsev–Petviashvili (KP) equation by the bifurcation method. First, the equation is converted into a planar dynamical system through fractional complex wave transformation. The phase portraits of the equation and qualitative analysis are presented under different bifurcation conditions. Then, the bounded and unbounded traveling wave solutions, including periodic, kink, anti-kink, dark-solitary, bright-solitary and breaking wave solutions, are acquired by integrating along different orbits. Finally, numerical simulations of the dynamic behaviors of the solutions obtained are graphically illustrated by choosing appropriate parameters.

We review results on GKSL-type equations with multi-modal generators which are quadratic in bosonic or fermionic creation and annihilation operators. General forms of such equations are presented. The Gaussian solutions are obtained in terms of equations for the first and the second moments. Different approaches for their solutions are discussed.

This paper presents exact solutions for the free vibrations and buckling of rectangular plates having two opposite, simply supported edges subjected to linearly varying normal stresses causing pure in-plane moments, the other two edges being free. Assuming displacement functions which are sinusoidal in the direction of loading (x), the simply supported edge conditions are satisfied exactly. With this the differential equation of motion for the plate is reduced to an ordinary one having variable coefficients (in y). This equation is solved exactly by assuming power series in y and obtaining its proper coefficients (the method of Frobenius). Applying the free edge boundary conditions at y=0, b yields a fourth order characteristic determinant for the critical buckling moments and vibration frequencies. Convergence of the series is studied carefully. Numerical results are obtained for the critical buckling moments and some of their associated mode shapes. Comparisons are made with known results from less accurate one-dimensional beam theory. Free vibration frequency and mode shape results are also presented. Because the buckling and frequency parameters depend upon the Poisson's ratio (ν), results are shown for 0≤ν≤0.5, valid for isotropic materials.

In this study, the longitudinal vibration of rods with variable cross-sections is studied. For the analytical solution of the problem, a new analytical method based on a recently developed method on the Riccati differential equation is utilized. The governing equation is reduced to Hill’s type second-order ordinary differential equation. The transformed equations can readily be solved analytically for various cases according to the method. Seven cases have been considered, and the frequency equations for each case have been obtained. According to the method developed, the problem is solved in the simplest way. By using the present method, the reader can readily decide whether the problem is solved analytically or numerically. The present method can solve the problem of longitudinal vibration of rods having cross-sections of arbitrary shape. Finally, the method is also applied to the longitudinal vibration of stepped rods. Mode shapes are plotted for special values.

This article focuses on the free vibration of single-degree-of-freedom (SDOF) systems with linear hysteretic damping and investigates the consistency and convergency of the solution to the free vibration in time domain. The criteria associated with the unit-impulse response function at $t=0$ are suggested to check the consistency of possible solutions. Exhaustive and systematic derivations are presented to clarify the confusing results in the literature. Subsequently, the integration in the Hilbert term proves to be nonconvergent when an exponential term (${\mathrm{\text{e}}}^{\bar{s}t}$) is assumed to be the general form of displacement solutions. Hence, it can be concluded that the exact solution to the free vibration of SDOF systems with linear hysteretic damping is still an open question. At last, a numerical verification method is proposed to assess the accuracy of possible solutions.

Within the Relativistic Theory of Gravitation it is shown that the equation of state p = ρ holds near the center of a black hole. For the stiff equation of state p = ρ − ρ_c the interior metric is solved exactly. It is matched with the Schwarzschild metric, which is deformed in a narrow range beyond the horizon. The solution is regular everywhere, with a specific shape at the origin. The gravitational redshift at the horizon remains finite but is large, z ~ 10²³ M_⊙/M. Time keeps its standard role also in the interior. The energy of the Schwarzschild metric, shown to be minus infinity in the General Theory of Relativity, is regularized in this setup, resulting in E = Mc².

The present studies deal with the peristaltic motion of an incompressible Williamson fluid model in an endoscope. The governing equations of Williamson fluid model are first simplify using the assumptions of long wavelength and low Reynolds number. The four types of solutions have been presented for velocity profile named (i) exact solution, (ii) perturbation solution, (iii) HAM solution, and (iv) numerical solutions. The comparisons of four solutions have been found a very good agreement between all the solutions. In addition, the expressions for pressure rise and velocity against various physical parameters are discussed through graphs.

In this study, the mathematical model of oscillating flow in an annular straight pipe due to an imposed oscillating pressure gradient and with harmonic boundary motion is established. So far, no literature is devoted to investigate the effect of harmonic boundary motion on the velocity profile and shear stress. The Navier–Stokes Equation in the cylindrical coordinate is the governing equation of fluid motion. The exact solution of this system in terms of real function only is presented. This method is more understood and helpful for deeply investigating the internal oscillating flow than the conventional complex method presented by Womersley. It is found that the effects of the boundary motion, the wave frequency, the Womersley number, and the radius ratio on the velocity profile and shear stress distribution are significant.

In this article, we considered the peristaltic flow of Newtonian incompressible fluid of chyme in small intestine. The analysis has been performed using an endoscope. The peristaltic flow of chyme is modeled by assuming that the peristaltic wave is formed in non-periodic mode comprising two sinusoidal waves of different wave lengths propagating with same speed along the outer wall of the tube. Heat transfer mechanisms have been taken into account, such that the constant temperature and are assigned to inner and outer tubes, respectively. A complex system of equations has been simplified using long wavelength and low Reynolds number approximation because such assumptions exist in small intestine. Exact solutions have been carried out for velocity temperature and pressure gradient. Graphical results have been discussed for pressure rise, frictional forces, temperature, and velocity profile. Comparison of present results with the results of the existing literature have been presented through figures. Trapping phenomena have been presented at the conclusion of the article.

The purpose of this paper is to develop a mathematical description of cargo transport in the entrance regions of axons and dendrites. The model accounts for the difference in microtubule (MT) orientation between axons and dendrites: axons have a uniform MT polarity orientation while dendrites have a mixed MT polarity orientation. Because of that, cargos pulled by dynein motors can enter dendrites but cannot enter axons. It is thus assumed that cargos are targeted to axons by associating them with kinesin motors and cargos are targeted to dendrites by associating them with dynein motors. Analytical solutions of the developed equations describing cargo concentrations in the entrance regions of axons and dendrites are obtained.

The viscous fluid model is considered in this article for the study of blood flow through an axis-symmetric stenosis with the effect of three distinct types of arteries i.e., diverging tapering arteries, converging tapering arteries and nontapered arteries. The Cauchy–Euler method has been used for the solution to velocity profile, resistance impedance to flow and the pressure gradient. The characteristics of viscous blood flow on velocity profile, impedance resistance to flow and pressure gradient have been discussed by plotting the graphs of various flow parameters and finally it is found that stenosis dominantes the curvature of curved artery.

A theoretical analysis of a multi-phase fluid problem is presented in this paper. Electro-osmotic flow with heat transfer in a divergent channel is modeled. Jeffrey fluid is taken as the base liquid, which suspends with the spherical gold particles. Electromagnetohydrodynamic (EHD) of a two-phase fluid through a porous medium is influenced by thermal radiation. Cumbersome mathematical manipulations leads to yield an exact solution for the set of strenuous differential equations. A comprehensive parametric study validates the accuracy and formidability of this intuitive analysis, by confirming the corresponding boundary conditions. The investigation articulately describes that the momentum of two-phase flow is supported by the Jeffrey fluid parameter. Additional golden particles increase the velocity of the base liquid. However, more thermal energy has contributed to the variation of the Brinkman number $B_{RM}$. Finally, thermal energy expunges from the diverse channel due to Jeffrey’s fluid parameter.

This paper considers reaction-diffusion equations from a new point of view, by including spatiotemporal dependence in the source terms. We show for the first time that solutions are given in terms of the classical Painlevé transcendents. We consider reaction-diffusion equations with cubic and quadratic source terms. A new feature of our analysis is that the coefficient functions are also solutions of differential equations, including the Painlevé equations. Special cases arise with elliptic functions as solutions. Additional solutions given in terms of equations that are not integrable are also considered. Solutions are constructed using a Lie symmetry approach.

Phylogenetic analysis has to overcome the grant challenge of inferring accurate species trees from evolutionary histories of gene families (gene trees) that are discordant with the species tree along whose branches they have evolved. Two well studied approaches to cope with this challenge are to solve either biologically informed gene tree parsimony (GTP) problems under gene duplication, gene loss, and deep coalescence, or the classic RF supertree problem that does not rely on any biological model. Despite the potential of these problems to infer credible species trees, they are NP-hard. Therefore, these problems are addressed by heuristics that typically lack any provable accuracy and precision. We describe fast dynamic programming algorithms that solve the GTP problems and the RF supertree problem exactly, and demonstrate that our algorithms can solve instances with data sets consisting of as many as 22 taxa. Extensions of our algorithms can also report the number of all optimal species trees, as well as the trees themselves. To better asses the quality of the resulting species trees that best fit the given gene trees, we also compute the worst case species trees, their numbers, and optimization score for each of the computational problems. Finally, we demonstrate the performance of our exact algorithms using empirical and simulated data sets, and analyze the quality of heuristic solutions for the studied problems by contrasting them with our exact solutions.

In this work, a three-dimensional (3D) nonlinear smoothed finite element method (S-FEM) solver is developed for large deformation problems. Node-based and face-based S-FEM using automatically generable four-noded tetrahedral elements (NS-FEM-Te4 and FS-FEM-Te4) are adopted to find the solution bounds in strain energy. The lower bound solutions are obtained using FEM-Te4 and FS-FEM-Te4, while the upper bound solutions are obtained using NS-FEM-Te4. A combined $\alpha$S-FEM-Te4 with a scaling factor $\alpha$ that controls the combination is constructed to find nearly exact solutions for the nonlinear solids mechanics problems through adjusting $\alpha$. This is achieved using the property that a successive change of scaling factor $\alpha$ can make the model transform from “overly-stiff” to “overly-soft”. Considering the properties of FS-FEM and NS-FEM, a selective FS/NS-FEM-TE4 is also used to solve 3D nonlinear large deformation problems, which produces a lower bound in strain energy. Hyperelastic Mooney–Rivlin and Ogden materials are both used in this study. Numerical examples reveal that S-FEM-Te4 is an effective method for obtaining solution bounds together with the standard FEM, and the FS-FEM-Te4, NS-FEM-Te4 and selective FS/NS-FEM-TE4 are robust with the high accuracy and computational efficiency for large deformation nonlinear problems.

In this paper we discuss a master equation applied to the two-level system of an atom and derive an exact solution to it in an abstract manner. We also present a problem and a conjecture based on the three-level system. Our results may give a small hint to understand the huge transition from Quantum World to Classical World. To the best of our knowledge this is the finest method up to the present.

A cosmology of Poincaré gauge theory is developed. An analytic solution is obtained. The calculation results agree with observation data and can be compared with the $\mathrm{\Lambda }$CDM model. The cosmological constant puzzle is the coincidence and fine tuning problem are solved naturally at the same time. The cosmological constant turns out to be the intrinsic torsion and curvature of the vacuum universe, and is derived from the theory naturally rather than added artificially. The dark energy originates from geometry, includes the cosmological constant but differs from it. The analytic expression of the state equations of the dark energy and the density parameters of the matter and the geometric dark energy are derived. The full equations of linear cosmological perturbations and the solutions are obtained.

In this paper, we study how to determine the unknown functions for the scalar tensor model $f(R,\varphi )$ where the Ricci scalar is allowed to appear in a nonlinear way. The methods followed to determine these functions are: the matter collineation approach, the Lie group method and the Lagrangian collineation approach. We find several exact analytical solutions for a cosmological model with a FRW metric. We determine that some of the results are also valid for some anisotropic metric (e.g. the self-similar ones).