Construction Of Analytical Solutions Research Articles

The common solution space of general relativity

We review the solution space for the field equations of Einstein's General Relativity for various static, spherically symmetric spacetimes. We consider the vacuum case, represented by the Schwarzschild black hole; the de Sitter-Schwarzschild geometry, which includes a cosmological constant; the Reissner-Nordström geometry, which accounts for the presence of charge. Additionally we consider the homogenenous and anisotropic locally rotational Bianchi II spacetime in the vacuum. Our analysis reveals that the field equations for these scenarios share a common three-dimensional group of point transformations, with the generators being the elements of the D⊗sT2 Lie algebra, known as the semidirect product of dilations and translations in the plane. Due to this algebraic property the field equations for the aforementioned gravitational models can be expressed in the equivalent form of the null geodesic equations for conformally flat geometries. Consequently, the solution space for the field equations is common, and it is the solution space for the free particle in a flat space. This approach open new directions on the construction of analytic solutions in gravitational physics and cosmology.

Numerical implementation of solving a boundary value problem with parameter for Fredholm integro-differential equation

The boundary value problem with parameter for Fredholm integro-differential equation with degenerate kernel is investigated in this paper. The aim of the paper is to establish the solvability conditions, to construct analytical and numerical solutions of the considered problem. The basis for achieving the goal is the ideas of Dzhumabayev parameterization method, classical numerical methods of solving Cauchy problems and numerical integration techniques. A problem with parameters is obtained by introducing an additional parameter and a new unknown function. A system of equations with respect to parameters is compiled according to the initial data of the considered equation and boundary conditions. The unknown function is found as a solution of the Cauchy problem for the ordinary differential equation. The equivalence of the original problem and the problem with parameters, the conditions of unique solvability are established and the formula for finding an analytical solution is obtained. Test examples of finding analytical and approximate solutions of the original problem are given.

Analytical solution for strongly nonlinear vibration of a stringer shell

In this paper, a new approach called the global residue harmonic balance technique is applied to construct analytical solutions for nonlinear oscillators that appear in cylindrical shells. Periodic solution is analytically proved, and consequently, the relation between amplitude and frequency is obtained in an analytical form. Numerical and analytical comparisons with the Runge–Kutta method and previously existing homotopy perturbation method (HPM) are given, to show the stability, quality, and efficiency of the present approach. The GRHBM provides an outstanding agreement with the numerical results for different amplitude values, leading to sufficiently accurate solutions for nonlinear oscillators.

Analytical and numerical solutions to the generalized Schrödinger equation with fourth-order dispersion and nonlinearity

This study aims to solve the generalized Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity (4-Order-GNLSE) using advanced analytical and numerical methods. The 4-Order-GNLSE is critical in understanding complex wave propagation phenomena in various physical contexts, including optical fibers and fluid dynamics, where higher-order dispersion and nonlinear effects are significant. We employ the Khater II (Khat II) and modified Rational (MRat) methods to construct analytical solutions. To validate these solutions, we utilize the trigonometric-quantic-B-spline (TQBS) method as a numerical scheme, demonstrating a close match between analytical and numerical results. Our findings indicate that the proposed methods effectively capture the intricate dynamics governed by the 4-Order-GNLSE. The results highlight the relevance of these solutions in practical applications, offering new insights into the wave propagation behaviors influenced by higher-order dispersion and nonlinearities. This research provides a significant contribution to the field of nonlinear wave equations, presenting novel solutions and validating methodologies that enhance our understanding and potential applications of these complex systems.

Higher-order hybrid rogue wave and breather interaction dynamics for the AB system in two-layer fluids

Rogue waves and breathers emerge as significant solitons, result from ubiquitous nonlinear effects of dynamics in the natural world, science and engineering. In this paper, we study the AB system, a model derived from a two-layer fluid that is widely used to investigate nonlinear fluid dynamics. We use the Darboux transformation to construct analytical solutions for higher-order hybrid rogue waves and breathers, which are expressed by symmetric algebraic matrices. We use these solutions and the spectrum parameters to explore the complex interaction dynamics of the hybrid rogue waves and breathers. We find that the interactions cause significant changes in the phases and shapes. Moreover, we discover an interesting phenomenon: the collisions between the rogue waves and breathers are semi-elastic, meaning that only the phases of the rogue waves change, while the speeds, shapes and phases of the breathers remain the same, before and after the collisions. This study provides new insights into the dynamics of the AB system, and helps us understand nonlinear phenomena in various two-layer fluid systems.

Momentum Shell in Quarkyonic Matter from Explicit Duality: A Dual Model for Cold, Dense QCD.

We present a model of cold QCD matter that bridges nuclear and quark matter through the duality relation between quarks and baryons. The baryon number and energy densities are expressed as functionals of either the baryon momentum distribution, f_{B}, or the quark distribution, f_{Q}, which are subject to the constraints on fermions, 0≤f_{B,Q}≤1. The theory is ideal in the sense that the confinement of quarks into baryons is reflected in the duality relation between f_{Q} and f_{B}, while other possible interactions among quarks and baryons are all neglected. The variational problem with the duality constraints is formulated and we explicitly construct analytic solutions, finding two distinct regimes: a nuclear matter regime at low density and a quarkyonic regime at high density. In the quarkyonic regime, baryons underoccupy states at low momenta but form a momentum shell with f_{B}=1 on top of a quark Fermi sea. Such a theory describes a rapid transition from a soft nuclear equation of state to a stiff quarkyonic equation of state. At this transition, there is a rapid increase in the pressure.

Construction of analytical solutions to systems of two stochastic differential equations

Abstract A scheme for the stochastization of systems of ordinary differential equations (ODEs) based on Itô calculus is presented in this article. Using the presented techniques, a system of stochastic differential equations (SDEs) can be constructed in such a way that eliminating the stochastic component yields the original system of ODEs. One of the main benefits of this scheme is the ability to construct analytical solutions to SDEs with the use of special vector-valued functions, which significantly differs from the randomization approach, which can only be applied via numerical integration. Moreover, using the presented techniques, a system of ODEs and SDEs can be constructed from a given diffusion function, which governs the uncertainty of a particular process.

Characteristics of dynamic waves in incompressible fluid regarding nonlinear Boiti-Leon-Manna-Pempinelli model

Distinct models involving nonlinearity are mostly appreciated for illustrating intricate phenomena arise in the nature. The new (3+1)-dimensional generalized nonlinear Boiti-Leon-Manna-Pempinelli (BLMP) model describes the dynamical behaviors of nonlinear waves arise in incompressible fluid. This present effort deals with the well-known governing BLMP equation by adopting two efficient schemes, namely improved tanh and improved auxiliary equation. As a result, a variety of appropriate wave solutions are made available in different type functions. The gathered solutions are figured out to characterize their internal properties for depicting the relevant phenomena. Diverse wave profiles are noticed in 3D, 2D and contour sense after assigning parameter’s values involved in the achieved solutions. The finding results are comparably different and general due to the existing wave solutions. The employed approaches perform in a great way to construct analytic wave solutions of considered evolution equation and deserve further use in relevant research area.

Analytical solutions of peridynamic equations. Part II: Elastic wave propagation

We use the separation of variables technique to construct analytical solutions for peridynamic models of dynamic elasticity. We show that, similar to the case of peridynamic models for transient diffusion, infinite series nonlocal solutions for peridynamic elasticity can be obtained directly from the solutions of the corresponding classical model by inserting “peridynamic/nonlocal factors” in the time-exponential part of the solution. The analytical solutions show that wave dispersion, caused by nonlocality, is contained in the horizon-dependent nonlocal factor. We obtain formulas for wave dispersion and group velocities for 1D and 2D peridynamic elastic wave propagation with three commonly-used peridynamic kernels. We observe interesting complexity in nonlocal solutions, generated by nonlocal wave dispersion, and “proportional” to the size of the nonlocal interaction region. Different from the diffusion case, as time goes to infinity, the nonlocal solution for elasticity does not converge to the classical one for a fixed horizon size, meaning that nonlocal effects persist in time. We solve several examples of wave propagation with Dirichlet boundary conditions and smooth or discontinuous initial conditions and compare these analytical solutions with those corresponding to the classical model, which is seen as a particular case of the PD model for horizon equal to zero. Interestingly, we find that in PD solutions, initial discontinuities in space persist at the same location, in time. While most of the analytical solutions we present here are formal, for some of the cases, we are able to prove uniform convergence of the series solutions. This work is the first presentation of a systematic analytical treatment of peridynamic problems in 2D finite domains.

Stochastic dispersive Schrödinger–Hirota equation having parabolic law nonlinearity with multiplicative white noise via Ito calculus

Purpose:The aim of this study is to introduce the stochastic dispersive Schrödinger–Hirota equation with parabolic law nonlinearity (SHEPL) with multiplicative white noise via Ito calculus and investigate its stochastic optical soliton solutions. For this purpose, we used unified Riccati equation expansion (UREEM) and a sub-version of an auxiliary methods to construct analytical solutions. Methodology:Firstly, we implemented traveling wave transform into SHEPL with multiplicative white noise via Ito calculus and constructed imaginary and real parts of its nonlinear ordinary differential equation (NODE) form. Then, we presented solution algorithms of UREEM and a sub-version of an auxiliary methods and successfully implemented to obtained NODE. We created analytical solutions of SHEPL with using the appropriate solution sets which involve unknown parameters and necessary transformations. Then we presented graphical representations of these solution functions by investigating the noise effect on the soliton structure and presented the gained results. Findings:We proposed the stochastic SHEPL equation for the first time in this article. We obtained stochastic optical soliton solutions, reflected their graphical representations and also observed effect of noise factor on the obtained soliton structure. Originality:The stochastic model of SHEPL has not been studied and the results have not been reported.

A hybrid analytical technique for solving multi-dimensional time-fractional Navier-Stokes system

<abstract> <p>In this research, a hybrid method, entitled the Laplace Residual Power Series technique, is adapted to find series solutions to a time-fractional model of Navier-Stokes equations in the sense of Caputo derivative. We employ the proposed method to construct analytical solutions to the target problem using the idea of the Laplace transform and the residual function with the concept of limit at infinity. A simple modification of the suggested method is presented to deal easily with the nonlinear terms constructed on the properties of the power series. Three interesting examples are solved and compared with the exact solutions to test the reliability, simplicity, and capacity of the presented method of solving systems of fractional partial differential equations. The results indicate that the used technique is a simple approach for solving nonlinear fractional differential equations since it depends only on the residual functions and the concept of the limit at infinity without needing differentiation or other complex computations.</p> </abstract>

Constructing Analytical Solutions of the Fractional Riccati Differential Equations Using Laplace Residual Power Series Method

In this article, a hybrid numerical technique combining the Laplace transform and residual power series method is used to construct a series solution of the nonlinear fractional Riccati differential equation in the sense of Caputo fractional derivative. The proposed method is implemented to construct analytical series solutions of the target equation. The method is tested for eminent examples and the obtained results demonstrate the accuracy and efficiency of this technique by comparing it with other numerical methods.

On the robustness of solitons crystals in the Skyrme model

In this work we analize how the inclusion of extra mesonic degrees of freedom affect the finite density solitons crystals of the Skyrme model. In particular, the first analytic examples of hadronic crystals at finite baryon density in both the Skyrme ω-mesons model as well as for the Skyrme ρ-mesons theory are constructed. These configurations have arbitrary topological charge and describe crystals of baryonic tubes surrounded by a cloud of vector-mesons. In the ω-mesons case, it is possible to reduce consistently the complete set of seven coupled non-linear field equations to just two integrable differential equations; one ODE for the Skyrmion profile and one PDE for the ω-mesons field. This analytical construction allows to show explicitly how the inclusion of ω-mesons in the Skyrme model reduces the repulsive interaction energy between baryons. In the Skyrme ρ-mesons case, it is possible to construct analytical solutions using a meron-type ansatz and fixing one of the couplings of the ρ-mesons action in terms of the others. We show that, quite remarkably, the values obtained for the coupling constants by requiring the consistency of our ansatz are very close to the values used in the literature to reduce nuclei binding energies of the Skyrme model without vector-mesons. Moreover, our analytical results are in qualitative agreement with the available results on the nuclear spaghetti phase.

A Lightweight Verification Method Based on Metamorphic Relation for Nuclear Power Software

The verification of nuclear design software commonly uses direct comparison methods. Benchmark questions, classical programs, experimental data, manual solutions, etc., would be used as expected results to compare with program outputs to evaluate the reliability of software coding and the accuracy of the numerical solution. Because nuclear power software numerically simulates complex physical processes, it involves many partial differential equations. It is usually challenging to construct analytical or accurate solutions and is expensive to develop benchmark questions and experimental data. Hence, the quantity of verification examples is small. By using the direct comparison method, verification is complicated, high cost, and inadequate. Entering the validation process without adequate proof will adversely impact the effectiveness and efficiency of validation. Metamorphic testing is an indirect verification technology that cleverly combines the nature of the model with software verification. It evaluates the correctness of the code by examining whether the program satisfies the metamorphic relation. Without manual solutions or benchmark examples, it has broad application prospects in the field of nuclear power. A lightweight verification method based on metamorphic relation has been produced here. Metamorphic relations are identified from physical equations, numerical algorithms, and program specifications. Next, they are explicitly used to system, integration, and unit tests to improve test adequacy. Because no need to develop verification examples, this method can detect code errors as soon as possible at a low cost, improve test efficiency, avoid mistakes remaining in subsequent stages and reduce the overall cost of verification.

An alternative technique for the symmetry reduction of time‐fractional partial differential equation

Based on the Lie symmetry method and taking the generalized time‐fractional Burgers equations involving the Riemann–Liouville derivative for example, we propose a new similarity transformation which converts the time‐fractional Burgers equation into a fractional ordinary differential equation without containing the Erdélyi–Kober fractional differential operator. Such type of reduced equations facilitates us to study the properties as well as construct analytical solutions. Consequently, we obtain an exact solution and a new power series solution of the time‐fractional Burgers equation via the reduced equation.

Phase Structure of Internal Gravity Waves in the Ocean with Shear Flows

Purpose. The description of the internal gravity waves dynamics in the ocean with background fields of shear currents is a very difficult problem even in the linear approximation. The mathematical problem describing wave dynamics is reduced to the analysis of a system of partial differential equations; and while taking into account the vertical and horizontal inhomogeneity, this system of equations does not allow separation of the variables. Application of various approximations makes it possible to construct analytical solutions for the model distributions of buoyancy frequency and background shear ocean currents. The work is aimed at studying dynamics of internal gravity waves in the ocean with the arbitrary and model distributions of density and background shear currents. Methods and Results. The paper represents the numerical and analytical solutions describing the main phase characteristics of the internal gravity wave fields in the stratified ocean of finite depth, both for arbitrary and model distributions of the buoyancy frequency and the background shear currents. The currents are considered to be stationary and horizontally homogeneous on the assumption that the scale of the currents' horizontal and temporal variability is much larger than the characteristic lengths and periods of internal gravity waves. Having been used, the Fourier method permitted to obtain integral representations of the solutions under the Miles – Howard stability condition is fulfilled. To solve the vertical spectral problem, proposed is the algorithm for calculating the main dispersion dependences that determine the phase characteristics of the generated wave fields. The calculations for one real distribution of buoyancy frequency and shear flow profile are represented. Transformation of the dispersion surfaces and phase structures of the internal gravitational waves’ fields is studied depending on the generation parameters. To solve the problem analytically, constant distribution of the buoyancy frequency and linear dependences of the background shear current on depth were used. For the model distribution of the buoyancy and shear flow frequencies, the explicit analytical expressions describing the solutions of the vertical spectral problem were derived. The numerical and asymptotic solutions for the characteristic oceanic parameters were compared. Conclusions. The obtained results show that the asymptotic constructions using the model dependences of the buoyancy frequency and the background shear velocities’ distribution, describe the numerical solutions of the vertical spectral problem to a good degree of accuracy. The model representations, having been applied for hydrological parameters, make it possible to describe qualitatively correctly the main characteristics of internal gravity waves in the ocean with the arbitrary background shear currents.

Fazovyye kharakteristiki poley vnutrennikh gravitatsionnykh voln v okeane so sdvigom skorosti techeniy

Purpose. The description of the internal gravity waves dynamics in the ocean with background fields of shear currents is a very difficult problem even in the linear approximation. The mathematical problem describing wave dynamics is reduced to the analysis of a system of partial differential equations; and while taking into account the vertical and horizontal inhomogeneity, this system of equations does not allow separation of the variables. Application of various approximations makes it possible to construct analytical solutions for the model distributions of buoyancy frequency and background shear ocean currents. The work is aimed at studying dynamics of internal gravity waves in the ocean with the arbitrary and model distributions of density and background shear currents. Methods and Results. The paper represents the numerical and analytical solutions describing the main phase characteristics of the internal gravity wave fields in the stratified ocean of finite depth, both for arbitrary and model distributions of the buoyancy frequency and the background shear currents. The currents are considered to be stationary and horizontally homogeneous on the assumption that the scale of the currents' horizontal and temporal variability is much larger than the characteristic lengths and periods of internal gravity waves. Having been used, the Fourier method permitted to obtain integral representations of the solutions under the Miles – Howard stability condition is fulfilled. To solve the vertical spectral problem, proposed is the algorithm for calculating the main dispersion dependences that determine the phase characteristics of the generated wave fields. The calculations for one real distribution of buoyancy frequency and shear flow profile are represented. Transformation of the dispersion surfaces and phase structures of the internal gravitational waves’ fields is studied depending on the generation parameters. To solve the problem analytically, constant distribution of the buoyancy frequency and linear dependences of the background shear current on depth were used. For the model distribution of the buoyancy and shear flow frequencies, the explicit analytical expressions describing the solutions of the vertical spectral problem were derived. The numerical and asymptotic solutions for the characteristic oceanic parameters were compared. Conclusions. The obtained results show that the asymptotic constructions using the model dependences of the buoyancy frequency and the background shear velocities’ distribution, describe the numerical solutions of the vertical spectral problem to a good degree of accuracy. The model representations, having been applied for hydrological parameters, make it possible to describe qualitatively correctly the main characteristics of internal gravity waves in the ocean with the arbitrary background shear currents.

Structural analysis of an L-infinity variational problem and relations to distance functions

In this work we analyse the functional ${\cal J}(u)=\|\nabla u\|_\infty$ defined on Lipschitz functions with homogeneous Dirichlet boundary conditions. Our analysis is performed directly on the functional without the need to approximate with smooth $p$-norms. We prove that its ground states coincide with multiples of the distance function to the boundary of the domain. Furthermore, we compute the $L^2$-subdifferential of ${\cal J}$ and characterize the distance function as unique non-negative eigenfunction of the subdifferential operator. We also study properties of general eigenfunctions, in particular their nodal sets. Furthermore, we prove that the distance function can be computed as asymptotic profile of the gradient flow of ${\cal J}$ and construct analytic solutions of fast marching type. In addition, we give a geometric characterization of the extreme points of the unit ball of ${\cal J}$. Finally, we transfer many of these results to a discrete version of the functional defined on a finite weighted graph. Here, we analyze properties of distance functions on graphs and their gradients. The main difference between the continuum and discrete setting is that the distance function is not the unique non-negative eigenfunction on a graph.

Integral transforms for flow and transport in discrete and continuum models of fractured heterogeneous porous media

The Generalized Integral Transform Technique (GITT) is employed in combination with a single domain reformulation strategy and a coupled eigenvalue problem expansion base to construct analytical or hybrid numerical-analytical solutions for flow and transport in fractured porous media. The problem formulation and provided formal solutions encompass both continuum (multi-porosity/multi-permeability) and discrete (fracture-matrix interaction) models for heterogeneous porous media. The single domain representation is written with the aid of space variable coefficients, that account for the abrupt physical properties and source terms transitions across the different subregions. The aim is to provide an unified formalism that includes a wide class of problems in fluid flow, heat and mass transfer in heterogeneous media, with the integral transformation process for the potentials (pressure, velocities, temperature, or concentrations) being applied over the whole physical domain at once, thus markedly reducing the mathematical manipulations. The space variable coefficients are then carried on to the eigenvalue problem formulation which provides the base for the eigenfunction expansion. In addition, a coupled eigenvalue problem alternative is proposed, which is particularly useful in dealing with multi-porosity/multi-permeability flow and transport models, collapsing the multiple coupled potentials equations into one single integral transformation process. Two representative applications are briefly considered, one dealing with contaminant transport with discrete fracture-matrix interaction in layered porous media and the other related to dual porosity/dual permeability model of flow in unsaturated soils.

Optimization Problems Arising from the Design of Pipes Made of Composite Materials

The work is devoted to the construction of analytical solutions for the stress-strain state of a cylindrical hollow elastic rod with a layered structure along the radius. Earlier, the problem of finding the stress-strain state of a rod of composite material fixed at one end with the applied forces and moments of forces at the other end was considered. An approximate representation of the solutions was given, which included auxiliary problems on one fragment of the cylinder, consisting of periodically repeating similar fragments. Such auxiliary problems in the general case do not have an analytical solution. In this paper it is shown that in the presence of radial symmetry of the rod section, it is possible to construct a stress-strain state in an analytical form. In addition, tensile and bending stiffness can be presented in an analytical form. The latter circumstance allows us to set a problem of optimizing the stiffness characteristics of a rod with its fixed weight. Optimization is carried out by varying the thickness of the layers of the same materials.