Explicit Form Of Solution Research Articles

Analytical evaluation of the elastic stresses in a multilayer spherical pressure vessel

An explicit-form solution is constructed for a problem on elastic deformation of a multilayer spherical vessel due to uniform internal pressure. The vessel is assembled of perfectly connected elastic layers whose material properties may arbitrarily vary across their thickness. The solution is constructed through the use of the single solid approach along with the modified scheme of the direct integration method which allow for the consideration of the multilayer vessel as a composite sphere with piecewise-variable profiles of the material properties. As a result, the original problem is reduced to solving a governing integral equation of the second kind whose solution is constructed in the form of the explicit dependence on the internal pressure. The solution is verified by the comparison with exact solutions derived by making use the method of tailored solutions for specific benchmark problems. By comparing our with the one for a functionally-graded sphere, whose material properties are evaluated through the use of the micromodular homogenization technique, the specific features of the circumferential stress are analyzed.

Stability analysis and dynamics of solitary wave solutions of the (3+1)-dimensional generalized shallow water wave equation using the Ricatti equation mapping method

Our aim is to examine the dynamic characteristics of the (3+1)-dimensional generalized equation governing shallow water waves. When the horizontal extent of the fluid significantly surpasses the vertical dimension, the employment of shallow water equations becomes appropriate. By employing an inventive Ricatti equation mapping approach, we have obtained a range of solitary wave solutions in both explicit and generalized forms. Solitons are particularly useful in signal and energy transmission due to their ability to preserve their shape during propagation. By studying soliton solutions in nonlinear systems, we can gain valuable knowledge applicable to practical fields. To deepen our grasp of the equation’s physical implications, we have presented some solutions through 3D, 2D, and contour graphics. Employing a linear stability approach, we confirm the equation’s stability. The method used distinguishes out for its simplicity, dependability, and ability to generate novel solutions for nonlinear partial differential equations in the area of mathematical physics. The research findings reported here demonstrate the viability of the used method in studying nonlinear phenomena in the studied equation as well as other nonlinear problems in mathematical physics. By employing this tool, academics have the chance to deepen and increase their grasp of the complex mathematical concepts underlying real-world problems.

Lipschitz Continuity and Explicit Form of Solution in a Class of Free Boundary Problem with Neumann Boundary Condition

Lipschitz Continuity and Explicit Form of Solution in a Class of Free Boundary Problem with Neumann Boundary Condition

An enormous diversity of soliton solutions to the (2+1)-dimensional extended shallow water wave equation using three analytical methods

In this paper, we obtain a variety of analytical wave solutions of a ([Formula: see text])-dimensional extended shallow water wave equation. The applications of the governing equation are enormous in ocean modeling and investigation of moist-convection properties in atmospheric dynamics. Two powerful mathematical approaches, the [Formula: see text]-expansion method (EEM) and the generalized projective Riccati equations method (GPREM), were used to find closed-form traveling wave solutions. In addition, the breather wave solutions were described using the Cole-Hopf transform, derived with the help of Hirota bilinear method (HBM). Eventually, we obtain 46 closed-form solutions in explicit form. The general form of obtained solutions include trigonometric, rational, exponential and hyperbolic function solutions. To illustrate the physical significance of some novel results, we provided contour, two- and three-dimensional graphics under the suitable free choices of unknown parameters. The dynamical shapes of these solutions are one soliton, multiple solitons, kink, anti-kink, lump and periodic solutions. We believe that the applied methods of this work are well-organized, genuine and powerful mathematical tools for solving the nonlinear evolution equations (NLEEs) occurring in the study of ocean science and engineering.

Classical Darboux transformation and exact soliton solutions of a two-component complex short pulse equation

<abstract><p>This paper investigates soliton solutions to a two-component complex short pulse (c-SP) equation. Based on the known Lax pair representation of this equation, we verify the integrability of a two-component c-SP equation and find an equivalent convenient Lax pair through hodograph transformation. The classical Darboux transformation (DT) is utilized to construct multi-soliton solutions for the two-component c-SP equation as an ordinary determinant. Furthermore, the details of one-soliton and two-soliton solutions are presented and generalized for $ N $-fold soliton solutions. We also derive exact soliton solutions in explicit form using suitable reduction constraints from various "seed" solutions and explore them via graphs.</p></abstract>

Dynamic Cournot Oligopoly Models of the State Promotion of Innovative Electronic Courses in Universities

We built and investigated a two-level dynamic game theoretic model of control of the state promotion of innovative electronic courses in universities based on Cournot oligopoly. A federal state is the Principal, and universities competing a la Cournot are agents. The agents invest in the development of new electronic educative courses that is considered as their innovative investments. The Principal gives subsidies to the agents for the promotion of innovations. The agents play a dynamic game in normal form that results in a Nash equilibrium, and the Principal solves an inverse Stackelberg game (a Germeier game of the type Γ2t). We investigated different types of strategies: (1) uniform strategies for all agents; (2) type-dependent (agent efficiency-dependent) strategies; (3) action-dependent strategies. For a specific form of the model functions we found a solution in explicit form, and in the general case we used a method of qualitatively representative scenarios in simulation modeling. We analyzed the results by means of the individual and collective relative efficiency indices.

New Analytical Method For Solution of Second Order Ordinary Differential Equations With Variable Coefficients

In this study, we propose four methods for the analytical solution of second order ordinary non-homogeneous differential equations with variable coefficients. In the first case, the method directly gives the solution in explicit or integral form. In the second method, the solution of the problem reduces to the solutions of adjoined second order ordinary differential equation of homogeneous type. As long as the analytical solution of two adjoined equations can be solved, the analytical solution can always be found. In the third method, the differential equation is transformed into Riccati equation. Riccati equation is solved by means of a method recently developed. In order to solve non-homogeneous differential equation, the fourth method is developed. The strategy is different but the solution is again based on the solution of Riccati equation.

Analytical studies of Eyring-Powell fluid models

PurposeWhen the literature is reviewed carefully, the analytical solutions of these types of models are missing. First using appropriate similarity transformation, the equations are reduced to dimensionless form (NODE). To solve the reduced models, ansatz-based methods are considered. Finally, the explicit form solutions are obtained and the effects of material parameters and Prandtl number on the velocity and temperature profiles are shown in figures by the exact solutions. This study aims to discuss the aforementioned solution.Design/methodology/approachOne of the non-Newtonian fluids is Eyring-Powell (EP) fluid which is derived from the kinetic theory of fluids. Two variations of EP model are considered to obtain the exact solutions that are missing in the literature. In order to obtain exact solutions, one of the ansatz-based methods is considered. The effects of material parameters and Prandtl number on the velocity and temperature profiles are shown in figures by the exact solutions. The results will guide to develop the model to predict the velocity profile and temperature profile when experimental data for dimensionless material parameters of EP fluid are available.FindingsFinally, the explicit form solutions are obtained and the effects of material parameters and Prandtl number are shown in the figures. The results will guide to develop of the model to predict the velocity profile and temperature profile when experimental data for dimensionless material parameters of EP fluid are available. For the modified EP models, only special cases are considered. The generalized form, i.e. the modified EP models, which include deformation parameters, will be considered in the authors’ future work.Originality/valueWhen the literature is reviewed carefully, the analytical solutions of these types of models are missing so by this work, the gap in the literature is filled. The explicit form solutions are obtained and the effects of material parameters and Prandtl number on the velocity and temperature profiles are shown in figures.

The h-Fourier sine-Laplace discrete generalized convolution on time scale

In this paper, we introduce h-Fourier sine-Laplace discrete generalized convolution. We study a class of generalized convolution transforms and give necessary and sufficient conditions so that the transformations are unitary. As application, we obtain solutions in explicit form of some classes of the Toeplitz plus Hankel-type equations related to the h-Fourier sine-Laplace generalized convolution.

MULTI-TERM TIME-FRACTIONAL DERIVATIVE HEAT EQUATION FOR ONE-DIMENSIONAL DUNKL OPERATOR

In this paper, we investigate the well-posedness for Cauchy problem for multi-term time-fractional heat equation associated with Dunkl operator. The equation under consideration includes a linear combination of Caputo derivatives in time with decreasing orders in (0, 1) and positive constant coefficients and one-dimensional Dunkl operator.  To show solvability of this problem  we use several important properties of multinomial Mittag-Leffler functions and Dunkl transforms, since various estimates follow from the explicit solutions in form of these special functions and transforms. Then we prove the uniqueness and existence results. To achieve our goals, we use methods corresponding to the different areas of mathematics such as the theory of partial differential equations, mathematical physics, hypoelliptic operators theory and functional analysis. In particular, we use the direct and inverse Dunkl transform to establish the existence and uniqueness of solutions to this problem on the Sobolev space. The generalized solutions of this problem are studied.

Multi-term fractional oscillation integro-differential equations

In this paper we study solvability of multi-term Caputo and Riemann-Liouville fractional oscillation integro-differential equations. We show that these equations have unique solutions in the space of functions with square average power growth and derive the solutions in explicit forms.

Invariance properties, exact and explicit solutions of time-fractional Gear–Grimshaw model

In this study, we consider a time-fractional shallow water wave model which was introduced by Gear and Grimshaw. This model represents the dynamics of two-layered shallow water waves in oceans and rivers. Also it covers many shallow water wave models with special values taken to parameters. By performing the group invariant analysis approach for the governing system with Riemann–Liouville derivative, symmetry generators were found. Similarity reduction was constructed by Erdelyi–Kober sense. In addition, by adapting the Sine-Gordon expansion method, we find some exact traveling wave solutions. Furthermore, explicit solutions in form of power series are examined via the residual power series method. Exact and approximate solutions are compared for some [Formula: see text] values and these are given as tables.

The New Wave Structures to the Fractional Ion Sound and Langmuir Waves Equation in Plasma Physics

In this paper, we consider the fractional ion sound and Langmuir waves (FISALWs) equation. We apply the unified solver technique in order to extract some new solutions for the FISALWs equation. The fractional derivative is defined in the sense of a conformable fractional derivative. The proposed solver is based on He’s semi-inverse method and gives beneficial solutions in explicit form. The recital of the method is trustworthy and useful and gives new, more general exact solutions. The constraint conditions for the existence of valid soliton solutions are reported. The enforcement of the presented solutions might be especially interesting in the applications of plasma physics such as bursty waves in cusp regions, Langmuir turbulence, and solar wind. Finally, the proposed solver can be extended to many other models in new physics and applied science.

Global propagation of massive quantum fields in the plane gravitational waves and electromagnetic backgrounds

The behavior of massive quantum fields in the general plane wave spacetime and external, non-plane, electromagnetic waves is studied. The asymptotic conditions, the ‘in’ (‘out’) states and the cross sections are analysed. It is observed that, despite of the singularities encountered, the global form of these states can be obtained: at the singular points the Dirac delta-like behavior emerges and there is a discrete change of phase of the wave function after passing through each singular point. The relations between these phase corrections and local charts are discussed. Some examples of waves of infinite range (including the circularly polarized ones) are presented for which the explicit form of solutions can be obtained. All these results concern both the scalar as well as spin one-half fields; in latter case the change of the spin polarization after the general sandwich wave has passed is studied.

Multiple double-pole bright-bright and bright-dark solitons and energy-exchanging collision in the M-component nonlinear Schrödinger equations.

Multiple double-pole bright-bright and bright-dark soliton solutions for the multicomponent nonlinear Schrödinger (MCNLS) system comprising three types of nonlinearities, namely, focusing, defocusing, and mixed (focusing-defocusing) nonlinearities, arising in different physical settings are constructed. An interesting type of energy-exchanging phenomenon during collision of these double-pole solitons is unraveled. To explore the objectives, we consider the general solutions of a set of generalized MCNLS equations and by taking the long-wavelength limit with proper parameter choices of single-pole bright-bright and bright-dark soliton pairs, the multiple double-pole bright-bright and bright-dark soliton solutions are constructed in terms of determinants. The regular double-pole bright-bright solitons exist in the focusing and focusing-defocusing MCNLS equations and undergo a particular type of energy-sharing collision for M≥2 in addition to the usual elastic collisions. A striking feature observed in the process of energy-sharing collisions is that the double-pole two-soliton possessing unequal intensities before collision indeed exactly exchange their intensities after collision. Further, the existence of double-pole bright-dark solitons in the MCNLS equations with focusing, defocusing, and mixed (focusing-defocusing) nonlinearities is analyzed by constructing explicit determinant form solutions, where the double-pole bright solitons exhibit elastic and energy-exchanging collisions while the double-pole dark solitons undergo mere elastic collision. The double-pole bright-dark solitons possess much richer localized coherent patterns than their counterpart double-pole bright-bright solitons. For particular choices of parameters, we demonstrate that the solitons would degenerate into the background, resulting in a lower number of solitons. Another important observation is the formation of doubly localized rogue waves with extreme amplitude, in the case of double-pole bright-dark four-solitons. Our results should stimulate interest in such special multipole localized structures and are expected to have ramifications in nonlinear optics.

Solution of the problems of quasi-statics for an elastic body with double porosity

The construction of solutions in explicit form is especially important from the point of view of its application, since it makes it possible to effectively carry out a quantitative analysis of the problem under study. This paper investigates the processes of deformation of solids in the quasi-static case. Two-dimensional boundary value problems of Dirichlet and Neumann for an elastic body with double porosity are considered. In Using the Laplace transform, these problems are reduced to auxiliary boundary value problems. Special representations of solutions to auxiliary boundary value problems are constructed using elementary functions that allow reducing the original system of equations to equations of a simple structure and facilitate the solution of the original problems. Auxiliary boundary value problems are solved for a specific elastic body - a porous disk. Solutions to these problems are obtained in the form of series. Conditions are provided that ensure the absolute and uniform convergence of these series and the use of the inverse Laplace theorem. It is proved that the inverse transforms provide a solution to the initial problems.

Solution of the optimal control problem by symbolic regression method

The paper introduces an approach to solve the optimal control problem applying symbolic regression methods. All known approaches to solve the optimal control problem are based on direct or indirect methods. Indirect methods may provide analytical solution but impose restrictions on the form and dimension of control object. In direct ones the optimal control problem is reduced to the nonlinear programming problem which provides general numerical solution. Also direct methods impose no restrictions on control object and have a wide range of application to complex applied problems. The approach proposed combines the advantages of both direct and indirect methods not involving the disadvantages mentioned previously. The suggested approach provides solution to the optimal control problem as a control function of time in explicit form. This approach benefits in case of applied optimal control problem requiring a solution in explicit form while indirect methods are inapplicable due to the restrictions imposed on the control system. The computational experiment of solving the optimal control problem for a mobile robot with phase constraints is presented. The results illustrate that the proposed approach finds an optimal control as a function of time which provides a mobile robot’s motion along the optimal trajectory avoiding phase constraints.

Construction of exact solutions to nonlinear PDEs with delay using solutions of simpler PDEs without delay

We describe new indirect methods for constructing exact solutions of nonlinear PDEs with delay. Features of the proposed methods are illustrated by nonlinear delay reaction-diffusion and wave-type equations with variable coefficients. All the presented equations contain from three to seven arbitrary functions, which depend on the spatial variable and/or unknown quantity. We obtain new generalized traveling-wave solutions and functional separable solutions in explicit or implicit form for equations with constant delay as well as exact solutions with time-varying delay (a special case of which are pantograph-type PDEs with proportional delay). We show that the proposed methods can also be applied to construct exact solutions for nonlinear systems of coupled delay PDEs and higher-order delay PDEs. The considered equations and their exact solutions can be used to formulate test problems to check the adequacy and estimate the accuracy of numerical and approximate analytical methods of solving nonlinear initial-boundary value problems for delay PDEs.

Holomorphic Solutions to Linear q-Difference Equations in a Banach Space

We prove the existence and uniqueness of a holomorphic solution to a linear q-difference equation for a closed operator with not necessarily dense domain in a Banach space. We consider holomorphic solutions in a neighborhood of zero in the case |q| 1. We obtain the solutions in an explicit form.

On boundary value problems of the Samarskii–Ionkin type for the Laplace operator in a ball

In this paper, we consider nonlocal boundary value problems for the Laplace operator in a ball, which are a multidimensional generalisation of the Samarskii–Ionkin problem. The well-posedness of the problems are investigated, and Fredholm property of the problems are studied. Moreover, we obtain integral representations of their solutions in explicit forms.