Exact Solutions Of Equations Research Articles

New Exact Solutions of Landau-Ginzburg-Higgs Equation Using Power Index Method

In the present study, the optimality approach is applied to find the exact solution of the Landau-Ginzburg-Higgs Equation (LGHE) using new transformations. This method is a direct algebraic method for obtaining exact solutions of nonlinear differential equations. We find suitable solutions of the LGHE in terms of elliptic Jacobi functions by applying transformations of basic functions. Exact solutions of the equations are obtained with the help of symbolic software (Maple) which allows the computation of equations with parameter constants. It is exposed that PIM is influential, suitable, and shortest and offers an exact solution of LGHE.

Numerical approach to evaluate the influence of water saturation on the strength of Neogene Tuff in Utsunomiya city, Japan

Evaluation of the deformation behavior due to drying is crucial in underground construction works. Especially, geological repository for High-Level Radioactive Waste (HLW) shall be carefully selected to act as a natural barrier to prevent radioactive nuclide leaks. Uniaxial Compressive Strength (UCS) is one such governing factors when designing or selecting a suitable location for such underground storage compartments or access tunnels. Since nuclide leakage through geological repositories, intact rocks UCS will vary with the drying conditions. Hence, it is vital to evaluate the behavior of UCS subject to the conditions prescribed. A variety of statistical models has been built to govern the interdependency between UCS and water content. However, there are no significant amount of research have been conducted to emphasize the UCS variation with the degree of saturation under the influence of drying condition. Numerically, Richard’s equation exact solution in Neuman boundary condition has been derived to exploit the degree of saturation with respect to time and position. Experimentally, obtain the UCS variation with the degree of saturation pertaining to the Neogene Tuff in Utsunomiya city, Japan. A best fit statistical model was generated for experimental data between UCS variation and the degree of saturation. Then, the experimental saturation degree was replaced by Richard’s equation exact solution saturation degree values. Surprisingly, produced graphs for experimental data and numerical analysis data for exact solution were perfectly overlapped. Results point out that change of UCS shows similar characteristics, which exponentially decrease, with both experimental saturation values and exact solution saturation values influenced the drying condition. Moreover, it suggests that UCS value can be predicted subject to the time and position in the sedimentary rocks under drying conditions.

New explicit solitons for the general modified fractional Degasperis–Procesi–Camassa–Holm equation with a truncated M-fractional derivative

Important analytical methods such as the methods of exp-function, rational hyperbolic method (RHM) and sec–sech method are applied in this paper to solve fractional nonlinear partial differential equations (FNLPDEs) with a truncated [Formula: see text]-fractional derivative (TMFD), which consist of exponential terms. A general modified fractional Degasperis–Procesi–Camassa–Holm equation (GM-FDP-CHE) is investigated with TMFD. The exp-function method is also applied to derive a variety of traveling wave solutions (TWSs) with distinct physical structures for this nonlinear evolution equation. The RHM is used to obtain single-soliton solutions for this equation. The sec–sech method is used to derive multiple-soliton solutions of the GM-FDP-CHE. These techniques can be implemented to find various differential equations exact solutions arising from problems in engineering. The analytical solution of the [Formula: see text]-fractional heat equation is found. Graphical representations are also given.

On exact solutions of equations of rotational motion of a rigid body under action of torque of circular-gyroscopic forces

Abstract. A nonlinear system of differential equations describing the rotational motion of a rigid body under the action of torque of potential and circular-gyroscopic forces is considered. For this torque, the system of differential equations has three classical first integrals: the energy integral, the area integral, and the geometric integral. For the analogue of the Lagrange case, when two moments of inertia coincide and the potential depends on one angle, an additional first integral is found and integration in quadratures is performed. A number of examples is considered where parametric families of exact solutions are considered. In these examples, polynomial or analytical functions were used as a potential. In particular, we construct families of periodic and almost periodic motions, as well as families of asymptotically uniaxial rotations. We also identified movements that have limit values of opposite signs for unlimited increase and decrease of time.

General Fractional Dynamics

General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described.

Exact Analytical Solutions of Generalized Fifth-Order KdV Equation by the Extended Complex Method

The recently introduced technique, namely, the extended complex method, is used to explore exact solutions for the generalized fifth-order KdV equation. Appropriately, the rational, periodic, and elliptic function solutions are obtained by this technique. The 3D graphs explain the different physical phenomena to the exact solutions of this equation. This idea specifies that the extended complex method can acquire exact solutions of several differential equations in engineering. These results reveal that the extended complex method can be directly and easily used to solve further higher-order nonlinear partial differential equations (NLPDEs). All computer simulations are constructed by maple packages.

Variable Coefficient Exact Solutions for Some Nonlinear Conformable Partial Differential Equations Using an Auxiliary Equation Method

The objective of this present paper is to utilize an auxiliary equation method for constructing exact solutions associated with variable coefficient function forms for certain nonlinear partial differential equations (NPDEs) in the sense of the conformable derivative. Utilizing the specific fractional transformations, the conformable derivatives appearing in the original equation can be converted into integer order derivatives with respect to new variables. As for applications of the method, we particularly obtain variable coefficient exact solutions for the conformable time (2 + 1)-dimensional Kadomtsev–Petviashvili equation and the conformable space-time (2 + 1)-dimensional Boussinesq equation. As a result, the obtained exact solutions for the equations are solitary wave solutions including a soliton solitary wave solution and a bell-shaped solitary wave solution. The advantage of the used method beyond other existing methods is that it provides variable coefficient exact solutions covering constant coefficient ones. In consequence, the auxiliary equation method based on setting all coefficients of an exact solution as variable function forms can be more extensively used, straightforward and trustworthy for solving the conformable NPDEs.

Exact Solutions of the Two Dimensional KdV-Burger Equation by Generalized Kudryashov Method

In this article, the generalized Kudryashov method which provides the exact solutions is examined. It is possible to obtain new exact solutions of the nonlinear differential equations with this method. By implementing this developed method to the two-dimensional KdV-Burger equation, new exact solutions of this equation are found. This new exact solutions are the solutions that are not in the literature. In addition, two and three-dimensional graphics of these new exact solutions have been drawn and their physical behavior has been demonstrated.

Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u=u(x,t), these equations contain the same function at a past time, w=u(x,t−τ), where τ>0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments, w=u(px,qt), where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.

Non-isothermal decomposition kinetics of nano-scale CaCO3 as a function of particle size variation

We report the synthesis of nanocrystalline calcium carbonate with varying particle sizes by precipitation techniques from an aqueous solution of calcium nitrate and sodium carbonate at controlled pH. The particle size of the carbonate powder was precisely controlled by changing the precursor concentration. The synthesized carbonate powders were characterized by using scanning electron microscopy, X-ray diffraction technique, and transmission electron microscopy. The particle size, along with the crystallite size of as-synthesized carbonate powder, decreases with increasing precursor concentration. The non-isothermal decomposition kinetics of the carbonate powder was also evaluated by using near to the modified Arrhenius equation's exact solution. The experimental results were best fitted at n = 0.5, and the one-dimensional diffusion-controlled transport process mechanism (D1) and one-dimensional phase boundary movement mechanism (R1) was found to be very close fit of the corresponding evaluated g(α) value. The apparent activation energy of the nano calcium carbonate decomposition was found in the range of 120–175 kJ/mol, which is also inherently functioning with the average particle size. The apparent activation energy of decomposition of CaCO3 found to be decreased with decreasing average particle size of nanocrystalline calcium carbonate.

New exact solutions of a nonlinear integrable equation

Analytic solutions of the partial differential equations are needed to explain many phenomena seen in thermodynamics, aerodynamics, plasma physics, and other fields. In this paper, variational principle is analyzed of the integrable nonlinear Korteweg–de Vries (KdV) typed equation. In addition, exact solutions of this equation are obtained by using various methods such as direct integration, homogeneous balance method, Exp‐function method, and Kudryashov method.

Group analysis and exact solutions of the dynamic equations of plane strain of an incompressible nonlinearly elastic body

Under study is the plane strain dynamics of an incompressible body in the framework of a nonlinear model of elasticity which uses the actual state variables. We obtain some system of nonlinear equations for displacement and pressure, give the exact solutions of the equations by using group analysis, and indicate the area of the possible application of these solutions.

Kink Type Traveling Wave Solutions of Right-handed Non-commutative Burgers Equations via Extended (G0/G)-Expansion Method

The extended (G0 /G)-expansion method is significant for finding the exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. In this paper, we inhanced new traveling wave solutions of right-handed non-commutative burg- ers equations via extended (G0 /G)-expansion. Implementation of the method for searching exact solutions of the equation provided many new solutions which can be used to exploy some practically physical and machanical phenomena. Moreover, when the parameters are re- placed by special values, the well-known solitary wave solutions of the equation rediscovered from the traveling wave solutions and included free parameters may imply some physical meaningful results in fluid mechanics, gas dynamics, and traffic flow.

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Exact Solutions of a Nonclassical Nonlinear Equation of the Fourth Order

For a nonclassical nonlinear partial differential equation of the fourth order, ten families of exact solutions expressed in terms of elementary and special functions are constructed. These results are compared with previously obtained exact solutions of the equation.

Exact Solution of the Equations of Axisymmetric Viscous Fluid Flow between Parallel Plates Approaching and Moving Apart from One Another

The exact solutions of the Navier-Stokes equations in a fluid layer in between parallel plates moving so that the distance varies in accordance with an arbitrary-power law are investigated. The no-slip condition is imposed on the plate boundary. The exact solutions of the Navier-Stokes equations are constructed as series in powers of the Reynolds number. The cases of the decelerated motion in accordance with the time-square-root law, the uniform motion, and the uniformly accelerated motion of the plates are studied in detail. In the first of the cases mentioned above the series converge and in the other cases the solution is determined by means of asymptotic series. The critical Reynolds number which corresponds to the development of backflow is determined.

Exact Solutions of Equations of a Nonstationary Front with Equilibrium Points of a Fractional Order

A family of exact solutions of an evolution equation describing the combustion process in a medium with a power-law temperature dependence of the source density is found. A formal asymptotics of the solution of the initial boundary value problem for the reaction–diffusion equation is constructed. The correctness of the partial sum of an asymptotic series is proved using the method of differential inequalities.

A class of exact solutions of equations for plane steady motion of incompressible fluids of variable viscosity in presence of body force

This paper determines a class of exact solutions for plane steady motion of incompressible fluids of variable viscosity with body force term in the Navier-Stokes equations. The class consists of stream function  characterized by equation , in polar coordinates ,  where ,  and  are continuously differentiable functions, derivative of  is non-zero but double derivative of  is zero. We find exact solutions, for a suitable component of body force, considering two cases based on velocity profile. The first case fixes both the functions ,  and provides viscosity as function of temperature. Where as the second case fixes the function , leaves  arbitrary and provides viscosity and temperature for the arbitrary function . In both the cases, we can create infinite set of expressions for streamlines, viscosity function, generalized energy function and temperature distribution in the presence of body force. Â

On the theory of the Schrödinger equation with the full set of relativistic corrections

All relativistic corrections to the Scr{\"o}dinger equation which determine the interlink between spin and orbit of moving particles, are directly calculated from the Dirac equation using the spin invariant operators. It is shown that among the second order corrections there are not only the well-known Darwin and Thomas terms, but also the new ones. Only with the account of the latter corrections the energies found with the obtained spin-orbit interaction operator, coincide with the energies of the Dirac equation exact solution. The problem of electron spectrum in the quantum well type structures is studied in details and the physical reasons for the appearance of spin-orbit interaction operators in the Dresselhaus or Rashba form, are analyzed.