Dimensional Equation Research Articles

Investigation of soliton solutions to the truncated M-fractional (3+1)-dimensional Gross-Pitaevskii equation with periodic potential

<p>This research explores some modernistic soliton solutions to the (3+1)-dimensional periodic potential the Gross–Pitaevskii equation with a truncated M-fractional derivative plays a significant role in Bose–Einstein condensation, which describes the dynamics of the condensate wave function. The obtained results include trigonometric, hyperbolic trigonometric and exponential function solutions. Three techniques named: the $ \exp_a $ function method, the Sardar sub-equation method, and the extended $ (G'/G) $-expansion approach are employed to achieve a variety of new solutions for the governing model. More comprehensive information about the dynamical representation of some of the solutions is being presented by visualizing the 2D, 3D and contour plots. This work reveals a number of new types of traveling-wave solutions, such as the double periodic singular, the periodic singular, the dark singular, the dark kink singular, the periodic solitary singular, and the singular soliton solutions. These novel solutions are not the same as those that were previously studied for this governing equation. The presented techniques demonstrate clarity, efficacy, and simplicity, revealing their relevance to diverse sets of dynamic and static nonlinear equations pertaining to evolutionary events in computational physics, in addition to other real-world applications and a wide range of study fields for addressing a variety of other nonlinear fractional models that hold significance in the fields of applied science and engineering.</p>

Global existence of weak solutions to a class of higher-order nonlinear evolution equations

<p>This paper deals with the initial boundary value problem for a class of <italic>n</italic>-dimensional higher-order nonlinear evolution equations that come from the viscoelastic mechanics and have no positive definite energy. Through the analysis of functionals containing higher-order energy of motion, a modified potential well with positive depth is constructed. Then, using the potential well method, and Galerkin method, it has been shown that when the initial data starts from the stable set, there exists a global weak solution to such an evolution problem.</p>

Lie symmetries, modulation instability, conservation laws, and the dynamic waveform patterns of several invariant solutions to a (2+1)-dimensional Hirota bilinear equation

Lie symmetries, modulation instability, conservation laws, and the dynamic waveform patterns of several invariant solutions to a (2+1)-dimensional Hirota bilinear equation

A Non-Local wave equation with General fractional derivatives and time delay

We analyze, spatially one dimensional, non-local elastic body with deformation measure given in the form of General fractional derivative of Riesz type with truncated power-law kernel. It is assumed that stress at the point x and time t depends on the strain at point ξ at time t−|x−ξ|/c, where c is a constant having dimension of time. We formulate spatially one dimensional wave equation for such a body and show the influence of the parameter c on the solution.

Elzaki projected differential transform method for multi-dimensional aggregation and combined aggregation-breakage equations

Numerous real-world fields, including planetary science, bio-pharmaceutical, chemical study, food processing industry, and many more are profoundly impacted by population balance equations. Model complexity limits the analytical investigations to a few aggregation-breakage parameters, although various numerical and semi-analytical schemes are available. This article proposes a new semi-analytical approach, the Elzaki integral transform as a pre-treatment to reinforce domain decomposition for better accuracy and convergence, in conjunction with the projected differential transform method for finding the closed form or approximated series solutions for non-linear aggregation, aggregation-breakage, and multi-dimensional aggregation equations. The technique’s key benefit over traditional numerical methods is its ability to solve linear or non-linear differential equations directly without discretization or linearization. Theoretical convergence analysis and error estimates of series solutions for both one and two dimensional aggregation models are of particular interest. Finally, several numerical examples of aggregation, aggregation-breakage, and two dimensional aggregation equations are taken to validate the accuracy of the proposed method by comparing numerical simulations with exact solutions. Interestingly, we have obtained closed form solutions for the pure aggregation equation when considering constant and product aggregation kernels. Additionally, error tables and graphs help to highlight the method’s innovative nature.

Peristaltic flow of Williamson nanofluid on a rough surface

The aim of this current research is to investigate the peristaltic flow of Williamson nanofluid across a rough surface in a non-uniform channel under the influence of inclined magnetic field. The Joule heating and viscous dissipation effects are also retained in the current scrutiny. The objective of studying peristaltic flow of Williamson nanofluid on a rough surface is to gain insights into the complex fluid dynamics and heat transfer phenomena occurring in such systems. This knowledge can be used to design more efficient and effective nanofluid-based devices and processes. In the context of mathematical modeling, the appropriate dimensional nonlinear equations for momentum, heat and mass transport are simplified into dimensionless equation by applying the essential estimation of long wavelength and low Reynolds number. The equations subjected to boundary conditions have solved numerically by the Mathematica software built-in numerical Solver ND_solve method. Various essential physical characteristics on velocity, temperature and concentration are presented graphically in the end. It can be seen that fluid velocity decreases at the central part of the channel for the escalting values Hartman number M. As Darcy number Da increases then velocity profile increases at the core part of the channel and the walls of the channel experiencing an opposite behavior. It is noticed that Higher value of Eckert number Ec enhances the temperature profile. When Weissenberg number We gets stronger then temperature profile decreases. It is observed that the temperature and concentration profiles show an opposite behavior for the rising values of thermophoresis parameter [Formula: see text].

(3+1)-Dimensional Gardner Equation Deformed from (1+1)-Dimensional Gardner Equation and its Conservation Laws

Through the application of the deformation algorithm, a novel (3+1)-dimensional Gardner equation and its associated Lax pair are derived from the (1+1)-dimensional Gardner equation and its conservation laws. As soon as the (3+1)-dimensional Gardner equation is set to be y or z independent, the Gardner equations in (2+1)-dimension are also obtained. To seek the exact solutions for these higher dimensional equations, the traveling wave method and the symmetry theory are introduced. Hence, the implicit expressions of traveling wave solutions to the (3+1)-dimensional and (2+1)-dimensional Gardner equations, the Lie point symmetry and the group invariant solutions to the (3+1)-dimensional Gardner equation are well investigated. In particular, after selecting some specific parameters, both the traveling wave solutions and the symmetry reduction solutions of hyperbolic function form are given.

Asymptotic model of long-wave vibrations of an ultrathin strip-beam taking into account surface effects

The paper is concerned with the derivation of asymptotically consistent equations governing long-wave vibrations of an ultrathin elastic strip-beam taking into account surface effects within the framework of the Gurtin-Murdoch theory of the surface elasticity. Two dimensional equations of motion of an elastic isotropic medium are used as the initial ones. In the general case, the strip-beam is under the action of variable unsteady surface forces. The presence of residual shear stresses is assumed on the face surfaces. The ratio of the strip thickness to the characteristic length of bending deformation is considered as a small parameter. Within the framework of the Gurtin-Murdoch theory, two cases are considered, providing for the presence of large a) residual stresses on the faces, b) effects of surface inertia. Using the method of asymptotic integration over the transverse direction, relations for displacements and stresses in an ultra-thin strip-beam are obtained, and equivalent one dimensional the Timoshenko type equations taking into account surface effects are derived. As an example, free vibrations of a simply supported beam accounting for the surface effects are considered.

New Exact Traveling Wave Solutions of the Non-linear (2+1)-dimensional Klein-gordon Equation

In this article, we discuss the nonlinear (2+1)-dimensional Klein-Gordon equation with an additional term. The functional variable method is used to construct exact solutions of the quadratic and cubic nonlinear (2+1)-dimensional Klein-Gordon equation. The exact solutions of these equations including soliton and periodic wave solutions are obtained. The advantage of the used method beyond other existing methods is that it provides more new exact solutions. Some selected solutions of the equations are presented graphically by Matlab program. This method is efficient and it can be successfully used to obtain another nonlinear wave equations in mathematical physics and engineering.

Soliton interactions, soliton bifurcations and molecules, breather molecules, breather-to-soliton transitions, and conservation laws for a nonlinear (3+1)-dimensional shallow water wave equation

Soliton interactions, soliton bifurcations and molecules, breather molecules, breather-to-soliton transitions, and conservation laws for a nonlinear (3+1)-dimensional shallow water wave equation

Different lump k-soliton solutions to (2+1)-dimensional KdV system using Hirota binary Bell polynomials

Abstract In this article, the (2+1)-dimensional KdV equation by Hirota’s bilinear scheme is studied. Besides, the binary bell polynomials and then the bilinear form is created. In addition, an interaction lump with k k -soliton solutions of the addressed system with known coefficients is presented. With the assistance of the stated methodology, a cloaked form of an analytical solution is discovered in expressions of lump-soliton rational functions with a few lovable parameters. Solutions to this study’s problems are identified specifically as belonging to the lump-one, two, three, and four soliton solutions. By defining the specific advantages of the epitomized parameters by the depiction of figures and by interpreting the physical occurrences are established acceptable soliton arrangements and dealt with the physical importance of the obtained arrangements. Finally, under certain conditions, the physical behavior of solutions is analyzed by using the mentioned method. Moreover, the graphs with high resolutions including three-dimensional plots, density plots, and two-dimensional plots to determine a deep understanding of plotted solutions that will arise in the applied mathematics and nonlinear physics are employed.

Numerical Solution for the Heat Equation Using Crank-Nicolson Difference Method

Abstract: In this paper, the Crank-Nicolson difference method is applied to a simple problem involving one dimensional heat equation. Numerical solution to the heat equation using Crank-Nicolson difference equation is obtained. The method of solving the problem is implemented by using Python Programming. We have discussed the local truncation error and stability analysis of Crank-Nicolson difference method difference method of heat equation.

Lump solution, interaction solution, and interference wave for the (3+1)-dimensional BKP-Boussinesq equation as well as analysis of BNNM model degradation

Lump solution, interaction solution, and interference wave for the (3+1)-dimensional BKP-Boussinesq equation as well as analysis of BNNM model degradation

Well-posedness and finite element approximation of mixed dimensional partial differential equations

In this article, a mixed dimensional elliptic partial differential equation is considered, posed in a bulk domain with a large number of embedded interfaces. In particular, well-posedness of the problem and regularity of the solution are studied. A fitted finite element approximation is also proposed and an a priori error bound is proved. For the solution of the arising linear system, an iterative method based on subspace decomposition is proposed and analyzed. Finally, numerical experiments are presented and rapid convergence using the proposed preconditioner is achieved, confirming the theoretical findings.

Lie symmetry analysis, numerical solutions by spectral method and travelling wave solutions of a (5+1)-dimensional partial differential equation

ABSTRACT The purpose of this research is to investigate the Lie symmetries and the numerical solution of the (5 + 1)-dimensional evolutionary differential equation using the Chebyshev spectral method. Also, the exact solution of this equation is presented by the traveling wave method. These equations play a very important role in mathematical physics and engineering sciences. The implemented algorithm is quite efficient and practically suitable for these problems. The use of Maple and MATLAB software allows us to perform complex and boring calculations. The most distinctive aspect of this method is that no prescribed assumptions are required, and enormous computational effort is reduced and rounding errors are avoided. Our proposed methods for checking and solving the considered equation show that this method is very accurate and practical.

Double-Periodic Soliton Solutions of the (2+1)-Dimensional Ito Equation

In this work, a (2 + 1)-dimensional Ito equation is investigated, which represents the generalization of the bilinear KdV equation. Abundant double-periodic soliton solutions to the (2 + 1)-dimensional Ito equation are presented by the Hirota bilinear form and a mixture of exponentials and trigonometric functions. The dynamic properties are described through some 3D graphics and contour graphics.

Heat Transfer Enhancement of Convective Casson Nanofluid Flow by CNTs over Exponentially Accelerated Plate

Carbon nanotubes (CNTs) nanofluids are gaining increased popularity among researchers due to their outstanding thermal properties, leading to numerous promising industrial applications. Analytical solutions discovered in the study of CNTs nanofluids, combined with a Casson-type fluid model, are extremely limited. Therefore, a study on the heat transfer analysis of an unsteady and incompressible Casson carbon nanofluid flow is conducted. Human blood-based single-walled carbon nanotubes (SWCNTs) and human blood-based multi-walled carbon nanotubes (MWCNTs) are considered as nanofluids that move beyond an exponentially accelerated vertical plate. A set of dimensional momentum and energy equations, along with their initial and exponentially accelerated boundary conditions, is employed to represent the problem. The transformation of these equations to the dimensionless expression is achieved by using suitable dimensionless variables. The resulting equations are then tackled using Laplace transformation to acquire the analytical solution for temperature and velocity. Figures and tables are produced for a further analysis of temperature and velocity characteristics. The study shows that an increase in nanoparticle volume fraction enhances nanofluid flow and heat transmission, proving highly beneficial for cancer treatment. However, the flow is retarded due to the increment of Casson parameter values, while an enhancement is observed with a superior accelerating parameter.

Exact solutions, symmetry groups and conservation laws for some (2+1)-dimensional nonlinear physical models

In this paper, the Bernoulli sub-equation function method is used to construct new exact travelling wave solutions for two important physical models: (2+1)-dimensional hyperbolic nonlinear Schrödinger (HNLS) equation and (2+1)-dimensional Heisenberg ferromagnetic spin chain (HFSC) equation. These solutions provide valuable insights into the behavior of these models, described in terms of exponential and hyperbolic tangent (tanh) functions. The study also involves an exploration of the infinitesimal generators and symmetry groups through the Lie symmetry method. In addition, by using multiplier approach, the conservation laws are established for these models. Graphical simulation of some solutions in the form of two-dimensional and three-dimensional are plotted to understanding of the underlying physical phenomena and mathematical properties of the (2+1)-dimensional HNLS and HFSC equations. The solutions and graphing are performed using Maple software.

KERNEL DETERMINATION PROBLEM FOR ONE PARABOLIC EQUATION WITH MEMORY

This paper studies the inverse problem of determining a multidimensional kernel function of an integral term which depends on the time variable \(t\) and \((n-1)\)-dimensional space variable \(x'= \left(x_1,\ldots, x_ {n-1}\right)\) in the \(n\)-dimensional diffusion equation with a time-variable coefficient at the Laplacian of a direct problem solution. Given a known kernel function, a Cauchy problem is investigated as a direct problem. The integral term in the equation has convolution form: the kernel function is multiplied by a solution of the direct problem's elliptic operator. As an overdetermination condition, the result of the direct question on the hyperplane \(x_n = 0\) is used. An inverse question is replaced by an auxiliary one, which is more suitable for further investigation. After that, the last problem is reduced to an equivalent system of Volterra-type integral equations of the second order with respect to unknown functions. Applying the fixed point theorem to this system in Hölder spaces, we prove the main result of the paper, which is a local existence and uniqueness theorem.

What is a real black hole?

Abstract Four dimensional Einstein equation with non-vanishing energy momentum tensor is solved to give a solution where the event horizon of a celestial body is outside of the body, which corresponds to a black hole by definition. There is no singularity at the center of the object unlike the case of Schwarzschild solution. We do not have the gravitational collapse. We do not have the gravitational collapse thus avoiding the Penrose-Hawking singularity theorems. The main feature of our black hole solution is that it has the horizon inside the black hole in addition to the usual outside horizon. This inner horizon singularity is not removable by a coordinate transformation because the determinant of metric tensor is singular at this point but there is no singularity neither in the density nor in the pressure indicating there is no gravitational collapse.