3D Graphical Representation Research Articles

Impact of White Noise on the Exact Solutions of the Stochastic Riemann Wave Equation in Quantum Mechanics

In this article, the stochastic Riemann wave equation (SRWE) forced by white noise in the Itô sense is considered. The extended tanh function and mapping methods are applied to obtain new elliptic, rational, hyperbolic, and trigonometric stochastic solutions. Furthermore, we generalize some previous studies. The obtained solutions are important in explaining some exciting physical phenomena, since the SRWE is required for describing wave propagation. We plot numerous 3D and 2D graphical representations to explain how the multiplicative white noise influences the exact solutions of the SRWE. We can infer that the introduction of multiplicative white noise disrupts the symmetry of the solutions and serves to stabilize the solutions of the SRWE.

Analysing soliton dynamics and a comparative study of fractional derivatives in the nonlinear fractional Kudryashov’s equation

In this scholarly exploration, we employ new mapping method to unveil new soliton solutions to the nonlinear fractional Kudryashov’s equation, using β-derivative and M-Truncated fractional derivatives. Soliton phenomena, invaluable for enhancing computational capabilities in computer systems, find particular utility in tasks like data analysis, image processing, and simulations across various computer science domains. Our research reveals diverse solution forms, encompassing dark, periodic, kink, singular, dark-bright, and bright solitons. These versatile soliton types offer adaptable tools for both modelling and simulation. To enhance understanding, we provide 3D and 2D graphical representations, which facilitate a comparative analysis of solutions derived from the two fractional derivatives. Additionally we compare the effects of different fractional derivative operators (β-derivative and M-Truncated derivative) involve in the nonlinear fractional Kudryashov’s equation. The incorporation of both derivatives allows for a broader exploration of solution forms and enhance the understanding of the equation’s behaviour and its solution. Our work bridges soliton physics and computer science, underscoring the significance of soliton phenomena in advancing computational modelling and simulations. This research contributes to the intersection of theoretical mathematics and practical computer science, highlighting the vital role of solitons in scientific fields.

A novel investigation of extended (3+1)-dimensional shallow water wave equation with constant coefficients utilizing bilinear form

The aim of this research is to investigate new soliton solutions for the shallow water wave equation in an extended (3+1) dimensional space with constant coefficients. The application of this equation is relevant to the analysis of water body dynamics in seas and oceans. Bäcklund transform for the system is constructed from the bilinear form and rational as well exponential traveling wave solutions are analyzed to obtain kink and singular kink solitons. The Hirota bilinear form is further utilized to construct complexiton solutions through extended transformed rational function method. Appropriate parameter values are selected to create 3D and 2D graphical representations, as well as corresponding density plots, for the physical characterization of certain reported results. The obtained results of this research demonstrate how to use discussed approaches for addressing various real world problems described by nonlinear models.

Comparative study of thermal effect on thermoelastic double porous solid by finite element method and finite-infinite element method

In the present study, we have made a comparative study between the well-established finite element method (FEM) and finite-infinite element method (FIEM). Nine-noded infinite elements have been developed to model the half-space. Further, we assume a thermal shock which is applied on the double porosity medium. This medium is assumed to be unbounded in terms of x and y directions. For sake of simplicity, we have converted our model equations in non-dimensional form. Then after, suitable boundary conditions have been imposed in the present study. Numerical validation of the present mathematical model has been made for copper-like material. Displacement components of solid particles, volume fractions fields of double porosity solid, temperature field and stress components have been computed for the domain . Due to limitation and a better understanding of the comparative study between these two methods, we are giving 2D-graphical representation of field variables with respect to x-space for fixed value of y=1 and 3D-graphical representation for fixed value of x = 1, respectively. From the results, we find advantage of use of infinite element method over finite element method.

Multiplicative Reverse Product Connectivity and Multiplicative Reverse Sum Connectivity of Silicate Network

The connectivity indices are helpful to estimate the chemical characteristics of the compounds in chemical graph theory. This report introduces the multiplicative reverse product connectivity index and the multiplicative sum connectivity index of the silicate network. Further, there 2D and 3D graphical representations are plotted.

New fractal–fractional parametric inequalities with applications

In the present paper, we first establish a general parameterized identity for local fractional twice differentiable functions involving extended fractal–fractional integral operators. Thus by employing generalized convexity on differentiable mappings along with Yang’s Power-mean, Hölder’s and improved fractal integral inequalities lead us to develop variety of new fractal–fractional parameterized inequalities. Several examples are provided with graphical illustrations to prove the validity of new results. We give error analysis of improved bounds numerically and also by 2D, 3D graphical representations. Finally, we show that our main results recapture fractal variants of trapezoid, midpoint, Simpson and Bullen-type inequalities. Some related applications to the fractal means, moment of random variables and wave equations are given as well.

A new implementation of a novel analytical method for finding the analytical solutions of the (2+1)-dimensional KP-BBM equation

In this work, we perform a comprehensive analytical study to find the novel exact traveling wave solutions of the (2+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation. The recently developed (G′G′+G+A) -expansion technique is a capable method for finding the new exact solutions of assorted nonlinear evolution equations. Some new analytical solutions are obtained by utilizing the aforementioned method. The obtained solutions are expressed as trigonometric functions and exponential functions. The extracted exact wave solutions are advanced and fully unique from the earlier literature Moreover, we have presented the contour simulations, 2D and 3D graphical representations of the solution functions and we have observed that the solutions obtained are periodic and solitary wave solutions. We have shown two soliton wave solutions and two singular periodic wave solutions for the particular values of the parameters graphically. As per our knowledge, we must say that the extracted solutions might be significant and essential for new physical phenomenon.

Novel analytical solutions of stochastic Ginzburg-Landau equation driven by Wiener process via the improved modified extended tanh function method

In this manuscript, the improved modified extended tanh integration technique is implemented to investigate the exact solutions of stochastic Ginzburg–Landau model driven by Wiener process which appeared in various fields of chemistry, mathematics and physics. Many types of stochastic wave solutions are raised such as hyperbolic stochastic solutions, trigonometric stochastic solutions, Weierstrass elliptic stochastic solutions and Jacobi elliptic stochastic solutions. Moreover, 3D and 2D graphical representations are depicted to show the effect of the multiplicative noise on the propagated waves. The results reveal that as the noise level increases, the wave degrades. The wave is distorted when σ⩾1. In addition, the 3D graphical representation show that the propagated wave is flattered when the noise strength is increased. These solutions may be applicable in various applications in applied science. The proposed method is efficient, direct and powerful.

Solitary Wave Solutions of the Fractional-Stochastic Quantum Zakharov–Kuznetsov Equation Arises in Quantum Magneto Plasma

In this paper, we consider the (3 + 1)-dimensional fractional-stochastic quantum Zakharov–Kuznetsov equation (FSQZKE) with M-truncated derivative. To find novel trigonometric, hyperbolic, elliptic, and rational fractional solutions, two techniques are used: the Jacobi elliptic function approach and the modified F-expansion method. We also expand on a few earlier findings. The extended quantum Zakharov–Kuznetsov has practical applications in dealing with quantum electronpositron–ion magnetoplasmas, warm ions, and hot isothermal electrons in the presence of uniform magnetic fields, which makes the solutions obtained useful in analyzing a number of intriguing physical phenomena. We plot our data in MATLAB and display various 3D and 2D graphical representations to explain how the stochastic term and fractional derivative influence the exact solutions of the FSEQZKE.

The soliton solutions for the (4 + 1)‐dimensional stochastic Fokas equation

In this paper, we look at the (4 + 1)‐dimensional stochastic Fokas equation (SFE) perturbed in the Itô sense by white noise. The tanh‐coth method and mapping method are used to acquire new trigonometric, hyperbolic, elliptic, and rational stochastic solutions. Also, we extend some earlier studies. Because the SFE equation is essential for describing different phenomena in ocean engineering, fluid mechanics and optics, the solutions obtained are useful in interpreting certain fascinating physical phenomena. To demonstrate how the multiplicative white noise affects the exact solutions of the SFE, we plot our figures in MATLAB and show several 3D graphical representations. We demonstrate how the solutions of SFE are stabilized by multiplicative white noise.

Solitary Wave Solutions for the Stochastic Fractional-Space KdV in the Sense of the M-Truncated Derivative

The stochastic fractional-space Korteweg–de Vries equation (SFSKdVE) in the sense of the M-truncated derivative is examined in this article. In the Itô sense, the SFSKdVE is forced by multiplicative white noise. To produce new trigonometric, hyperbolic, rational, and elliptic stochastic fractional solutions, the tanh–coth and Jacobi elliptic function methods are used. The obtained solutions are useful in interpreting certain fascinating physical phenomena because the KdV equation is essential for understanding the behavior of waves in shallow water. To demonstrate how the multiplicative noise and the M-truncated derivative impact the precise solutions of the SFSKdVE, different 3D and 2D graphical representations are plotted.

Optical soliton solutions for the integrable Lakshmanan-Porsezian-Daniel equation via the inverse scattering transformation method with applications

The Lakshmanan-Porsezian-Daniel equation (LPD) is a nonlinear Schrödinger type (NLS) equation which describe the effect on the integrable properties of Heisenberg bilinear spin chains under the classical limit by the biquadratic interactions. In this paper, the integrability of LPD equation is illustrated by obtaining the Lax pair via the AKNS scheme. The exact soliton solutions via the inverse scattering transformation (IST) method are obtained and illustrated in 3D graphical representation.

The Analytical Solutions of the Stochastic mKdV Equation via the Mapping Method

Here, we analyze the (2+1)-dimensional stochastic modified Kordeweg–de Vries (SmKdV) equation perturbed by multiplicative white noise in the Stratonovich sense. We apply the mapping method to obtain new trigonometric, elliptic, and rational stochastic fractional solutions. Because of the importance of the KdV equation in characterizing the behavior of waves in shallow water, the obtained solutions are beneficial in interpreting certain fascinating physical phenomena. We plot our figures in MATLAB and show several 3D and 2D graphical representations to show how the multiplicative white noise affects the solutions of the SmKdV. We show that the white noise around zero stabilizes SmKdV solutions.

Theoretical Analysis of Superphenalene Using Different Kinds of VDB Indices

Topological indices (TIs) are numerical quantities that enable theoretical chemists to analyse molecular structures mathematically. These TIs are essential to exploring chemical compounds using theoretical techniques like QSAR/QSPR methods. Superphenalene is a large polycyclic aromatic hydrocarbon molecule which has been quickly gaining importance as a building block for alternate energy providers due to its photovoltaic properties. The exciting features of this compound, coupled with its potential applications, warrant an investigation of its nature and properties from a structural perspective. The objective of this research is to compute the proper analytical expressions of four kinds of vertex degree-based (VDB) indices for superphenalene. The numerical values of these indices and 3D graphical representations also help in understanding the relationship between the VDB indices of the compound and its underlying chemical structure quantitatively.

Optical solitons and qualitative analysis of nonlinear Schrodinger equation in the presence of self steepening and self frequency shift

In this manuscript, the optical solitons of nonlinear Schrödinger equation (NLSE) with cubic–quintic law nonlinearity, in the presence of self-frequency shift and self-steepening, has been studied. The ultrahigh capacity propagation and transit of laser light pulses in optical fibres were described using this form of equation. To extract new results, two strong methodologies has been used. To extract the exact solution unified method has been employed. The solutions obtained by this analytical method, are in form of polynomial and rational function solution. Moreover, the validity of non-singular solutions has guaranteed by a limitation condition that is graphically illustrated in 3D. The 2D graphical representation are also used to demonstrate the influence of parameters on the predicted non-singular solutions. The other technique, used for qualitative analysis, is bifurcation. The system has been transformed into a planer dynamical system, which has been transformed into a hamiltonian system. All the possible phase portrait has been plotted by complete discrimination method. The acquired results are novel and have not been recorded before and they indicate that the proposed methodologies may be used to investigate innovative soliton solutions and phase portraits for any NLSE.

Soliton solutions for nonlinear variable-order fractional Korteweg–de Vries (KdV) equation arising in shallow water waves

Nonlinear fractional differential equations provide suitable models to describe real-world phenomena and many fractional derivatives are varying with time and space. The present study considers the advanced and broad spectrum of the nonlinear (NL) variable-order fractional differential equation (VO-FDE) in sense of VO Caputo fractional derivative (CFD) to describe the physical models. The VO-FDE transforms into an ordinary differential equation (ODE) and then solving by the modified (G′/G)-expansion method. For accuracy, the space-time VO fractional Korteweg-de Vries (KdV) equation is solved by the proposed method and obtained some new types of periodic wave, singular, and Kink exact solutions. The newly obtained solutions confirmed that the proposed method is well-ordered and capable implement to find a class of NL-VO equations. The VO non-integer performance of the solutions is studied broadly by using 2D and 3D graphical representation. The results revealed that the NL VO-FDEs are highly active, functional and convenient in explaining the problems in scientific physics.

Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method

The fractional massive Thirring model is a coupled system of nonlinear PDEs emerging in the study of the complex ultrashort pulse propagation analysis of nonlinear wave functions. This article considers the NFMT model in terms of a modified Riemann–Liouville fractional derivative. The novel travelling wave solutions of the considered model are investigated by employing an effective analytic approach based on a complex fractional transformation and Jacobi elliptic functions. The extended Jacobi elliptic function method is a systematic tool for restoring many of the well-known results of complex fractional systems by identifying suitable options for arbitrary elliptic functions. To understand the physical characteristics of NFMT, the 3D graphical representations of the obtained propagation wave solutions for some free physical parameters are randomly drawn for a different order of the fractional derivatives. The results indicate that the proposed method is reliable, simple, and powerful enough to handle more complicated nonlinear fractional partial differential equations in quantum mechanics.

High-Definition Survey of Architectural Heritage Fusing Multisensors—The Case of Beamless Hall at Linggu Temple in Nanjing, China

Following the development of digital measurement technology in recent years, the information contained in the measurement outcomes have become increasingly rich. However, the traditional graphical representation method based on vector graph needs to be updated. In this study, we use the Beamless Hall of Linggu Temple as an example. Measurements are conducted by using digital techniques, including three-dimensional (3D) laser scanning, close-range photogrammetry, and infrared thermal imaging. The pseudocolours that express spatial information and moisture distribution are calculated and generated through point clouds, which are used to express the land subsidence, wall deformation, moisture distribution, and other effects of the Beamless Hall. Furthermore, combining it with two-dimensional (2D) graphical representation, such as the plan, elevation, and section, damage-related information can be expressed intuitively and efficiently. This method can combine the advantages of graphics and images to provide a comprehensive and intuitive representation of the digital measurement results of brick architecture heritage. It can also provide a reference for surveying similar monuments and buildings of our architectural heritage.

The Influence of Noise on the Solutions of Fractional Stochastic Bogoyavlenskii Equation

We look at the stochastic fractional-space Bogoyavlenskii equation in the Stratonovich sense, which is driven by multiplicative noise. Our aim is to acquire analytical fractional stochastic solutions to this stochastic fractional-space Bogoyavlenskii equation via two different methods such as the exp(−Φ(η))-expansion method and sine–cosine method. Since this equation is used to explain the hydrodynamic model of shallow-water waves, the wave of leading fluid flow, and plasma physics, scientists will be able to characterize a wide variety of fascinating physical phenomena with these solutions. Furthermore, we evaluate the influence of noise on the behavior of the acquired solutions using 2D and 3D graphical representations.

Topological Properties of Sierpinski Network and its Application.

Sierpinski graphs S(n, k) are largely studied because of their fractal nature with applications in topology, chemistry, mathematics of Tower of Hanoi and computer sciences. Applications of molecular structure descriptors are a standard procedure which are used to correlate the biological activity of molecules with their chemical structures, and thus can be helpful in the field of pharmacology. The aim of this article is to establish analytically closed computing formulae for eccentricity-based descriptors of Sierpinski networks and their regularizations. These computing formulae are useful to determine a large number of properties like thermodynamic properties, physicochemical properties, chemical and biological activity of chemical graphs Methods: At first, vertex sets have been partitioned on the basis of their degrees, eccentricities and frequencies of occurrence. Then these partitions are used to compute the eccentricity-based indices with the aid of some combinatorics. The total eccentric index and eccentric-connectivity index have been computed. We also compute some eccentricity-based Zagreb indices of the Sierpinski networks. Moreover, a comparison has also been presented in the form of graphs. These computations will help the readers to estimate the thermodynamic properties and physicochemical properties of chemical structure which are of fractal nature and can not be dealt with easily. A 3D graphical representation is also presented to understand the dynamics of the aforementioned topological descriptors.