Symbolic Software Mathematica Research Articles

Exploring the nonlinear behavior of solitary wave structure to the integrable Kairat-X equation

This study presented various types of soliton solutions for the nonlinear integrable Kairat-X equation by utilizing the improved F-expansion technique with symbolic computational software Mathematica. Explored results for the nonlinear integrable Kairat-X equation are interesting, novel, and more general with different physical structures of solitary waves and solitons, such as kink wave, mixed dark–bright, peakon, anti-kink wave, bright, anti-kink dark, periodic, and dark solitons. With numerical simulations, the secured soliton solutions visualized in two-dimensional, three-dimensional, and contour graphs represent the physical phenomena of the demonstrated results. The explored soliton solutions will be helpful to comprehend interesting physical structures in fiber optics, nonlinear optics, ferromagnetic dynamics, and many other scientific fields. The extracted soliton structure sheds light that the enhanced technique is effective, powerful, concise, and reliable. We can also investigate the soliton results of other nonlinear integrable partial and fractional equations.

Analyzing the impact of convective boundary conditions on fluid–particle mixture flow in a ciliated walled horizontal tube filled with Rabinowitsch fluid

This study investigates the impact of convective boundary conditions, thermal radiation, and viscous dissipation on the flow of fluid–particle mixture in a ciliated walled horizontal tube filled with Rabinowitsch fluid. A mathematical model is developed using partial differential equations of continuity, motion, and energy, which are solved using the symbolic software MATHEMATICA 12.0 with convective boundary conditions. Exact solutions for velocity, temperature distribution, pressure gradient, pressure rise, stream function, and heat transfer rate are obtained and analyzed for dilatant, Newtonian, and pseudoplastic fluids. Computational results reveal that the velocity and temperature profiles are highest at the center of the tube for pseudoplastic and Newtonian fluids compared to dilatant fluid. Additionally, pressure gradient fluctuates near the tube walls. The study also explores special cases where the Rabinowitsch fluid model is reduced to dilatant, Newtonian, and pseudoplastic fluids. The findings have potential applications in medical treatments for respiratory system tumors, airway cleaning, physiological fluids, and manufacturing processes involving peristaltic flow. This original research contributes to the understanding of fluid dynamics in complex systems and has implications for various practical applications.

Thermal transport characteristics of fluid-particle suspension in a vertical uniform tube: A computational study on Rabinowitsch fluid with uniform heat source

Problem statement The thermal transport of fluid-particle suspension in Rabinowitsch fluid is discussed theoretically in a vertical uniform tube under the consideration of a uniform heat source. Methodology The fluid and particle phase models are used to develop the mathematical model in the form of partial differential equations which are solved through the symbolic software MATHEMATICA version 12.0 and present the exact solution of temperature distribution, velocity of fluid and particle phases, wall shear stress, pressure gradient, pressure rise, and stream function, etc. Computational results and findings The volume fraction density of the particles slows down the motion of the fluid particles. The velocity of Newtonian fluid is maximum as compared to pseudo-plastic and Dilatant fluids. The fluid phase velocity is superior to particle phase velocity distribution for all pertinent parameters of the analysis. The pressure rise increases in the case of Newtonian and pseudo plastic fluid and decreases in Dilatant fluid. The trapping phenomena are strongly associated with the volume fraction density of the particle. The shear stress shows a decreasing trend against the volume fraction density of the particles, Grashoff number, and heat source parameter. Special case The current fluid model (Rabinowitsch fluid) can be reduced in Newtonian, pseudo plastic, and Dilatant fluid for taking κ → 0 , κ < 0 , and κ > 0 , respectively. Applications The current analysis can be useful in biomedical engineering to observe the blood supply through the arteries, the removal of kidney stones, and also the passage of urine to the bladder from the ureter. Originality The present research is original and has not been published before.

Dynamical analysis of exact optical soliton structures of the complex nonlinear Kuralay-II equation through computational simulation

In this paper, we successfully extracted the various types of soliton solutions for the complex nonlinear Kuralay-II equation through the improved F-expansion method with symbolic computational software Mathematica. The extracted soliton solutions for the Kuralay-II equation are interesting, novel and more general such as anti-kink wave solitons, dark solitons, kink wave solitons, bright solitons, periodic wave solitons, mixed solitons in bright-dark soliton shape, peakon solitons, and solitary wave structures. The graphical structure of some extracted solutions is visualized in 2D, 3D and contour plottings with imaginary, real, and absolute values of the functions by using the numerical simulation. The proposed research will contribute to advancing our knowledge about the complex nonlinear Kuralay-II equation and demonstrating the applicability to the proposed approach to investigate other higher-order complex nonlinear equations. The successful investigation demonstrated that the proposed method is effective, simple, more powerful, efficient and can be utilized on a variety of other nonlinear equations. The explored solitary waves and optical solitons will play an important role in the investigation of nonlinear phenomena in various domains of science and engineering.

Unraveling metachronal wave effects on heat and mass transfer in Non-Newtonian fluid

Non-Newtonian fluids flow generated by “cilia” are critical in medicine and bio-medical engineering. Such investigations are created by the back-and-forth movement of a microscopic hair-like structure connected to the walls, which causes a metachronal wave to form and drive biological fluids. Motivated by a wide range of biological applications, this study aims to explain the incompressible flow of Ellis fluid caused by the propagation of an infinite metachronal wave train traveling along channel walls owing to constantly beating cilia. The problem is simplified by low Reynolds number and long wavelength assumption. The mathematical model is solved with the aid of symbolic computational software Mathematica 13 version. The consequences of emerging parameters are then shown in graphical form and discussed comprehensively. From the study, it is worth mentioning that velocity declines with increasing material constants. Temperature distribution is also improved in the core sector of the channel and reduced at the walls. It is predicted that this approach will make an essential contribution to the progress and enhancement of various types of drug delivery systems in the biomedical industry and biomechanics.

Weakly restoring forces and shallow water waves with dynamical analysis of periodic singular solitons structures to the nonlinear Kadomtsev–Petviashvili-modified equal width equation

In this study, the nonlinear two-dimensional Kadomtsev–Petviashvili-modified equal width (KP-MEW) equation is under investigation, which is described as a nonlinear wave model in weakly restoring forces and shallow water waves in the way of ferromagnetic, solitary waves in two dimensions with short amplitude and long wavelength and are also helpful in the investigation of various behaviors in nonlinear sciences. We successfully found the interesting and novel solitary wave results in bright solitons, kink solitons, dark solitons, anti-kink solitons, periodic singular solitons for nonlinear two-dimensional KP-MEW equation on the base of extension of simple equation technique under symbolic computational software Mathematica. The created solitons play an important role in nonlinear engineering and physics such as nonlinear optics, nonlinear dynamics, soliton wave theory, optical fiber and communication system. We are sure that these constructed results are novel and interesting which does not exist in the previous literature works. The graphical representation of constructed results is shown by 2D, 3D and contour graphs. The investigated research work proved that utilized technique is more concise, straightaway and efficient for study of other nonlinear partial differential equations.

A Novel Approximation Approach for the Analytical Solution of the Flow of Micropolar Fluid Through a Permeable Channel

An attempt is made in this study to investigate the problem of micropolar fluid flow in a porous medium theoretically. Employing the Berman’s similarity solution, the model equations governing the flow is transformed into a set of nonlinear ordinary differential equation and solved using Temimi-Ansari method. Expressions for the velocity and micro-rotation profiles are obtained under the impressions of diverse parameters affecting the flow problem. Using symbolic computation software Mathematica, the nondimensional equations are solved numerically using the Keller Box scheme. Comparison between the analytical solution obtained by TAM and the numerical result are compared with results in literature to observe rapid convergence. Findings from the study showed in the presence of

Newly formed center-controlled rouge wave and lump solutions of a generalized (3+1)-dimensional KdV-BBM equation via symbolic computation approach

In this article, we investigate the generalized (3+1)-dimensional KdV-Benjamin-Bona-Mahony equation governed with constant coefficients. It applies the Painlevé analysis to test the complete integrability of the concerned KdV-BBM equation. The symbolic computational approach provides first-order, second-order rogue wave and lump solutions with center-controlled parameters. The rogue waves localized in space and time have a significant amplitude, and lumps are of rational form solution, localized decaying solutions in all space directions rationally. Utilizing a symbolic computation approach, we get the bilinear equation of the KdV-Benjamin-Bona-Mahony equation and show the center-controlled rogue waves and lumps. We employ the symbolic system software Mathematica to do the symbolic computations, form the first and second-order rogue waves, and lump solutions with appropriate values of constant coefficients. The KdV-Benjamin-Bona-Mahony equation analyses the evolution of long waves with modest amplitudes propagating in plasma physics and the motion of waves in fluids and other weakly dispersive mediums. Moreover, rogue waves and lumps occur in several scientific areas, such as fluid dynamics, optical fibers, dusty plasma, oceanography, water engineering, and other nonlinear sciences.

Analysis of new soliton type solutions to generalized extended (2 + 1)-dimensional Kadomtsev-Petviashvili equation via two techniques

In this paper, we examine the solitary wave solutions of the generalised (2 + 1) extended Kadomtsev–Petviashvili (eKP) equation, which is used as model for the surface waves and internal waves in straits or channels and describes the dynamics of nonlinear waves in plasma physics and fluid dynamics. We secure distinct soliton solutions by employing two amelioration methodologies, namely the improved F-expansion method and the new Kudryashov method. These methods explore several unique forms of solitary wave solutions, including dark, rational, singular, periodic, and exponential ones. In order to illustrate propagation of some reported wave solutions, the graphs associated with these solutions are presented by choosing appropriate parametric values in 2D, 3D, contour, and density plots with the use of the symbolic software Mathematica 13.0. The presented solutions realize very significant fact for investigation the qualitative interpretations for various phenomena in our nature. The computed solutions revealed that the applied methods are influential, efficious and skilful and can be a best way to handle other higher order nonlinear evolution equations.

Effect of Double-Diffusive Stagnation point flow of Eyring– Powell Nanofluid on a Slender Stretching Sheet with Non-uniform heat Source Sink and Inclined Magnetic Field

In the present paper, the effects of an inclined magnetic field and a non-uniform heat source sink on a double diffusive convective stagnation point flow in a slender stretching sheet are studied. The suitable similarity transformation is utilized for the conversion of nonlinear differential equations. These converted equations are solved by means of Differential Transformation method (DTM) with the support of symbolic software Mathematica. Further, the effects of appropriate parameters on velocity profile, solute, nanoparticle concentration and temperature profiles are shown graphically with some suitable discussions. It is found that velocity decreases with a rise of magnetic parameter. Because applying the uniform magnetic field normal to the flow direction gives rise to Lorentz force. This force has the tendency to slow down the velocity of the fluid in the boundary layer. Also non-uniform heat source sink parameters enhance the thermal profile of the system. But, opposite behavior can be seen in solute and nanoparticle concentration profiles. Such results can be useful in design and structure of materials, where implementation of variable thickness decreases the weight of stretched element and boosts the usage of materials.

The exact solutions of the [formula omitted]–dimensional Kadomtsev–Petviashvili equation with variable coefficients by extended generalized [formula omitted]-expansion method

In this paper 2+1–dimensional Kadomtsev–Petviashvili (KP) equation with variable coefficients is investigated through the extended generalized G′G–expansion technique. One of the most universal model is KP equation, which is used to explain the ion acoustic waves in plasma physics, to model two dimensional shallow water waves, and in ferromagnetic, Bose–Einstein condensation and string theory. The obtained exact solutions of KP equation are in the form of hyperbolic function, trigonometric function, and rational function. With the aid of symbolic computational software Mathematica, the three dimensional surface plots with corresponding contour plots are provided for the obtained closed from solutions, which are of the form of solitary waves, multi solitons and periodic solitary wave like dynamical structures.

Using the improved $$\exp ( - \Phi (\xi ))$$ expansion method to find the soliton solutions of the nonlinear evolution equation

With the help of the symbolic calculation software Mathematica, the nonlinear coupled Schrodinger system, (2 + 1)-dimensional dispersive long-wave system and (2 + 1)-dimensional Nizhnik–Novikov–Veselov system are solved by using the improved $$\exp ( - \Phi (\xi ))$$ expansion method. Diverse new soliton solutions are obtained, and the structure and properties of soliton solutions are studied. We derive seven sets of exact solutions under different conditions for the nonlinear coupled Schrodinger system and the (2 + 1)-dimensional dispersive long wave equation, respectively, and for the (2 + 1)-dimensional Nizhnik–Novikov–Veselov system, we find the inelastic interaction solitary waves and fussion soliton.

Thermal Analysis for Peristaltic Flow of Nanofluid under the Influence of Porous Medium and Double Diffusion in a Non-Uniform Channel Using Sumudu Transformation Method

The present study aims to investigate the study of double-diffusive convection on peristaltic flow under the assumption of long wavelength and low Reynolds number. The mathematical modelling for a two-dimensional flow, along with double diffusion in nanofluids is considered. The motivation of the present research work is to analyze the effects of thermal radiation on a peristaltic flow through a porous medium in a nonuniform channel. The heat flux of the linear approximation employs the thermal radiation of the flow problem. The governing equations are analytically solved by using Homotopy Perturbation Sumudu Transformation method (HPSTM) with the help of the symbolic software Mathematica. The results of the velocity, pressure rise, temperature, solutal (species) concentration and nanoparticle volume fraction profiles are graphically shown.

Double diffusion on peristaltic flow of nanofluid under the influences of magnetic field, porous medium, and thermal radiation

AbstractThe present article investigates the study of double‐diffusive convection on peristaltic flow under the assumption of long wavelength and low Reynolds number. The mathematical modeling for a two‐dimensional flow, along with double diffusion in nanofluids, is considered. The motivation of the present research work is to analyze the effects of the magnetic field and thermal radiation on a peristaltic flow through a porous medium in an asymmetric channel. The heat flux of the linear approximation employs the thermal radiation of the flow problem. The effect of thermal radiation and double diffusion is an important aspect of research due to its application in public health potential. Infrared radiation techniques are indeed used to treat many skin‐related diseases. It can also be used as a measure of thermotherapy in some bones to enhance the blood circulation. It is found that approximately 80% of blood flow increases with radiation. The governing equations are analytically solved by using Homotopy analysis method with the help of the symbolic software Mathematica. The results of the velocity, pressure rise, temperature, solutal (species) concentration, and nanoparticle volume fraction profiles are graphically shown.

On Tzit&amp;#233;eica type nonlinear equations for multiple soliton solutions in nonlinear optics

Nonlinear equations personate a consequential role in scientific fields such as nonlinear optics, solid state physics and quantum field theory. This article studies the Tzitzeica-Dodd-Bullough-Mikhailov and Tzitzeica-type equations that appear in nonlinear optics. The Painleve and traveling wave transformations both play a key role in revamping the aforementioned equations into nonlinear ordinary differential equations. Then, the simple ansatz approach is followed to seize complex singular, complex bright solitons and other kink type solutions. The existing literature unveils that this study is a novel contribution in the literature and the proposed approach is forthright and simple to implement for solving nonlinear problems. This approach has no extra condition as compared to many other techniques have. We also verify and interpret graphically the secured solutions through symbolic software Mathematica and MatLab respectively.

Investigation of Forced Convective Heat Transfer in Nanofluids

The present paper concerns a theoretical study of heat transfer of the laminar two dimensional flows of various nanofluids taking into account the dissipation due to viscous term past a 2-D flat plate had a different temperatures. The steady incompressible flow equations were used and transformed to a nonlinear Ordinary Differential Equation (ODE) using a similarity variable. These equations were solved numerically using implicit finite difference method in which the partial derivatives were replaced by appropriate central differences patterns and using Newton’s method to linearize the resulting algebraic equations. Finally, the block-tridiagonal-elimination technique was used to solve that linear system. Three types of nanoparticles namely, Cu-water, Al2O3-water, and TiO2-water in the base flow of water were considered. The symbolic software Mathematica was used in the present study. Different types of nanoparticles, different values of, nanoparticle volume fraction, Eckart and Prandtl number were tested and analyzed at different wall temperature. The effect of these parameters on the flow behaviour, the local skin friction coefficient, Nusselt number, the velocity and the temperature profiles were presented and investigated. It is concluded that these parameters affect the fluid flow behaviour and heat transfer parameters especially nanoparticle concentration. The presence of nanoparticles showed an enhancement in the heat transfer rate moreover its type has a significant effect on heat transfer enhancement.

The impacts of varying magnetic field and free convection heat transfer on an Eyring–Powell fluid flow with peristalsis: VIM solution

A mathematical model of a non-Newtonian fluid (Eyring–Powell fluid) flow with peristalsis under the effect of varying magnetic field and free convection heat transfer has been discussed. Most of the previous attempts of peristaltic flow problems for a non-Newtonian fluid have been solved by using the perturbation technique for a small non-Newtonian parameter, which gives some limits for the results. To avoid such restriction, the variational iteration method (VIM) is applied to solve the current model. A code is established by using the symbolic software MATHEMATICA to get the successive solutions. Semi-analytical solutions are found by VIM for the velocity, heat transfer, pressure gradient and stream function, which include Eyring–Powell fluid parameters. Moreover, a comparison between VIM and numerical solutions is discussed. The obtained results show that the variation of the heat transfer for the Newtonian fluid is large compared with the Eyring–Powell fluid. In addition, it is noteworthy that the peristaltic transport overcomes on Lorentz force nearby the peristaltic walls.

Dispersion relations of periodic quantum graphs associated with Archimedean tilings (II)

We continue the work of a previous paper (Luo et al 2019 J. Phys. A. Math. Theor. 52 165201) to derive the dispersion relations of the periodic quantum graphs associated with the remaining 5 of the 11 Archimedean tilings, namely the truncated hexagonal tiling (3,122), rhombi-trihexagonal tiling , snub square tiling (32,4,3,6), snub trihexagonal tiling (34,6), and truncated trihexagonal tiling . The computation is done with the help of the symbolic software Mathematica. With these explicit dispersion relations, we perform more analysis on the spectra.

Dispersion relations of periodic quantum graphs associated with Archimedean tilings (I)

There are 11 kinds of Archimedean tilings for the plane in total. Applying the Floquet–Bloch theory, we derive the dispersion relations of the periodic quantum graphs associated with a number of Archimedean tilings, namely the triangular tiling (36), the elongated triangular tiling , the trihexagonal tiling and the truncated square tiling (4,82). The derivation makes use of characteristic functions, with the help of the symbolic software Mathematica. The resulting dispersion relations are surprisingly simple and symmetric. They show that in each case the spectrum is composed of point spectrum and an absolutely continuous spectrum. We further analyzed the structure of absolutely continuous spectra. Our work is motivated by the studies on the periodic quantum graphs associated with hexagonal tiling in Kuchment and Post (2007 Commun. Math. Phys. 275 805–26) and Korotyaev and Lobanov (2007 Ann. Henri Poincaré 8 1151–76).

Dynamic Modeling and Development of Symbolic Calculation Software for N‐DOF Flexible‐Link Manipulators Incorporating Lumped Mass

Considering the complex dynamic modeling of multi‐DOF planar flexible manipulators, a general‐purpose method for the rigid‐flexible coupling dynamic modeling of N‐DOF flexible manipulators is proposed in this paper, and symbolic calculation software is developed. The modeling method is based on the Lagrange equation and assumed mode method (AMM). First, the N‐DOF flexible manipulator is divided into two parts, which are assumed to be rigid and flexible. On this basis, the rigid part and the flexible part are coupled, and the calculation process of the model is further simplified. Then, the simplest general symbolic expression of the dynamic model of the N‐DOF flexible manipulator is obtained with the induction method. According to the modeling method, “symbolic expression computation software for dynamic equations of N‐DOF flexible manipulators” is developed using the symbolic calculation software Mathematica. Finally, the dynamic modeling method and the symbolic calculation software are verified by a trajectory tracking experiment with a PD control applied to a 2‐DOF flexible manipulator. Compared with the traditional modeling method, the calculation time can be reduced by more than 90% using the modeling method proposed in this paper, which significantly reduces the complexity of the modeling process.