Existence of Approximate Solutions to Nonlinear Lorenz System under Caputo-Fabrizio Derivative

In this article,we developed sufficient conditions for the existence and uniqueness of an approximate solution to a nonlinear system of Lorenz equations under Caputo-Fabrizio fractional order derivative(CFFD).The required results about the existence and uniqueness of a solution are derived via the fixed point approach due to Banach and Krassnoselskii.Also,we enriched our work by establishing a stable result based on the Ulam-Hyers(U-H)concept.Also,the approximate solution is computed by using a hybrid method due to the Laplace transform and the Adomian decomposition method.We computed a few terms of the required solution through the mentioned method and presented some graphical presentation of the considered problem corresponding to various fractional orders.The results of the existence and uniqueness tests for the Lorenz system under CFFD have not been studied earlier.Also,the suggested method results for the proposed system under the mentioned derivative are new.Furthermore,the adopted technique has some useful features,such as the lack of prior discrimination required by wavelet methods.our proposed method does not depend on auxiliary parameters like the homotopy method,which controls the method.Our proposed method is rapidly convergent and,in most cases,it has been used as a powerful technique to compute approximate solutions for various nonlinear problems.

Analysis and Numerical Computations of the Multi-Dimensional,Time-Fractional Model of Navier-Stokes Equation with a New Integral Transformation

The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is analyzed by employing Caputo fractional derivative.We deliberated three stimulating examples that correspond to the triple and quadruple Elzaki transform decomposition methods,respectively.The findings illustrate that the established approaches are extremely helpful in obtaining exact and approximate solutions to the problems.The exact and estimated solutions are delineated via numerical simulation.The proposed analysis indicates that the projected configuration is extremely meticulous,highly efficient,and precise in understanding the behavior of complex evolutionary problems of both fractional and integer order that classify affiliated scientific fields and technology.

Simulating the Turbulent Hydrothermal Behavior of Oil/MWCNT Nanofluid in a Solar Channel Heat Exchanger Equipped with Vortex Generators

Re-engineering the channel heat exchangers(CHEs)is the goal of many recent studies,due to their great importance in the scope of energy transport in various industrial and environmental fields.Changing the internal geometry of the CHEs by using extended surfaces,i.e.,VGs(vortex generators),is the most common technique to enhance the efficiency of heat exchangers.This work aims to develop a newdesign of solar collectors to improve the overall energy efficiency.The study presents a new channel design by introducing VGs.The FVM(finite volume method)was adopted as a numerical technique to solve the problem,with the use of Oil/MWCNT(oil/multi-walled carbon nano-tubes)nanofluid to raise the thermal conductivity of the flow field.The study is achieved for a Re number ranging from12×10^(3) to 27×10^(3),while the concentration(φ)of solid particles in the fluid(Oil)is set to 4%.The computational results showed that the hydrothermal characteristics depend strongly on the flow patterns with the presence of VGs within the CHE.Increasing the Oil/MWCNT rates with the presence of VGs generates negative turbulent velocities with high amounts,which promotes the good agitation of nanofluid particles,resulting in enhanced great transfer rates.

Singular solitons and other solutions to a couple of nonlinear wave equations

This paper addresses the extended （G′/G）-expansion method and applies it to a couple of nonlinear wave equations. These equations are modified the Benjamin-Bona-Mahoney equation and the Boussinesq equation. This extended method reveals several solutions to these equations. Additionally, the singular soliton solutions are revealed, for these two equations, with the aid of the ansatz method.

New Compacton Solutions of Nonlinearly Dispersive R（m, n） Equations

在这篇论文，我们为 R 建立准确答案(m， n ) 由使用一个 sn-cn 方法的方程。作为结果，丰富的新紧缩的 ons，即有无限的翅膀， Jacobi 椭圆形的功能的新类型，独居的波浪和周期的波浪解决方案的缺席的 solitons，这，方程随最小的计算被获得。R 的性质(m， n ) 方程在图被显示出。

Inclined magnetic field and variable viscosity effects on bioconvection of Casson nanofluid slip flow over non linearly stretching sheet

In pursuit of improved thermal transportation,the slip flow of Casson nanofluid is considered in the existence of an inclined magnetic field and radiative heat flux flow over a nonlinear stretching sheet.The viscosity of the fluid is considered as a function of temperature along with the convective thermal boundary condition.Numerical solutions are obtained via Runge-Kutta along with the shooting technique method for the chosen boundary values problem.To see the physical insights of the problem,some graphs are plotted for various flow and embedded parameters on temperature function,micro-organism distribution,velocity,and volume fraction of nanoparticles.A decline is observed in the velocity and the temperature for Casson fluid.Thermophoresis and Brownian motion incremented the temperature profile.It is also found that thermal transportation can be enhanced in the presence of nanoparticles and the bioconvection of microorganisms.Present results are useful in the various sectors of engineering and for heat exchangers working in various technological processors.The main findings of the problem are validated and compared with those in the existing literature as a limiting case.

Analytical and Approximate Solutions for Complex Nonlinear Schr?dinger Equation via Generalized Auxiliary Equation and Numerical Schemes

This article studies the performance of analytical, semi-analytical and numerical scheme on the complex nonlinear Schr¨odinger(NLS) equation. The generalized auxiliary equation method is surveyed to get the explicit wave solutions that are used to examine the semi-analytical and numerical solutions that are obtained by the Adomian decomposition method, and B-spline schemes(cubic, quantic, and septic). The complex NLS equation relates to many physical phenomena in different branches of science like a quantum state, fiber optics, and water waves. It describes the evolution of slowly varying packets of quasi-monochromatic waves, wave propagation, and the envelope of modulated wave groups, respectively. Moreover, it relates to Bose-Einstein condensates which is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero. Some of the obtained solutions are studied under specific conditions on the parameters to constitute and study the dynamical behavior of this model in two and three-dimensional.

Optical Solitons and Stability Analysis in Ring-Cavity Fiber System with CarbonNanotube as Saturable Absorber

This paper addresses the ring-cavity fiber laser system. A class of gray and black soliton solutions of the model are reported by adopting an appropriate envelope ansatz. Further more, the modulation instability (MI) of the equation is studied using the linear-stability analysis (LSA) technique and the MI gain spectrum is derived. Some physical interpretations and analysis of the results obtained are also presented.

Fractional soliton dynamics of electrical microtubule transmission line model with local M-derivative

In this paper,two integrating strategies namely exp[-Ф(Х)]and (G'/G^(2))-expansion methods together with the attributes of local-M derivatives have been acknowledged on the electrical microtubule(MT)model to retrieve soliton solutions.The said model performs a significant role in illustrating the waves propagation in nonlinear systems.MTs are also highly productive in signaling,cell motility,and intracellular transport.The proposed algorithms yielded solutions of bright,dark,singular,and combo fractional soliton type.The significance of the fractional parameters of the fetched results is explained and presented vividly.

Investigation of adequate closed form travelling wave solution to the space-time fractional non-linear evolution equations

This work aims to construct exact solutions for the space-time fractional(2+1)-dimensional dispersive longwave(DLW)equation and approximate long water wave equation(ALW)utilizing the twovariable(G′/G,1/G)-expansion method and the modified Riemann-Liouville fractional derivative.The recommended equations play a significant role to describe the travel of the shallow water wave.The fractional complex transform is used to convert fractional differential equations into ordinary differential equations.Several wave solutions have been successfully achieved using the proposed approach and the symbolic computer Maple package.The Maple package program was used to set up and validate all of the computations in this investigation.By choosing particular values of the embedded parameters,we pro-duce multiple periodic solutions,periodic wave solutions,single soliton solutions,kink wave solutions,and more forms of soliton solutions.The achieved solutions might be useful to comprehend nonlinear phenomena.It is worth noting that the implemented method for solving nonlinear fractional partial dif-ferential equations(NLFPDEs)is efficient,and simple to find further and new-fangled solutions in the arena of mathematical physics and coastal engineering.

Nonclassical Lie symmetry and conservation laws of the nonlinear timefractional Korteweg–de Vries equation

In this paper,we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann–Liouville concept.In this study,first,we employ the classical and nonclassical Lie symmetries(LS)to acquire similarity reductions of the nonlinear fractional far field Korteweg–de Vries(KdV)equation,and second,we find the related exact solutions for the derived generators.Finally,according to the LS generators acquired,we construct conservation laws for related classical and nonclassical vector fields of the fractional far field KdV equation.

Diverse chirped optical solitons and new complex traveling waves in nonlinear optical fibers

This paper studies chirped optical solitons in nonlinear optical fibers.However,we obtain diverse soliton solutions and new chirped bright and dark solitons,trigonometric function solutions and rational solutions by adopting two formal integration methods.The obtained results take into account the different conditions set on the parameters of the nonlinear ordinary differential equation of the new extended direct algebraic equation method.These results are more general compared to Hadi et al(2018 Optik 172545–53)and Yakada et al(2019 Optik197163108).

Insights of numerical simulations of magnetohydrodynamic squeezing nanofluid flow through a channel with permeable walls

An analysis related to the transport of nanofluid that is confined in a channel between two orthogonal permeable walls will be investigated with the help of two numerical techniques.With the aid of the reasonable similarity changeover;the said model will be shaped into the desired non-linear equation whose behavior will be explained analytically as well as graphically where the functioning of three disparate variables such as magnetic parameter,permeable Reynold number,and channel permeable ratio will be explained precisely.The said model has been controlled by the homotopy analysis method(HAM)and then,compared through an efficient numerical method named shooting method(ShM).The profiles show that increases in the magnetic parameter decline the nanofluid flow velocity,whereas magnitude-wise raise is shown in each axial velocity profile.The radial profile raises while getting variation in wall expansion parameter from negative to positive.For the entire domain,the rate of change in velocity description enhances at the center while reduces at the surfaces.And the outcomes disclose that the wall permeable ratio has a significant impact on the solutions.Since the zero value of wall ratio refers to the particular type that Terrill has discussed.

The new soliton solutions for long and short-wave interaction system

The goal of this paper is to discover modern soliton solutions to long and short-wave interaction system by procedures called extended rational sine-cosine and rational sinh-cosh methods.We assume that the equation has a hypothetical soliton solutions.By reorganizing the resulting equations,we obtain a system of equations.Using Maple software,we get unknown coefficients in the system and writing them in the original equation,we obtain new solition solutions of the equation.The results show that the soliton solutions generated by the method for the long and short-wave interaction system are bright,kink type,bright periodic and dark solutions.We provided 3-D figures to illustrate the solutions.Computational results indicate that the method employed in this paper is superior than some other methods used in the literature to solve the same system equations.