Nonlinear Differential Equations Research Articles

Generalized kudryashov and extended auxiliary equation methods for novel solitons solutions to (1+1)-dimensional doubly dispersive equation of murnaghan’s rod

Abstract In this study, the Generalized Kudryashov method and the Extended Auxiliary Equations method are employed to investigate the strongly nonlinear (1+1)-dimensional Doubly Dispersive equation model of inhomogeneous Murnaghan’s rod for developing novel soliton solutions. These symbolic methods are famous for solving various problems involving nonlinear partial differential equations. The study finds novel solitons like hyperbolic, exponential, rational, and trigonometric. Moreover, 2D, 3D, and contour plots under tunable parameters depict graphical representations of the solutions that furnish dynamic wave behavior and insights into the material’s elastic properties under strain and stress.

A comparative study of numerical methods for approximating the solutions of a macroscopic automated-vehicle traffic flow model

In this paper, a particle method is used to approximate the solutions of a “fluid-like” macroscopic traffic flow model for automated vehicles. It is shown that this method preserves certain differential inequalities that hold for the macroscopic traffic model: mass is preserved, the mechanical energy is decaying and an energy functional is also decaying. To demonstrate the advantages of the particle method under consideration, a comparison with other numerical methods for viscous compressible fluid models is provided. Since the solutions of the macroscopic traffic model can be approximated by the solutions of a reduced model consisting of a single nonlinear heat-type partial differential equation, the numerical solutions produced by the particle method are also compared with the numerical solutions of the reduced model. Finally, a traffic simulation scenario and a comparison with the Aw-Rascle-Zhang (ARZ) model are provided, illustrating the advantages of the use of automated vehicles.

Thermal instability of three-dimensional boundary layer stagnation point flow towards a rotating disc

In this paper, the thermal instability of a three-dimensional boundary layer axisymmetric stagnation point flow towards a heated horizontal rotating disk is considered. A large number of works have been done on stability analysis. However, they did not check the thermal stability of the non-parallel-flow in the face of small disturbances that occur in the vicinity of the heated rotating disk. The governing equations of the basic flow are reduced to three coupled nonlinear partial differ-ential equations, and solved numerically with the fourth-order Runge-Kutta method. Thermal stability is examined by making use of linear stability theory based on the decomposition of the normal mode of Görtler-Hammerlin. The resulting eigenvalue problem is solved numerically using a pseudo-spectral method based on the expansion of Laguerre’s polyno-mials. The obtained results are discussed in detail through multiple configurations. As the main result, for large Prandtl numbers (Pr), the rotation disk parameter (Ω) has a destabilizing effect while for small Pr (around the unity) it tends to stabilize the basic flow. It was found that as the disk radius r→0, the flow is linearly stable, and the disturbances grow rapidly away from the stagnation point. For low values of Pr, the flow becomes more stable, and strong thermal gradients are necessary to destabilize it. However, an increase in Pr leads to a significant expansion of the instability region.

Lie symmetry method to discuss the optimal system of Lie subalgebras and similarity solutions for shock propagation in a perfectly conducting dusty gas with magnetic field in rotating medium

In this paper, the similarity solutions for the shock wave propagation in a perfectly conducting dusty gas under the effect of axial or azimuthal magnetic field in the case of rotating medium by using Lie symmetry method have been discussed. The optimal system of one-dimensional Lie subalgebras related to the hyperbolic system of nonlinear partial differential equations had been constructed by using general invariant function and general adjoint transformation matrix. The optimal system is the collection of inequivalent optimal classes, which provides crucial information regarding different types of the similarity solutions. On the basis of the optimal system, we have discussed all possible similarity solutions for inequivalent optimal classes individually. Also, we have found that the similarity solutions exist in two cases for power law shock path and in one case with exponential law shock path. The shock wave decay is due to an increment in the physical parameter such as shock Cowling number or non-idealness parameter or adiabatic exponent or rotational parameter. For smaller value of the ratio of density of solid particles to initial species density of the gas i.e. [Formula: see text], the shock strength decreases with an increment in the mass fraction of solid particles in the mixture [Formula: see text], but the shock strength decreases as [Formula: see text] increases for [Formula: see text]. Also, the shock strength increases with an increment in [Formula: see text] at constant [Formula: see text]. It is observed that the shock wave is stronger in the presence of axial magnetic field as compared to azimuthal magnetic field.

Evaluating the mixed convection flow of Casson fluid from the semi-infinite vertical plate with radiation absorption effect

This study examines a steady laminar Casson fluid flow induced by a semi-infinite vertical plate under the impact of the Darcy-Forchheimer relation and thermal radiation. The features of mixed convection, cross-diffusion, radiation absorption, heat generation, chemical reactions and viscous dissipation are also considered to explain the transport phenomenon. The resultant system of equations, concerned with the problem under consideration, is transformed into a group of non-linear ordinary differential equations (ODEs) by means of similarity variables. The bvp4c method, an instrument popular for its numerical accomplishments, is utilized to solve this problem. The effect of flow parameters on heat transfer, concentration and velocity is evaluated via diagrams. To validate our code, we have compared the present outcomes to the prevenient literature, and stable consent has been detected. Moreover, the friction coefficient Cfx, Nusselt number Nux, and Sherwood number Shx are also computed to assess velocity gradient, efficiency of heat transfer and mass transfer process, respec-tively.

Parameter-dependent fractional boundary value problems: analysis and approximation of solutions

We study a parameter-dependent non-linear fractional differential equation, subject to Dirichlet boundary conditions. Using the fixed point theory, we restrict the parameter values to secure the existence and uniqueness of solutions, and analyse the monotonicity behaviour of the solutions. Additionally, we apply a numerical-analytic technique, coupled with the lower and upper solutions method, to construct a sequence of approximations to the boundary value problem and give conditions for its monotonicity. The theoretical results are confirmed by an example of the Antarctic Circumpolar Current equation in the fractional setting.

Classical-quantum simulation of non-equilibrium Marshak waves

In the radiation hydrodynamic simulations used to design inertial confinement fusion (ICF) and pulsed power experiments, nonlinear radiation diffusion tends to dominate CPU time. This raises the interesting question of whether a quantum algorithm can be found for nonlinear radiation diffusion which provides a quantum speedup. Recently, such a quantum algorithm was introduced based on a quantum algorithm for solving systems of nonlinear partial differential equations (PDEs) which provides a quadratic quantum speedup. Here, we apply this quantum PDE (QPDE) algorithm to the problem of a non-equilibrium Marshak wave propagating through a cold, semi-infinite, optically thick target, where the radiation and matter fields are not assumed to be in local thermodynamic equilibrium. The dynamics is governed by a coupled pair of nonlinear PDEs which are solved using the QPDE algorithm, as well as two standard PDE solvers: (i) Python's py-pde solver; and (ii) the KULL ICF simulation code developed at Lawrence-Livermore National Laboratory. We compare the simulation results obtained using the QPDE algorithm and the standard PDE solvers and find excellent agreement.

Mathematical modeling of an articulated flying vehicle for precise control analysis in grasping highly oscillatory objects

In this research work, a detailed mathematical modeling approach has been taken in order to develop a complete package for the investigation of phenomena associated with the flight dynamics and control of an articulated aerial robot in grasping high weight oscillatory objects such as living creatures. To this end, the origin of forces and moments generated by a moving target is analyzed and the nonlinear differential equations of the system are derived. Then, the model is verified and the effect of target oscillations on the whole system dynamics is investigated. In the next step, due to the nonlinear nature of this under-actuated system and the presence of the force-generating target with unknown dynamics and model parametric and non-parametric uncertainties, two versions of adaptive sliding mode control (A-SMC) are designed to enable the system follows the nominal trajectories in a desirable way without needing to know the upper bounds of the time-varying forces and uncertainties. In this regard, the stability of closed-loop systems is proved based on Lyapunov theory. The performance and robustness of the designed controllers are evaluated and compared through numerical and Monte Carlo simulations with some standard criteria. At the end, the conclusion of this work is presented.

A Novel Approach for Time-Local Fractional Solutions of Certain Nonlinear Partial Differential Equations in Fractal Dimension

Time-local fractional approaches for nonlinear partial differential equations in fractal dimensions are essential for capturing the complex, irregular behaviors found in fractal systems. In this paper, a new modification of the local fractional Laplace variational iteration method (MLFLVIM) for obtaining analytical approximate solutions to the fractional gas dynamics equation, fractional Stefan equation, and fractional Newell-Whitehead-Segel equation within the context of fractal time space is presented. The proposed method (MLFLVIM) elegantly combines the local fractional Laplace transform (LFLT) with modified variational iteration method. Specifically, we first apply the (LFLT) to the given local fractional PDEs, yielding a transformed system of equations. We then apply modified variational iteration to this system. Finally, we use the inverse of (LFLT) to obtain the desired solution. To demonstrate the effectiveness of this approach, we implement it on three numerical physical problems. The results show that the (MLFLVIM) can successfully handle these nonlinear LFPDEs and provide accurate analytical approximation solutions.

Simulation of blood flow in a tapered artery with ternary hybrid nanofluid and Jeffrey model

This research simulates flow of blood in a tapered peristaltic artery integrating ternary hybrid nanofluid using the Jeffrey fluid model. The simulation examines the effects of Au, Fe3O4, and Ag nanoparticles on the flow. The artery, modeled as a peristaltic channel, is subjected to nonlinear thermal radiation, magnetic forces, and heat source. The problem is formulated using non-linear Cartesian partial differential equations, which are then transformed into dimensionless form without approximations. Adomian Decomposition Method (ADM) is employed to solve these equations, revealing how normal and shear stress, normal and axial velocities, induced magnetic fields, and temperature profiles vary with changes in relevant parameters. Additionally, biological factors are considered to graph streamlines, providing a comprehensive analysis of the flow dynamics. Some notable results include that adding ternary nanoparticles increases the Nusselt number by 26.03% in Newtonian fluids and 158.69% in Jeffrey fluids, and increasing tapering parameters and wavenumber improves axial and normal velocities, heat transfer, and flow complexity.

New analytical wave structures for generalized B-type Kadomtsev–Petviashvili equation by improved modified extended tanh function method

Abstract Recently, solving the complicated nonlinear partial differential equations has become very important demand in order to simulate their physical phenomena. This manuscript focuses on extracting the wave solutions of (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equation (GBKPE), which demonstrates the behavior of nonlinear waves in fluid mechanics. The improved modified extended Tanh function (IMETF) method is the suggested method to do this task as it gives different types of solutions. This method enables us to obtain many solutions, such as Jacobi elliptic, dark soliton, and singular soliton, exponential, and singular periodic wave solutions. Additionally, for more illustrations graphical visual representations of some solutions are provided.

Practical stability of stochastic differential delay equations driven by G-Brownian motion with general decay rate

This article is concerned with the quasi sure practical stability of nonlinear stochastic differential delay equations driven by G-Brownian motion (G-SDDEs) with a general decay rate. Sufficient conditions are established by constructing appropriate G-Lyapunov functionals. Moreover, we provide some numerical examples to demonstrate the effectiveness of the obtained results. For more information see https://ejde.math.txstate.edu/Volumes/2024/70/abstr.html

Investigation of activation energy of Maxwell fluid with Newtonian heat effect at the boundary layer flow on a cylinder embedded in porous medium

AbstractThis study aims to investigate the flow of Maxwell fluid over a cylinder, considering the combined effect of Magnetohydrodynamics (MHD) and a permeable medium on the equation of momentum, heat generation and radiation on the equation of energy, and activation energy on the mass equation. The objective is to analyze the impression of these physical phenomena on the fluid's behavior under specified boundary conditions, with a focus on the Newtonian heating effect. The significance of this research lies in its application to industrial processes, polymer extrusion and cooling of metallic sheets, where controlling fluid flow and energy transfer is crucial. The dimensional governing equations for momentum, energy, and concentration were changed into non‐dimensional forms via appropriate dimensionless quantities and similarity variables. The subsequent nonlinear ordinary differential equations were answered using the BPV4C inbuilt MATLAB solver. The key findings reveal that the concentration outline rises with a rise in activation energy and declines with a rise in the chemical reaction. Additionally, the Nusselt number profile displays an increasing trend with the Newtonian heating parameter, indicating enhanced heat transfer. Quantitative results demonstrate that the occurrence of a porous medium and MHD significantly impacts the velocity and temperature contours. The real‐time application of this study can be seen in the optimization and design of chemical reactors, heat exchangers, and cooling systems in engineering processes. The developed model delivers insights into controlling energy and mass transfer in processes involving stretching surfaces, which is essential for improving efficiency and product excellence in various industrial applications.

Soret and Dufour effects for unsteady flow with shapes and (Cu+Fe3O4)/(ethelene–glycol+methanol) between two stretchable rotating disks

The ongoing study looks at the multiple shapes of nanoparticles in an unsteady flow between two stretchable rotating disks. Unsteady flow of hybrid nanofluid takes place between the stretchable disks, using (ethelene–glycol+methanol) as the base fluid and (Cu+Fe3O4) as the hybrid nanofluid. This study uses three distinct forms of (Cu+Fe3O4) hybrid nanofluids: brick, blade, and cylinder. Using chosen similarity transmutations, corresponding mathematical expressions (partial differential equations) are transformed into nonlinear ordinary differential equations. The whole set of equations takes the shape of an ordinary differential equation depending on the boundary conditions, and the solution is numerically computed using the bvp4c MATLAB solver. The influence of multiple factors across the velocity field is illustrated graphically. To acquire tiny quantities of the unsteady factors S, the momentum boundary layer for all types of nanofluids becomes less and more conspicuous. We compared our findings to previous findings and discovered an intriguing pattern. The main discovery of this work is Dufour and Soret effect on the heat and concentration profile. The rise in the rotation and the unsteady parameter ends up with an increase in the tangential velocity. The temperature rises as the Brinkman number and heat source factors rise, and falls as the unsteady parameter S rises. The concentration falls as the Schmidt number and unsteady factor increase, but it increases as the activation energy increases.

Study of magnetic field dependent viscosity and temperature on Jenkins model fluid flow over a rotating disk

The present work involves the study of ferrofluid flow over a stretchable rotating disk maintained at an uniform temperature in the presence of magnetic field dependent viscosity. The system is analyzed through the Jenkins model and the simplified governing equations are represented in the cylindrical co-ordinates. The obtained nonlinear coupled partial differential equations are reduced to a coupled nonlinear ordinary differential equations using the well known Von Karman transformation for rotating disk and solved numerically by the shooting method. The velocity profiles, pressure and temperature are studied by varying the stretching parameter, magnetic field dependent viscosity parameter, material constant, and Prandtl number. The comparison for limiting case of the present problem finds good agreement with the existing literature.

Lie-symmetry analysis of a three dimensional flow due to unsteady stretching of a flat surface with non-uniform temperature distribution

Various mathematical frameworks have been developed to study dynamics of the fluid flow in stretching films. In this work, a three–dimensional flow induced by an unsteady stretching of an infinite flat sheet is analyzed through the Lie symmetry approach. Twelve Lie point symmetries for the nonlinear partial differential equations describing the considered flow and heat transfer phenomena are derived. Invariants for these Lie symmetries are obtained to construct a new class of similarity transformations. With the deduced similarity transformations, the flow equations are converted to ordinary differential equations which are solved using the Homotopy analysis method. The deduced analytic solution enables an exploration of the effects of various parameters on the flow and heat transfer, which have not been revealed previously using the Lie method as per the authors’ knowledge. The influences of the stretching rate, parameters that maintain the non-uniformity and unsteadiness of sheet temperature, and the Prandtl number, are demonstrated with the help of graphs and tables. These results show that for certain values of the system parameters, heat transfer reverses its direction and occurs from the fluid to the sheet. At these values, maximum heat transfer does not occur at the sheet but rather slightly above it.

Some Exact Green Function Solutions for Non-Linear Classical Field Theories

We consider some non-linear non-homogeneous partial differential equations (PDEs) and derive their exact Green function solution as a functional Taylor expansion in powers of the source. The kind of PDEs we consider are dispersive ones where the exact solution of the corresponding homogeneous equations can have some known shape. The technique has a formal similarity with the Dyson–Schwinger set of equations to solve quantum field theories. However, there are no physical constraints. Indeed, we show that a complete coincidence with the statistical field model of a quartic scalar theory can be achieved in the Gaussian expansion of the cumulants of the partition function.

Computational study with neural networks to double diffusion in Prandtl thermal nanofluid flow adjacent to a stretching surface design with numerical treatment

The study focuses on the implementation of neural networks in analyzing the behavior of Prandtl nanofluid near a extending surface in the existence of dual diffusion. The research investigates the merged consequence of thermal and concentration gradients on the fluid flow and heat transfer characteristics using advanced computational techniques. The investigation delves into the phenomenon of double diffusion in the flow of Prandtl nanofluid near a stretching surface (DD-PNSS), employing the Levenberg-Marquardt scheme with Artificial Neural Networks (LMS-ANNs). The application of similarity variables converts the non-linear partial differential equations into non-linear ordinary differential equations. Through the application of the Lobatto IIIa formula in a three-stage process, various data sets are generated for the LMS-ANNs by varying parameters such as the Prandtl fluid parameter ([Formula: see text]), Prandtl number (Pr), Brownian motion parameter (Nb), thermophoresis parameter (Nt), and Dufour-solutal Lewis number (Ld). The range of parameter α is from 2 to 3.5, parameter β from 0.1 to 0.9, Nb is from 0.1 to 0.4, Nt is from 0.1 to 0.7, Le is from 1 to 3 and Ld is from 0.1 to 0.5. The intrinsic behavior of embedded parameters on dimensionless velocity, temperature, solutal concentration, and nanoparticle concentration profiles is graphically evaluated. The proposed LMS-ANNs model is meticulously tested, validated, and trained using a multi-stage approach, and its performance is compared to established references to ensure its reliability. The effectiveness of the suggested LMS-ANNs model is further affirmed through regression analysis, Mean Squared Error (MSE) evaluation, and histogram studies, showcasing an exceptional accuracy level ranging from 10−08 to 10−10, setting it apart from alternative approaches and reference models.

A Finite Difference-Based Adams-Type Approach for Numerical Solution of Nonlinear Fractional Differential Equations: A Fractional Lotka–Volterra Model as a Case Study

Abstract In this paper, we developed an efficient Adams-type predictor–corrector (PC) approach for the numerical solution of fractional differential equations (FDEs) with a power law kernel. The main idea of the proposed approach is to use a linear approximation to the nonlinear problem and then implement finite difference approximations of derivatives. Numerical comparisons with the fractional Adams method are made and simulation results are demonstrated to evaluate the approximation error of the proposed approach. The efficiency of this approach has been depicted by presenting numerical solutions of some test fractional calculus models. Numerical simulation of a fractional Lotka–Volterra model is provided, as a case study, using the proposed approach. The advantage of the proposed approach lies in its flexibility in providing approximate numerical solutions with high accuracy.

Computational Technology for Shell Models of Magnetohydrodynamic Turbulence Constructing

The paper discusses the computational technology for constructing one type of small-scale magnetohydrodynamic turbulence models – shell models. Any such model is a system of ordinary quadratic nonlinear differential equations with constant coefficients. Each phase variable is interpreted in absolute value as a measure of the intensity of one of the fields of the turbulent system in a certain range of spatial scales (scale shell). The equations of any shell model must have several quadratic invariants, which are analogues of conservation laws in ideal magnetohydrodynamics. The derivation of the model equations consists in obtaining such expressions for constant coefficients for which the predetermined quadratic expressions will indeed be invariants. Derivation of these expressions «manually» is quite cumbersome and the likelihood of errors in formula transformations is high. This is especially true for non-local models in which large-scale shells that are distant in size can interact. The novelty and originality of the work lie in the fact that the authors proposed a computational technology that allows one to automate the process of deriving equations for shell models. The technology was implemented using computer algebra methods, which made it possible to obtain parametric classes of models in which the invariance of given quadratic forms is carried out absolutely accurately – in formula form. The determination of the parameter values in the resulting parametric class of models is further carried out by agreement with the measures of the interaction of shells in the model with the probabilities of their interaction in a real physical system. The idea of the described technology and its implementation belong to the authors. Some of its elements were published by the authors earlier, but in this work, for the first time, its systematic description is given for models with complex phase variables and agreement of measures of interaction of shells with probabilities. There have been no similar works by other authors previously. The technology allows you to quickly and accurately generate equations for new non-local turbulence shell models and can be useful to researchers involved in modeling turbulent systems.