Nonlinear Differential Equations Research Articles

Existence and Stability for Fractional Differential Equations with a ψ–Hilfer Fractional Derivative in the Caputo Sense

This article aims to explore the existence and stability of solutions to differential equations involving a ψ-Hilfer fractional derivative in the Caputo sense, which, compared to classical ψ-Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing with initial conditions. We discovered that the ψ-Hilfer fractional derivative in the Caputo sense can be represented as the inverse operation of the ψ-Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense. Additionally, we applied Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense, and further discussed the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of these solutions. Finally, we illustrated our results through case studies.

Entire Solutions of Some Nonlinear Homogeneous Differential Equations

Entire Solutions of Some Nonlinear Homogeneous Differential Equations

Assisting and opposing mixed convective flow over a solid sphere at its bottom stagnation point with a constant surface heat, mass and motile microorganism flux

This study examined the laminar mixed convective flow around the lower point of stagnation for a solid sphere with a steady flux of gyrotactic microbial motility, mass and temperature. Biological convection phenomena have been considered in this study due to their intermixing traits and capacity in order to enhance mass transit in numerous biotechnology and biological systems. The first objective of this study is to examine and establish the mathematical formulation comprising partial differential equations for energy, momentum, mass conservation and mobile microorganism conservation balances. To do numerical calculations, the controlling partial differential equations are initially transformed by similarity transformations into a collection of interconnected non-linear ordinary differential equations. These equations are then solved by using the MATLAB Bvp4c scheme. The results indicate that several controlling parameters, particularly bioconvection parameters such as the Lewis number ( Le ), Bioconvection Lewis parameter ( Lb ) and Bioconvection Peclet number ( Pe ), have significant effects on flow transfer rates. A dual solution is observed in a certain region of mixed convection λ , where the mixed convection parameter is within the range of 0.41–0.65. The findings reveal significant results from dual solution analysis, highlighting the occurrence of unstable and stable phenomena for specific values of bioconvection parameters exclusively in the case of opposing flow. The assisting flow demonstrates only a single solution that is physically stable and unique. Observing lower stagnation point flow is essential for predicting and analysing complex fluid–microorganism interactions in various scientific and engineering applications such as designing spherical-shaped microbial fuel cell. Optimising flow conditions can enhance the efficiency of processes like wastewater treatment, where microorganisms play a crucial role in breaking down pollutants.

A finite-difference-based Adams-type approach for numerical solution of nonlinear fractional differential equations: A fractional Lotka-Volterra model as a case study

A finite-difference-based Adams-type approach for numerical solution of nonlinear fractional differential equations: A fractional Lotka-Volterra model as a case study

Influence of non-linear thermal radiation on the dynamics of homogeneous and heterogeneous chemical reactions between the cone and the disk

Abstract Purpose The current work presents a theoretical framework to boost heat transmission in a ternary hybrid nanofluid with homogeneous and heterogeneous reactions in the conical gap between the cone and disk apparatus. Furthermore, the impacts of non-linear thermal radiation on the ternary hybrid nanofluid composed of white graphene, diamond, and titanium dioxide dispersed in water are analyzed. Originality/value The combination of cone and disk systems is crucial for designing efficient heat exchange devices in the field of biomedical science for various purposes. For instance, in medical devices, the cone–disk apparatus is used to study the flow and heat transfer characteristics for better design and functionality. Hence, a sincere attempt has been made to study the impact of homogeneous and heterogeneous reactions on the nanofluid flow between the cone and disk in the presence of non-linear thermal radiation. Design/methodology/approach The mathematical model’s governing equations are partial differential equations (PDEs) which are then transformed into non-linear ordinary differential equations through appropriate similarity transformations. These transformed resultant equations are approximated by the Runge–Kutta–Fehlberg fourth/fifth order (RKF45) technique. The influence of essential aspects on the flow field, heat, and mass transfer rates was analyzed using a graphical representation. Findings The interesting part of this research is to discuss the power of parameters in three cases, namely, (1) rotating cone/disk, (2) rotating cone/stationary disk, and (3) stationary cone/rotating disk. Furthermore, the thermal variation of the fluid is analyzed by an artificial neural network with the help of the Levenberg–Marquardt backpropagation algorithm. The regression analysis, mean square error, and error histogram of the neural network are analyzed using this algorithm. From the graph, it is perceived that the flow field climbed up significantly with an increase in the values of radiation parameters in all cases. Also, it is noticed that temperature upsurges significantly by upward values of solid volume fraction of the nanoparticles ( ϕ \phi ).

Modified Hermite Wavelet Discrete Matrix Approach for the Brusselator Chemical Model

AbstractThe primary goal of this study is to create a wavelet collocation technique that can be used to solve nonlinear fractional order systems of ordinary differential equations, which are equations that arise in modeling problems related to auto‐catalytic chemical reactions. Using the Hermite wavelet collocation method (HWCM), the system of nonlinear ordinary differential equations of integer and fractional order is numerically solved. The nonlinear Brusselator system is transformed into an algebraic equation system using the collocation method and the fractional derivative operational matrices. The Newton‐Raphson method is used to solve these algebraic equations, and the approximate values of the derived unknown coefficients are substituted. Through the numerical examples, the method's computational effectiveness and correctness are illustrated with various model constraints. A numerical comparison is made between the current approach ND solver, RK method, and Haar wavelet method (HWM). The efficiency and reliability of the developed strategy's performance are shown in graphs and tables. The created Hermite wavelet collocation method is resilient and has good accuracy compared to current methods found in the literature. Numerical computations are performed through Mathematica, a mathematical software.

MHD double diffusive convective squeezing ternary nanofluid flow between parallel plates with activation energy and viscous dissipation

Purpose This study aims to present the consequences of activation energy and the chemical reactions on the unsteady MHD squeezing flow of an incompressible ternary hybrid nanofluid (THN) comprising magnetite (FE3O4), multiwalled carbon nano-tubes (MWCNT) and copper (Cu) along with water (H2O) as the base fluid. This investigation is performed within the framework of two moving parallel plates under the influence of magnetic field and viscous dissipation. Design/methodology/approach Due to the complementary benefits of nanoparticles, THN is used to augment the heat transmit fluid’s efficacy. The flow situation is expressed as a system of dimensionless, nonlinear partial differential equations, which are reduced to a set of nonlinear ordinary differential equations (ODEs) by suitable similarity substitutions. These transformed ODEs are then solved through a semianalytical technique called differential transform method (DTM). The effects of several changing physical parameters on the flow, temperature, concentration and the substantial measures of interest have been deliberated through graphs. This study verifies the reliability of the results by performing a comparison analysis with prior researches. Findings The enhanced activation energy results in improved concentration distribution and declined Sherwood number. Enhancement in chemical reaction parameter causes disparities in concentration of the ternary nanofluid. When the Hartmann number is zero, value of skin friction is high, but Nusselt and Sherwood numbers values are small. Rising nanoparticles concentrations correspond to a boost in overall thermal conductivity, causing reduced temperature profile. Research limitations/implications Due to its firm and simple nature, its implications are in various fields like chemical industry and medical industry for designing practical problems into mathematical models and experimental analysis. Practical implications Deployment of the squeezed flow of ternary nanofluid with activation energy has significant consideration in nuclear reactors, vehicles, manufacturing facilities and engineering environments. Social implications This study would be contributing significantly in the field of medical technology for treating cancer through hyperthermia treatment, and in industrial processes like water desalination and purification. Originality/value In this problem, a semianalytical approach called DTM is adopted to explore the consequences of activation energy and chemical reactions on the squeezing flow of ternary nanofluid.

The discrete fractional Karamata theorem and its applications

Abstract We establish a fractional extension of the discrete Karamata integration theorem for two types of fractional sum operators. This result, along with other properties of regularly varying sequences and the tools such as comparison theorems and a fixed point theorem in FK-space, is then used to study asymptotic properties of solutions to a nonlinear fractional difference equation. We also establish and utilize a fractional extension of the Stolz–Cesáro theorem, i.e., of the discrete l’Hospital rule. Both, the discrete fractional Karamata theorem and the discrete fractional l’Hospital rule, are believed to find applications in a broader context within the discrete fractional calculus.

Existence and forms of entire solutions to system of non-linear partial differential equations

Existence and forms of entire solutions to system of non-linear partial differential equations

Investigation of oceanic wave solutions to a modified (2+1)-dimensional coupled nonlinear Schrödinger system

This paper explores the oceanic wave characteristics exhibited by a modified integrable generalized [Formula: see text]-dimensional nonlinear Schrödinger system of equations through variable coefficients. Two newly modified methods, specifically the improved nonlinear Ricatti equation method and the improved sub-equation method, have been proposed to investigate the aforementioned nonlinear system. Through the utilization of these methods, we successfully obtain traveling and solitary waves solutions for this nonlinear system. We emphasize several constraint conditions that serve to guarantee the existence of these solutions. More comprehensive information about the physical dynamical representation of some of the solutions presented is illustrated through graphical depictions. The Mathematica software package is employed to produce both three-dimensional and their corresponding contour plots, thereby improving the visualization and comprehension of the solutions. This paper illustrates that the two proposed approaches provide straightforward and efficient means of acquiring various types of soliton, rational, trigonometric, hyperbolic and exponential solutions. Moreover, they present a more potent mathematical tool for addressing a variety of other nonlinear partial differential equations that hold significance in the field of applied science and engineering.

Galerkin’s Method and Multivariate Newton’s Method for the Nonlinear Deformation of the Magneto-Electro-Elastic Bi-Layered Laminates

In this paper, mathematical modelling for the large deformation of a magneto-electro-elastic rectangular bi-layered laminate with general boundary conditions is presented. Constitutive equations involving the magneto-electro-elastic (MEE) material properties are introduced, Maxwell equations accounts for the electric and magnetic effects are also utilized. First-order shear deformation theory (FSDT) considering the von Karman nonlinear strain is adopted, and the plain strain/stress assumption applicable for thin plate analysis is used. A rather compact set of governing equations related to kinematical variables, electric/magnetic potentials and the Airy stress function is obtained as a consequence of the preliminary condensation for the electro-magnetic state to the plate kinematics. Semi-analytic solution for a bi-layered BaTiO3-CoFe2O4 laminate with specified boundary conditions subjected to various external applied loads is performed. By employing the Bubnov-Galerkin method, the set of nonlinear partial differential equations is transformed to a set of third-order nonlinear algebraic equations for the static deformation due to applied load. Numerical results are carried out by using the multivariate Newton's method with respect to various volume fractions indicating the volume ratio between piezoelectric BaTiO3 layer and piezomagnetic CoFe2O4. From the result, the nonlinearity of the von Karman strain appears to enhance system rigidity as smaller deformations will be detected when external load is applied. Also, some other interesting results are obtained which could be useful to future analysis and design of multiphase composite plates.

A novel efficient analytical technique and its application to fractional Cauchy reaction–diffusion equations

In this research paper, we suggest a novel analytical method to obtain the approximate solutions of linear and nonlinear differential equations of arbitrary order. The proposed technique is a good combination of Kharrat–Toma transform and q-homotopy analysis method. Additionally, we also find out the solution of Cauchy reaction–diffusion equation associated with regularized version of Hilfer–Prabhakar fractional derivative. The Cauchy reaction–diffusion equation is broadly used to describe the dynamical processes in physics, geology, biology, ecology, etc. Some characteristic properties of the considered model and existence as well as the uniqueness of the solutions are also discussed. The outcomes of the considered model are exhibited graphically to show the accuracy and efficiency of the suggested method.

Other fundamental systems of solutions of the differential equation (D 2 – 2αD + α 2 + β 2) n y = 0, β ≠ 0

Abstract This paper generalizes the results of the first part of the paper [Fecenko, J.—Diekema, E.: On the linear (in)dependence of sequences of derivatives of the functions xn sin x and xn cos x, http://arxiv.org/abs/2305.11184]. The main goal is to prove that the sequence of functions f(x), Df(x), …, D 2n–1 f(x), where f(x) is x n−1e αx sin βx or x n−1 e αx cos βx for β ≠ 0, forms a fundamental system of solutions of the differential equation (D 2 – 2αD + α 2 + β 2) n y = 0. Or more generally, that the sequence of functions Dkf(x), D k+1 f(x), …, D 2n+k–1 f(x), k ∈ ℕ also forms a fundamental system of solutions of the aforementioned differential equation. The problem is solved using a suitable transformation of the Wronskian matrix, by solving a nonlinear differential equation and then calculating the determinant of the modified Wronskian matrix by applying Laplace’s generalized expansion.

Multiple solitons, multiple lump solutions, and lump wave with solitons for a novel (2+1)-dimensional nonlinear partial differential equation

The primary aim of this paper is to explore exact solutions to a novel (2+1)-dimensional water wave equation that models oceanic wave phenomena. We begin by applying the Hirota bilinear transformation method to derive multi-soliton solutions, including 3-soliton and 4-soliton solutions. Then, utilizing the bilinear form of the equation and the long-wave limit method, we identify multiple lump solutions and interaction solutions between lumps and solitons. These include 1-lump, 2-lump, and 3-lump solutions, as well as interactions between a 1-lump and a 1-soliton, and between a 1-lump and 2-solitons. The physical dynamics of these solutions are visually represented, offering insight into the corresponding oceanic wave phenomena.

Autonomous longitudinal separation control of aircraft in random wind fields based on MPC

Aiming at the problem of autonomous longitudinal separation under random disturbance of flight path, the random factor of high-altitude wind often leads to poor robustness of longitudinal separation between two aircraft. This paper proposes an autonomous longitudinal separation control method for aircraft based on model predictive control. Firstly, the nonlinear kinematic differential equation of the wind field difference between the two aircraft and the longitudinal separation is established, and the linear time-varying prediction model is derived. The longitudinal separation and path deviation distance of the two aircraft are selected as the optimization objectives, the high-altitude wind is the random disturbance, and the true airspeed and yaw angle of the leading aircraft are measurable. Terminal equality constraints are added to the air safety and aircraft performance constraints to maintain the stability of the system. To verify the effectiveness of the algorithm, within the prescribed simulation time of 120 seconds, this paper sets three groups of different expected intervals 12, 13, 14km. Through the designed MPC controller, by controlling the true airspeed and yaw angle of the trailing aircraft within the rolling time domain period, the interval curve between the two aircraft is relatively smooth and is always not less than the minimum safety interval of 10 km. It stabilizes at the expected target interval at the 74th, 90th and 118th seconds respectively, and begins to return to the route at the 58th, 74th and 95th seconds, and finally returns to the center line of the route; two groups of wind field control groups are set up, one group is the forecast wind intensity increased by 2 times, and the other group is the turbulent wind disturbance intensity increased by 8 times. Both can smoothly and stably establish the expected interval of 12 km at the 61st and 72nd seconds respectively.

Investigation of periodic heat transfer on the transient thermal characteristics of fully wet moving semi-spherical porous fin composed of functionally graded material

This study investigates the thermal behaviour of a fully wet, moving semi-spherical porous fin made of linear Functionally Graded Material (FGM). This investigation examines the fin response to convective-radiative heat transfer under periodic variations in base temperature across three different FGM scenarios: Homogeneous material (HM), Functionally Graded Material I (FGM I) and Functionally Graded Material II (FGM II). The resultant nonlinear partial differential equation is accurately solved using the Finite Difference Method (FDM) and outcomes are benchmarked against existing literature. This research pioneers an investigation into the effects of periodic heat transfer on the detailed thermal profiles of FGM fin, an area not been explored in existing literature. It significantly enhances understanding of the influence of oscillatory base temperatures on thermal management within FGMs, uncovering pivotal insights into the interaction between material grading, amplitude of temperature oscillation and their collective effects on thermal efficiency. Additionally, this research assesses the influence of variables such as the amplitude of input temperature, frequency of oscillation, thermal conductivity grading parameter and others on temperature distribution along the fin length and over dimensionless time. Notably, periodic heat transfer induces a dynamically wavy thermal profile in the fin over time, attributable to oscillating base temperatures. Moreover, the analysis demonstrates that FGM II fin exhibit the most significant temperature distribution, followed by FGM I and HM. The fin heat transfer rate is profoundly influenced by the amplitude of input temperature and the thermal conductivity grading parameter. These insights are pivotal for optimizing fin designs in critical applications, including electronics cooling, HVAC systems, automotive engine cooling and solar energy collection, substantially improving upon traditional designs.

A new approach to determine occupational accident dynamics by using ordinary differential equations based on SIR model

The motivation of this study is to develop and establish an occupational accident dynamical model (OA model) based on Susceptible-Infected-Recovered framework. In order to investigate the dynamics of the OA model, monthly occupational accident data from Turkey between 2013 and 2020 has been selected as dataset. The OA model is defined by a coupled first-order ordinary nonlinear differential equation with four variables. In addition, the relationships between these variables are described with ten parameters. The OA model’s characterization of the equilibrium points is analyzed by investigating the behaviors of these points according to the eigenvalues derived from the Jacobian matrix. Also, the stability of these points is obtained according to the eigenvalues. These results show the behavior of the system near equilibrium points. After that, the reproduction number is computed by using the next-generation matrix method. The calculated reproduction number for given parameters reveals that the OA model is unstable. The OA model is numerically solved in 96 steps, with a time interval of 1 month, using the ODE45 Matlab routine based on the explicit Runge–Kutta algorithm. In addition, a modified OA model is developed by adding the Occupational Health and Safety (OHS) re-training parameter to the OA model to observe a reducing effect on occupational accident numbers. The main results of this study provide a new approach about the future estimation of the number of occupational accidents. Furthermore, through the comparison of numerical results from both models, the study demonstrates that national safety policies, particularly those enhancing the efficacy of OHS training, can effectively mitigate accidents.

Conformal symplectic and constraint-preserving model order reduction of constrained conformal Hamiltonian systems

Dissipative and constrained dynamical systems model various physical phenomena comprising damping and constraining forces. These forces give rise to specific geometric properties of the dynamical system's governing equations. In this paper, we reduce and solve high-dimensional parameterized constrained and damped nonlinear differential equations with structure-preserving algorithms. These algorithms preserve the constrained equation's geometric properties, such as conformal symplecticness and phase-space, through model order reduction and numerical integration. We develop a reduced model based on an orthosymplectic reduced basis and establish the well-posedness and conformal symplecticness of the reduced model. Moreover, we derive a priori error bounds for state space and constraints errors. Subsequently, we utilize two RATTLE-based integrators to simulate the full and reduced models while preserving their geometric properties. Numerical experiments on constrained damped oscillators and a constrained lattice grid justify theoretical results and demonstrate the advantages of the structure-preserving computational methods for constrained dynamical systems.

Novel results from quadratically nonlinear elastic wave models using Murnaghan’s potential

AbstractIn this article, we study one, two and three-dimensional nonlinear elastic wave equations using quadratically nonlinear Murnaghan potential. We employ two effective methods for obtaining approximate series solutions the Adomian decomposition and the variational iteration method. These methods have the advantage of not requiring any physical parametric assumptions in the problem. Finally, these methods can generate expansion solutions for linear and nonlinear differential equations without perturbation, linearization, or discretization. The results obtained using the adopted methods along various initial and boundary conditions are in excellent agreement with the numerical results on MATLAB, which show the reliability of our methods to these problems. We came to the conclusion that our methods are accurate and simple to use.

Data-driven solutions and parameter estimations of a family of higher-order KdV equations based on physics informed neural networks

Physics informed neural network (PINN) demonstrates powerful capabilities in solving forward and inverse problems of nonlinear partial differential equations (NLPDEs) through combining data-driven and physical constraints. In this paper, two PINN methods that adopt tanh and sine as activation functions, respectively, are used to study data-driven solutions and parameter estimations of a family of high order KdV equations. Compared to the standard PINN with the tanh activation function, the PINN framework using the sine activation function can effectively learn the single soliton solution, double soliton solution, periodic traveling wave solution, and kink solution of the proposed equations with higher precision. The PINN framework using the sine activation function shows better performance in parameter estimation. In addition, the experiments show that the complexity of the equation influences the accuracy and efficiency of the PINN method. The outcomes of this study are poised to enhance the application of deep learning techniques in solving solutions and modeling of higher-order NLPDEs.