RK4 Method Research Articles

The Homotopy Analysis Method for Solving Differential Equations With Generalized Caputo-Type Fractional Derivatives

Abstract This study expands and modifies the homotopy analysis method to handle differential equations with generalized Caputo-type fractional derivatives. Analytical approximate solutions for such models were successfully provided using the proposed modification. The determination of the valid region of convergence for the proposed method, with respect to the auxiliary control parameter, was discussed when using fractional operators. Then, mainly, the accuracy and effectiveness of the proposed method was verified through illustrative examples and comparisons with the predictor corrector method and RK4 method. Finally, it is expected that the studied generalized operators and the suggested method can be widely applied in the field of fractional calculus.

Thermal and multi-boiling analysis of a rectangular porous fin: A spectral approach

Abstract Fins are commonly utilized to enhance (dissipate) heat in various engineering systems that include heat exchangers. In the present investigation, the impact of multi-boiling and thermo-geometric factors on a convective–radiative rectangular porous fin subjected to the temperature-dependent thermal conductivity of linear and non-linear variations is discussed extensively. The governing equations describing the problem were formulated with the aid of Darcy law. Similarity variables were employed to reduce the models to non-dimensional form. The solution of the governing dimensionless equation is approximated using the RK4 and spectral local linearization methods. Before parametric analysis, the agreement between the two numerical methods was established. Findings reveal that the non-linear variation of thermal conductivity shows better thermal efficiency than the linear variation. An improvement in the multi-boiling heat transfer parameter retards the temperature distribution of the fin. Furthermore, increasing the thermo-geometric parameter will result in a progressive decrease in the temperature of the fin. The results obtained in this work will aid in the design of heat exchangers and other heat transfer equipments.

Computational Framework of the SVIR Epidemic Model with a Non-Linear Saturation Incidence Rate

In this study, we developed an autonomous non-linear epidemic model for the transmission dynamics of susceptible, vaccinated, infected, and recovered individuals (SVIR model) with non-linear saturation incidence and vaccination rates. The non-linear saturation incidence rate significantly reduces the death ratio of infected individuals by increasing human immunity. We discuss a detailed explanation of the model equilibrium, its basic reproduction number R0, local stability, and global stability. The disease-free equilibrium is observed to be stable if R0<1, while the endemic equilibrium exists and the disease exists permanently in the population if R0>1. To approximate the solution of the model, the well-known Runge–Kutta (RK4) methodology is utilized. The implications of numerous parameters on the population dynamics of susceptible, vaccinated, infected, and recovered individuals are addressed. We discovered that increasing the value of the disease-included death rate ψ has a negative impact on those affected, while it has a positive impact on other populations. Furthermore, the value of interaction between vaccinated and infected λ2 has a decreasing impact on vulnerable and vaccinated people, while increasing in other populations. On the other hand, the model is solved using Euler and Euler-modified techniques, and the results are compared numerically and graphically. The quantitative computations demonstrate that the RK4 method provides very precise solutions compared to the other approaches. The results show that the suggested SVIR model that approximates the solution method is accurate and useful.

Influence of variable thermal conductivity and dissipation on magnetic Carreau fluid flow along a micro-cantilever sensor in a squeezing regime

Mathematical modeling of squeezing flows finds numerous applications in biological, mechanical, and medical engineering. Sensors feature such flows and can be effectively utilized to control vibrations and regulate lubrication. Magnetic fluids are critical to modern sensor systems. Micro-cantilevers are utilized in biomedical applications as biological, physical, or chemical sensors and operate via the detection of variations in the vibrational frequency or cantilever bending. Hence, a theoretical study is conducted to explore the dissipative flow and thermal characteristics in non-Newtonian boundary layer flow along a micro-cantilever sensor surface suspended in a squeezing regime between parallel plates. To achieve a more refined simulation, the effects of variable thermal conductivity and Joule magnetic dissipation are incorporated. The governing conservation equations for unsteady magnetic Carreau squeezing flow and heat transfer are rendered dimensionless and self-similar via appropriate scaling transformations. Emerging nonlinear coupled boundary value problem is then solved by using RK-4 method. The magnifying Weissenberg parameter decays the flow field. Skin friction is elevated with increasing Carreau power-law index and Weissenberg number. The novelty of the present study is the inclusion of dissipation and thermal conductivity variation effects which extends previous investigations and provides a more accurate appraisal of thermal characteristics in sensor squeezing flows.

A Computational Approach to a Model for HIV and the Immune System Interaction

This study deals with the numerical solution of the human immunodeficiency virus (HIV) infection model, which is a significant problem for global public health. Acquired immunodeficiency syndrome (AIDS) is a communicable disease, and HIV is the causative agent for AIDS, which damages the ability of the body to fight against disease and easily usual innocuous infections attack the body. On entering the body, HIV infects a large amount of CD4+ T-cells and disturbs the supply rate of these cells from the thymus. Herein, we consider the model with variable source terms in which the production of these cells is a monotonically decreasing function of viral load. Based on the reproduction number, we describe the stability of free equilibrium. The continuous Galerkin–Petrov method, in particular the cGP(2)-method, is implemented to determine the numerical solutions of the model. The influence of different parameters on the population dynamics of healthy/infected CD4+ T-cells and free HIV particles are examined, and the results are presented graphically. On the other hand, the model is solved using the fourth-order Runge–Kutta method, and briefly, the RK4-method, and the results of the proposed schemes are compared with those obtained from other classical schemes such as the Bessel collocation method (BCM), Laplace Adomian decomposition method (LADM), perturbation iteration algorithm (PIA), modified variational iteration method (MVIM), differential transform method (DTM), and exponential Galerkin method (EGM), numerically. Furthermore, absolute errors relative to the RK4 method are computed to describe the accuracy of the proposed scheme. It is presented that the cGP(2)-method gains accurate results at larger time step sizes in comparison with the results of the aforementioned methods. The numerical and graphical comparison reveals that the proposed scheme yields more accurate results relative to other traditional schemes from the literature.

Radiative couple stress Casson hybrid nanofluid flow over an inclined stretching surface due to nonlinear convection and slip boundaries

The study of fluid dynamics due to the stretching surface is one of the most eminent topics due to its potential industrial applications viz. drawing wire and plastic films, metal and polymer extrusion, fiber and glass production. In the present article, the author is going to study the effects of hybrid nanofluids flow on an inclined plate including CuO (Copper Oxide), and Cu (Copper). The Casson fluid with a couple-stress term has been used in the flow analysis. The surface of the plate is considered slippery. The convection has been taken nonlinear with thermal radiation. The governing equation of the flow of hybrid nanofluids with energy equation has been transformed into highly nonlinear ODEs through similarity transformation. The proposed model has been solved through a numerical RK-4 method. Significant variables of the physical process such as solar radiation, nonlinear convection parameters, heat transfer rates, and their effect on the solar power plant have been noticed.

Second harmonic generation of zeroth-order Bessel-Gaussian laser beam in collisionless plasma

The objective of present work is to investigate second harmonic generation (SHG) of zeroth-order Bessel-Gaussian (B-G) laser beam in collisionless plasma. The ponderomotive force produces density gradients in plasma by expelling the plasma electrons from high field portion to low field portion thereby modifying plasma’s refractive index. This modification causes self-focusing of main beam. The density gradient further causes excitation of electron plasma wave (EPW) at the main beam frequency. The excited EPW couples with input beam thereby generating second harmonics. The well known WKB approximation and moment theory are employed for obtaining non-linear differential equation for laser beam’s spot size and expression for SHG yield. Further, numerical solutions of these equations are obtained by well known RK-4 method. It has been concluded from the present analysis that transverse component of wave parameter μ, normalized laser intensity and plasma density plays a significant role in affecting laser beam self-focusing and its SHG yield. Results from the present study are important for the laser induced plasma.

The analysis of solitonic, supernonlinear, periodic, quasiperiodic, bifurcation and chaotic patterns of perturbed Gerdjikov–Ivanov model with full nonlinearity

In this manuscript, the perturbed Gerdjikov–Ivanov equation is considered for the investigating model. It is studied to govern the dynamics of soliton propagation through optical fibers, metamaterials or PCF. The observed model is subjected to an extended simple equation technique, which disclosed an abundant of exact solutions including trigonometric function, singular, kink, peakon and compacton solutions. These solutions are made available with their essential conditions, which guarantee the persistence of such optical solitons and portraits using appropriate physical parameters in 3D, 2D and density plots. After that, the traveling wave transformation is used to turn the mth-order nonlinear perturbed Gerdjikov–Ivanov equation into a planar dynamical system. The dynamical and chaotic behaviors of the considered equation are also discussed. The qualitative analysis of the dynamical system and the chaotic behaviors of the perturbed system are investigated using the theory of plane dynamic systems. In addition, we utilize the RK4 method to discover patterns in the dynamical system that seem to be super nonlinear, periodic, and quasiperiodic. The reported results are new and have not been investigated. They can be used in the explanation of the physical phenomena modeled and will give information about the long-term dynamic behavior. Numerical simulations reveal that by changing the frequency and amplitude parameters have an impact on the dynamic behaviors of the system. It is established that the extended simple equation method and dynamical observations offer further influential mathematical tools for constructing exact solutions and their qualitative analysis in NLEEs in mathematical physics.

Exponential space and thermal-dependent heat source effects on electro-magneto-hydrodynamic Jeffrey fluid flow over a vertical stretching surface

This paper deals with the study of an incompressible electro-magneto-hydrodynamic (EMHD) Jeffrey fluid flow over a vertical nonlinear stretching surface of variable thickness. Heat and mass transfer effects are analyzed by considering different source terms like viscous dissipation, Ohmic heating, thermophoresis, Brownian motion, thermal heat source, exponential heat source and activation energy. Governing equations for the flow system are converted into dimensionless forms using appropriate similarity transformations. The solution for the resulting governing equations is obtained by using the shooting technique with RK-4 method. The effects of various physical parameters such as magnetic field parameter [Formula: see text], Grashof number (Gr), solutal Grashof number [Formula: see text], Brownian diffusion parameter [Formula: see text], thermophoresis diffusion parameter [Formula: see text], thermal heat source parameter [Formula: see text], exponential heat source parameter [Formula: see text], Prandtl number (Pr) and Lewis number (Le) are presented with the help of graphs. It is observed that the heat transfer effects increase by increasing thermal and exponential heat sources, and mass transfer effects enhance by increasing the activation energy. Entropy generation for this flow system is also analyzed. Entropy decreases with an increase in the electric field parameter. In contrast, the Bejan number initially increases with an increase in the electric field parameter. After some particular value of electric field parameter, it changes its behavior in the boundary layer and decreases with an increase in the electric field parameter. Entropy and Bejan number increase with an increment in the concentration difference parameter. The accuracy of the results is validated by those of published literature and found in reasonable justification. The present results may be helpful in many engineering and industrial applications like manufacturing lubrication, natural gas networks, cooling nuclear reactors and spray processes.

The significance of Magnetohydrodynamics sutterby nanofluid flow with concentration depending properties across stretching/ shrinking sheet and porosity

In this study, the impact of activation energy and transiting parameters on the two-dimensional stagnation point flow of magnetized Sutterby nanofluid of nano-biofilm incorporating with microorganisms over the porous surface has been examined. Prior research suggests that both the fluid viscosity and thermal conductance are temperature-dependent. Also, the effects of solutal concentration on fluid viscosity, heat capacity and nanofluid properties have been elaborated. In recent years, numerous technical strategies comprising hydromagnetic fluxes and thermal intensification in porosity media, like molding, condenser heat exchanger, liquefied metal filtration, fusion control and nuclear reactor coolant, have been addressed. According to numerous empirical studies, the viscosity and thermal conductivity of the nanoparticles are largely dependent on the intensity of nanoparticles rather than the temperature. The classical RK-4 method with shooting technique has been used. The significance of involving parameters in the domains of heat, velocity, density and concentration has been illustrated. The effect of nondimensional parameters on the skin friction factor, Nusselt number and Sherwood number has been discussed. The nanoparticle density increases with the activating energy effects and thermophoresis factor and rapidly decreases for Lewis number and Brownian factor. The velocity, temperature and concentration profiles increase as the concentration-dependent properties do, but all physical quantities deteriorate for all concentration-varying factors.

High randomness hyperchaos-based parameterizable TRNG: Design, FPGA implementation and exhaustive security analysis

Recently, chaotic systems have become a thrilling discipline for information security applications, particularly in designing entropy sources. This paper presents a novel technique to tame an optimized 4D hyperchaotic Lorenz system by building a reconfigurable high randomness and low-cost hardware true random number generator (HC-HTRNG). Moreover, the proposed architecture has been modeled using the fourth-order Runge–Kutta method (RK4) and the 32 bits (10Q22) fixed-point arithmetic representation. The adopted methodology consists of two crucial points. First, we proposed using both intermediate and final solutions of the RK4 method as a raw data source for our HTRNG. The provided raw data are used to generate random numbers with parameterizable sizes. This approach reduces the design latency from four clock cycles to only one cycle while increasing the generation speed and the entropy value. Second, the designed HC-HTRNG integrates a FIPS 140-2-based built-in self-security test module (BISSTM). The BISSTM is used as an environment failure protection and testing mechanism (EFPTM) to guarantee online and continuous control of the proposed HC-HTRNG’s reliability and availability. Furthermore, the proposed HC-HTRNG has been implemented on the Xilinx ML605 FPGA platform using an RTL design based on the VHDL description of the RK4 method. A panoply of online/offline investigations and experiments were carried out intensely, deeply, and thoroughly to analyze, evaluate and validate the robustness and security aspects of the proposed HTRNG regarding all the aspects related to embedded system security. Notably, the evaluations were conducted regarding of chaos validation, security analysis, statistical tests, design performances, and comparisons. The investigations and implementation findings validate that the proposed architecture can attain high performance in terms of maximum post place and route operating frequency (MPRF) and throughput while occupying low FPGA space. Furthermore, timing and power efficiencies result presents an excellent trade-off between design efficiency and high-performance achievement. To the best of our knowledge, the proposed HC-HTRNG is the only one in the literature whose investigation has been subjected to 25 analyses, including 17 related to security aspects. Moreover, among all TRNGs, our generator is the only one that successfully passed seven statistical test suites, as well as the hardest and the most complex.

Symplectic Dynamics and Simultaneous Resonance Analysis of Memristor Circuit Based on Its van der Pol Oscillator

A non-autonomous memristor circuit based on van der Pol oscillator with double periodically forcing term is presented and discussed. Firstly, the differences of the van der Pol oscillation of memristor model between Euler method and symplectic Euler method, four-order Runge–Kutta method (RK4) and four-order symplectic Runge–Kutta–Nyström method (SRKN4), symplectic Euler method and RK4 method, and symplectic Euler method and SRKN4 method in preserving structure are compared from theoretical and numerical simulations, the symmetry and structure preserving and numerical stability of symplectic scheme are demonstrated. Moreover, the analytic solution of the primary and subharmonic simultaneous resonance of this system is obtained by using the multi-scale method. Finally, based on the resonance relation of the system, the chaotic dynamics behaviors with different parameters are studied.

A Modified Iterative Algorithm for Numerical Investigation of HIV Infection Dynamics

The human immunodeficiency virus (HIV) mainly attacks CD4+ T cells in the host. Chronic HIV infection gradually depletes the CD4+ T cell pool, compromising the host’s immunological reaction to invasive infections and ultimately leading to acquired immunodeficiency syndrome (AIDS). The goal of this study is not to provide a qualitative description of the rich dynamic characteristics of the HIV infection model of CD4+ T cells, but to produce accurate analytical solutions to the model using the modified iterative approach. In this research, a new efficient method using the new iterative method (NIM), the coupling of the standard NIM and Laplace transform, called the modified new iterative method (MNIM), has been introduced to resolve the HIV infection model as a class of system of ordinary differential equations (ODEs). A nonlinear HIV infection dynamics model is adopted as an instance to elucidate the identification process and the solution process of MNIM, only two iterations lead to ideal results. In addition, the model has also been solved using NIM and the fourth order Runge–Kutta (RK4) method. The results indicate that the solutions by MNIM match with those of RK4 method to a minimum of eight decimal places, whereas NIM solutions are not accurate enough. Numerical comparisons between the MNIM, NIM, the classical RK4 and other methods reveal that the modified technique has potential as a tool for the nonlinear systems of ODEs.

Numerical Solutions of a Differential System Considering a Pure Hybrid Fuzzy Neutral Delay Theory

In this paper, we propose and derive a new system called pure hybrid fuzzy neutral delay differential equations. We apply the classical fourth-order Runge–Kutta method (RK-4) to solve the proposed system of ordinary differential equations. First, we define the RK-4 method for hybrid fuzzy neutral delay differential equations and then establish the efficiency of this method by utilizing it to solve a particular type of fuzzy neutral delay differential equation. We provide a numerical example to verify the theoretical results. In addition, we compare the RK-4 and Euler solutions with the exact solutions. An error analysis is conducted to assess how much deviation from exactness is found in the two numerical methods. We arrive at the same conclusion for our hybrid fuzzy neutral delay differential system since the RK-4 method outperforms the classical Euler method.

Active and Passive Controls of Nanoparticles in Bioconvection Nanofluid Flow Containing Gyrotactic Microorganisms

The present work deals with an investigation for the bioconvective flow of nanofluid containing gyrotactic microorganisms over a permeable stretching surface embedded in a non-Darcy porous medium. Two varied cases of controlling nanoparticles say active and passive control have been studied. By introducing a similarity transformation, the governing boundary layer equations are converted into non-linear ordinary equations with pertinent boundary conditions and then solved numerically by a powerful programming MAPLE-18 which includes RK-4 method accompanied by shooting criteria. It is remarkably observed that the fluid velocity is decreased for the increasing value of the porosity parameter while it increases the fluid temperature for both types of control. Also, the present results divulge that the rate of heat transfer in the passive control model is remarkably higher than the active control model whereas the mass flux at the wall for the passive control model is lesser than the active control model.

Quantitative Features Analysis of Water Carrying Nanoparticles of Alumina over a Uniform Surface.

Little is known about the rising impacts of Coriolis force and volume fraction of nanoparticles in industrial, mechanical, and biological domains, with an emphasis on water conveying 47 nm nanoparticles of alumina nanoparticles. We explored the impact of the volume fraction and rotation parameter on water conveying 47 nm of alumina nanoparticles across a uniform surface in this study. The Levenberg–Marquardt backpropagated neural network (LMB-NN) architecture was used to examine the transport phenomena of 47 nm conveying nanoparticles. The partial differential equations (PDEs) are converted into a system of Ordinary Differential Equations (ODEs). To assess our soft-computing process, we used the RK4 method to acquire reference solutions. The problem is investigated using two situations, each with three sub-cases for the change of the rotation parameter K and the volume fraction . Our simulation results are compared to the reference solutions. It has been proven that our technique is superior to the current state-of-the-art. For further explanation, error histograms, regression graphs, and fitness values are graphically displayed.

Local radial basis function-finite difference based algorithms for singularly perturbed Burgers’ model

This work analyzes singularly perturbed Burgers’ model by developing two meshfree algorithms based on local radial basis function-finite difference approximation. The main goal of this task is to present computational modeling of the model when perturbation parameter ɛ→0 where most of the traditional numerical methods fail. In the evolvement of the first algorithm, time derivative is discretized by forward finite difference and then truncation error, stability and convergence analysis are discussed for the semi-discrete model. After that, local radial basis function-finite difference approximation is used for spatial discretization. In the second numerical algorithm, local radial basis function-finite difference and RK4 method are applied for spatial and fully discretization respectively. Also, the stability of the scheme is discussed via matrix method.

Numerical computation of 3D Brownian motion of thin film nanofluid flow of convective heat transfer over a stretchable rotating surface

This research examines the thin-film nanomaterial movement in three dimensions over a stretchable rotating inclined surface. Similarity variables are used to transform fundamental systems of equations into a set of first-order differential equations. The Runge–Kutta Fourth Order approach is utilized for numerical computations. The impact of embedded parameters (variable thickness, unsteadiness, Prandtl number, Schmidt number, Brownian-motion, and thermophoretic) is examined carefully. Physically and statistically, the indispensable terms namely Nusselt and Sherwood numbers are also investigated. Results indicated that, as the dimensionless parameter S raises, the temperature field decreases. In reality, as the values of S increases, heat transmission rate from the disc to the flowing fluid reduces. Internal collisions of liquid particles are physically hampered at a low rate. The momentum boundary layer is cooled when the parameter S is increased, as a consequence local Nusselt number rises. Sherwood number decreases as the parameter S increases because of inter collision of the microscopic fluid particles. Enhancing in the apparent viscosity and concentrations of the chemical reactions, a higher Schmidt number, Sc, lowers the Sherwood number. With increasing values of Prandtl number the Nusselt number decreases. For validation purpose, the RK4 method is also compared with homotopy analysis method (HAM). The results are further verified by establishing an excellent agreement with published data.

Switch controllers of an n-link revolute manipulator with a prismatic end-effector for landmark navigation.

Robotic arms play an indispensable role in multiple sectors such as manufacturing, transportation and healthcare to improve human livelihoods and make possible their endeavors and innovations, which further enhance the quality of our lives. This paper considers such a robotic arm comprised of n revolute links and a prismatic end-effector, where the articulated arm is anchored in a restricted workspace. A new set of stabilizing switched velocity-based continuous controllers was derived using the Lyapunov-based Control Scheme (LbCS) from the category of classical approaches where switching of these nonlinear controllers is invoked by a new rule. The switched controllers enable the end-effector of the robotic arm to navigate autonomously via a series of landmarks, known as hierarchal landmarks, and finally converge to its equilibrium state. The interaction of the inherent attributes of LbCS that are the safeness, shortness and smoothness of paths for motion planning bring about cost and time efficiency of the controllers. The stability of the switched system was proven using Branicky’s stability criteria for switched systems based on multiple Lyapunov functions and was numerically validated using the RK4 method (Runge–Kutta method). Finally, computer simulation results are presented to show the effectiveness of the continuous time-invariant velocity-based controllers.

Galerkin time discretization scheme for the transmission dynamics of HIV infection with non-linear supply rate

<abstract><p>The present work implements the continuous Galerkin-Petrov method (cGP(2)-method) to compute an approximate solution of the model for HIV infection of $ \text{CD4}^{+} $ T-cells. We discuss and analyse the influence of different clinical parameters on the model. The work also depicts graphically that how the level of $ \text{CD4}^{+} $ T-cells varies with respect to the emerging parameters in the model. Simultaneously, the model is solved using the fourth-order Runge Kutta (RK4) method. Finally, the validity and reliability of the proposed scheme are verified by comparing the numerical and graphical results with those obtained through the RK4 method. A numerical comparison between the results of the cGP (2) method and the RK4 method reveals that the proposed technique is a promising tool for the approximate solution of non-linear systems of differential equations. The present study highlights the accuracy and efficiency of the proposed schemes as in comparison to the other traditional schemes, for example, the Laplace adomian decomposition method (LADM), variational iteration method (VIM), homotopy analysis method (HAM), homotopy perturbation method (HAPM), etc. In this study, two different versions of the HIV model are considered. In the first one, the supply of new $ \text{CD4}^{+} $ T-cells from the thymus is constant, while in the second, we consider the production of these cells as a monotonically decreasing function of viral load. The experiments show that the lateral model provides more reasonable predictions than the former model.</p></abstract>