RK4 Method Research Articles

Solution of Nonlinear Reaction-Diffusion Model in Porous Catalysts Arising in Micro-Vessel and Soft Tissue Using a Metaheuristic

The steady-state reactive transport model (RTM) is a generalization of the nonlinear reaction-diffusion model in porous catalysts. The RTM is expressed as a non-linear ordinary differential equation of second-order with boundary conditions. Artificial neural network (ANN), Particle swarm optimization (PSO), and hybrid of PSO-SQP (Sequential Quadratic Programming) are used to obtain accurate, approximate solutions to the non-linear RTM. The proposed technique is applied to three different cases of non-linear RTM. The properties of the nonlinear reactive transport model in porous catalysts are investigated by considering various cases based on variation in the half-saturation concentration “ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> ” and the characteristic reaction rate “ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> .” The stability, reliability, and exactness of the proposed technique are established through comparison with the outcomes of the standard numerical procedure with the RK4 method and along with the different performance indices, which are Root-Mean-Square Error (RMSE), (TIC), Absolute Error (AE), and Mean Absolute Deviation (MAD).

Dynamic analysis of a predator-prey model with Holling type-II functional response and prey refuge by using a NSFD scheme

This study proposes and analyzes a nonstandard finite difference (NSFD) scheme to retain the fundamental qualitative aspects of a predator-prey interaction with a Holling type-II functional response and prey refuge. The existence and stability of fixed points are examined. A few numerical examples are presented to back up the theoretical findings. The results show that the numerical simulations match well with the theoretical outcomes. Moreover, the model experiences transcritical, period-doubling, and Neimark-Sacker bifurcation. Furthermore, numerical simulations show that standard numerical methods like the Euler method and the classical RK4 method are not dynamically consistent with the continuous model. As a result, they do not accurately reflect the continuous model’s behavior. The proposed NSFD scheme, on the other hand, is shown to be suitable and adequate for solving the continuous model.

Radial basis functions based meshfree schemes for the simulation of non-linear extended Fisher–Kolmogorov model

This work offers two radial basis functions (RBFs) based meshfree schemes for the numerical simulation of non-linear extended Fisher–Kolmogorov model. In the development of the first scheme, first of all, time derivative is discretized by forward finite difference and then stability and convergence of the semi-discrete model is analyzed in L2 and H02 spaces. After that, RBF-differential quadrature method (RBF–DQM) is applied for fully discretization. Then, the obtained system of algebraic linear equations is solved by Gauss-elimination method. In the second numerical scheme, first RBF–DQM is applied for spatial derivatives approximation and then we obtained a system of nonlinear ODEs. After that, the system of ODEs is solved by RK4 Method. Also, the stability of the scheme is discussed via matrix method. In order to prove the meshfree property of the proposed methods, we considered non-uniform angular and rectangular domains with different radius. In the supporting domain, we used 5,10,15,20 and 25 supporting nodes for each nodes. Six instances of the model are considered to examine the reliability and chastity of the proposed schemes and found accurate results.

THREE POINTS BLOCK EMBEDDED DIAGONALLY IMPLICIT RUNGE-KUTTA METHOD FOR SOLVING ODES

Block Embedded Diagonally Implicit Runge-Kutta (BEDIRK4(3)) me- thod derived using Butcher analysis and equi-distribution of error approach is outperformed standard Runge-Kutta (RK) formulae. BEDIRK4(3) method produces approximation to the solution of initial value problem (IVP) at a block of three points simultaneously. The standard one step RK3(2) method is used to approximate the solution at the first point of the block. At the second points the solution is approximated using RK4(2) method which is generated by the previous research. The same approach is used to obtain the solution at the third point. The code for this method was built and the algorithm developed is suitable for solving stiff system. The efficiency of the method is supported by some numerical results.

High‐order implicit time‐stepping with high‐order central essentially‐non‐oscillatory methods for unsteady three‐dimensional computational fluid dynamics simulations

AbstractThis article develops high‐order implicit time‐stepping methods combined with the fourth‐order central essentially‐non‐oscillatory (CENO) scheme for stiff three‐dimensional computational fluid dynamics problems having disparate characteristic time scales. Both aerodynamic and magnetohydrodynamic problems are considered on three‐dimensional multiblock body‐fitted grids with hexahedral cells. Several implicit time integration methods of third‐ and fourth‐order accuracy are considered, including the multistep backward differentiation formulas (BDF4), multistage explicitly singly diagonally implicit Runge‐Kutta (ESDIRK4), and Rosenbrock‐type methods (ROS34POW2). The resulting nonlinear algebraic system of equations is solved via a preconditioned Jacobian‐free inexact Newton–Krylov method with additive Schwarz preconditioning using block‐based incomplete LU decomposition. The performance of the high‐order implicit time‐stepping methods on smooth and stiff problems is compared with a standard fourth‐order explicit Runge‐Kutta (RK4) method. It is shown that the Rosenbrock methods, despite their advantage of only requiring the solution of linear systems, have significant drawbacks in terms of robustness issues for highly nonlinear compressible flows. The implicit BDF4 and ESDIRK4 methods are found to be much more efficient than the explicit fourth‐order RK4 method for a stiff resistive magnetohydrodynamic (MHD) problem discretized with the fourth‐order CENO method. When applied to the problem of vortex shedding governed by the Navier–Stokes equations, an A‐stable ESDIRK4 scheme proved to be the more robust and accurate implicit time‐marching scheme and was able to offer significant speedup compared with the RK4 method. Initial results are also shown for high‐order implicit time integration applied to two problems with discontinuities. The current study represents the first to achieve high‐order implicit time integration for MHD, enabling large time steps and substantial speedups for stiff MHD problems with high‐order accuracy, and it also represents the first to establish high‐order implicit time integration for high‐order CENO in space.

Dynamical behaviour of HIV Infection with the influence of variable source term through Galerkin method

In this paper, we formulate the intricate process of Human Immunodeficiency Virus (HIV) infection in a mathematical model. In the formulation of the model, the density of CD4+ T-cells is categorized into healthy and infected classes while the density of free HIV viruses in the human host’s blood is considered a separate class. It is reported that when HIV enters a human’s body, it infects a large amount of CD4+ T-cells and unbalance the supply of new cells from the thymus. Therefore, we incorporate variable source term relying on the viral load for the supply of new cells instead of using a fixed value for the supply of these cells. We implement a continuous Galerkin Petrov time discretization scheme, particularly cGP(2)-scheme to visualize the dynamical behavior of the mentioned model and a detailed description of the effects of different physical parameters of interest are depicts graphically and discuss that how the level of healthy, infected CD4+ T-cells and free HIV particles varies concerning the emerging parameters in the model. In the proposed cGP(2)-method, two unknowns terms can be found by solving a 2×2 block system. This method is accurate of order 3 in the whole time interval and shows even superconvergence of order 4 in the discrete-time points. Furthermore, determine the solution of the model utilizing the fourth-order Runge-Kutta (RK4)-method and find out the absolute errors between the solutions obtained from both approaches. Finally, the validity and reliability of the proposed scheme are verified by comparing the numerical and graphical results with the RK-4 method. We examine the accuracy of the cGP(2)-sheme and observe that it produces more accurate results at relatively larger step sizes as in comparison to RK-4 scheme. Correspondence between the findings of cGP(2)-method and RK4-method reveal that the new technique is a promising tool for obtaining approximate solutions to the nonlinear systems of real-world problems.

Non-linear interaction of Cosh-Gaussian beam in thermal quantum plasma under combined influence of relativistic-ponderomotive force

Non-linear interaction of Cosh-Gaussian beam in thermal quantum plasma (TQP) under combined influence of relativistic-ponderomotive force (RP Force) is explored. The authors have made use of parabolic equation approach for obtaining 2nd order differential equation governing behavior of beam waist of laser beam against normalized propagation distance through WKB and paraxial theory approaches. RK4 method is utilized for carrying out numerical simulations of this differential equation. Well established laser plasma parameters are taken for investigating response of beam waist with normalized distance. The comparison of present results is made with simple quantum plasma and relativistic plasma cases.

The Multiplicative Base Solution to the Differential Equations in Analog Circuits

This work focuses to solve any order of scalar differential equation involved in analog circuit representation. These types of mathematical representations have many applications in analysis and processing such as noise, filter, audio, RLC distributed interconnection (nodes) and transmission lines. Such systems are represented with scalar type differential equations and use numerical method to find a solution. One of the most successful methods is the fourth-order Runge–Kutta. This study introduced a multiplicative version of Runge–Kutta (MRK4) method. The performance analysis of the MRK4 is examined based on the error analysis method. The MRK4 method is applied to solve equations representing the linear and the nonlinear type systems. Results indicate the MRK4 to be superior with respect to the RK4 method.

The extinction and persistence of a stochastic model of drinking alcohol

The dynamical behavior of stochastic model for drinking evolution is researched in this paper. The drinking evolution criteria is comprised of three distinctive compartments namely, susceptible population S (non-consumers), Risk drinkers R and moderate drinkers M. At first, the Lyapunov function is constructed and analyzed followed by the feasibility and positivity of the solution of the proposed model. The sufficient conditions for the extinction and the persistence of the stochastic drinking model are derived through Lyapunov function. Furthermore, the proposed model is tested for an approximate solution using well known RK4 method. The proposed scheme has been simulated deterministically as well as stochastically. The obtained solutions shows strong convergence to the stochastic solution.

Magneto hydrodynamic and dissipated nanofluid flow over an unsteady turning disk

A modern development in the field of fluid dynamics emphasizes on nanofluids which maintain remarkable thermal conductivity properties and intensify the transport features of heat in fluids. The present communication provides an innovative idea of MHD (magneto-hydrodynamics) unsteady, incompressible nanoliquid flow due to the stretching rotating disk with the effect of Joule heating and dissipation. The engine oil is used as a carrier fluid for an immersed rotating disk. A special type of nanoliquid, which consists of cylindrical shape nano materials CNTs (Carbon Nanotubes), is being taken into an account. The CNTs are assembled of both single and multi-walled carbon nanotubes. Some other factors such as the effect of joule heating and viscus dispassion are also used in this investigation. The foremost set of PDEs (Partial Differential Equations) of our model is reformed to the dimensionless form via invoking suitable variables. The resultant set of equations is sketched out through RK-4 method. Furthermore, the velocity profile and energy distribution versus dimensionless flow factors have been sketched and discussed. Also, the outcomes of important engineering curiosity like Nusselt number and skin friction are depicted and interpreted by taking various model factors. Radial and transverse velocity fields, declines via [Formula: see text] (magnetic factor), while the temperature field enhances. Furthermore, the larger estimation of [Formula: see text] leads to enhance the velocity field while higher estimation of [Formula: see text] reducing the velocity and temperature fields. The current work has an extensive verity use such as nano-mechanics and electro-magnetic micro pumps.

Numerical Simulation for the Treatment of Nonlinear Predator–Prey Equations by Using the Finite Element Optimization Method

This article aims to introduce an efficient simulation to obtain the solution for a dynamical–biological system, which is called the Lotka–Volterra system, involving predator–prey equations. The finite element method (FEM) is employed to solve this problem. This technique is based mainly upon the appropriate conversion of the proposed model to a system of algebraic equations. The resulting system is then constructed as a constrained optimization problem and optimized in order to get the unknown coefficients and, consequently, the solution itself. We call this combination of the two well-known methods the finite element optimization method (FEOM). We compare the obtained results with the solutions obtained by using the fourth-order Runge–Kutta method (RK4 method). The residual error function is evaluated, which supports the efficiency and the accuracy of the presented procedure. From the given results, we can say that the presented procedure provides an easy and efficient tool to investigate the solution for such models as those investigated in this paper.

The accuracy comparison of the RK-4 and RK-5 method of SEIR model for tuberculosis cases in South Sulawesi

This research discusses about the accuracy of the 4th and 5th order Runge-Kutta methods (RK-4 and RK-5) as a numerical solution in the Susceptible-Exposed-Infected-Recovered (SEIR) mathematical model of Tuberculosis (TB) transmission in South Sulawesi. The data used are secondary data on the number of TB patients in South Sulawesi sourced from the South Sulawesi Health Office. The research begins by further examining the SEIR model on TB transmission, then looking for general solutions of the SEIR model with the 4th-order and 5th-order Runge-Kutta methods, parameter determination, simulation and results analysis. In this research, the movement graph is obtained from the SEIR model with real data. After analysing the numerical simulation, the accuracy of the 4th-order and 5th-order Runge-Kutta methods is compared, then the two methods are compared to observe a more accurate method. The result of this research shown that the simulation of SEIR model is accurate to predict TB cases in South Sulawesi. The Runge-Kutta method can used to observe trends in the spread of TB in South Sulawesi. The numerical simulation shown that the 5th-order more accurate than the 4th-order Runge-Kutta methods. The results of the modelling are simulated using Maple can predict the number of TB cases which can be used by the government as a consideration to prevent the spread of TB in South Sulawesi.

ρ-PROJECTIVE AND σ-INVOLUNTARY VARIATIONAL INEQUALITIES AND IMPROVED PROJECTION METHOD FOR PROJECTED DYNAMICAL SYSTEM

Purpose of study: To introduce the concept of projective and involuntary variational inequality problems of order and respectively. To study the equivalence theorem between these problems. To study the projected dynamical system using self involutory variational inequality problems.&#x0D; Methodology: Improved extra gradient method is used.&#x0D; Main Finding: Using a self-solvable improved extra gradient method we solve the variational inequalities. The algorithm of the projected dynamical system is provided using the RK-4 method whose equilibrium point solves the involutory variational inequality problems.&#x0D; Application of this study: Runge-Kutta type method of order 2 and 4 is used for the initial value problem with the given projected dynamical system with the help of self involutory variational inequality problems.&#x0D; The originality of this study: The concept of self involutory variational inequality problems, projective and involuntary variational inequality problems of order and respectively are newly defined.

Dynamic Analysis of the Symbiotic Model of Commensalism and Parasitism with Harvesting in Commensal Populations

This article discussed about a dynamic analysis of the symbiotic model of commensalism and parasitism with harvesting in the commensal population. This model is obtained from a modification of the symbiosis commensalism model. This modification is by adding a new population, namely the parasite population. Furthermore, it will be investigated that the three populations can coexist. The analysis carried out includes the determination of all equilibrium points along with their existence and local stability along with their stability requirements. From this model, it is obtained eight equilibrium points, namely three population extinction points, two population extinction points, one population extinction point and three extinction points can coexist. Of the eight points, only two points are asymptotically stable if they meet certain conditions. Next, a numerical simulation will be performed to illustrate the model’s behavior. In this article, a numerical simulation was carried out using the RK-4 method. The simulation results obtained support the results of the dynamic analysis that has been done previously.This article discussed about a dynamic analysis of the symbiotic model of The dynamics of the symbiotic model of commensalism and parasitism with harvesting in the commensal population. is the main focus of this study. This model is obtained from a modification of the symbiosis commensalism model. This modification is by adding a new population, namely the parasite population. Furthermore, it will be investigated that the three populations can coexist. The analysis carried out includes the determination begins by identifying the conditions for the existence of all equilibrium points along with their existence and local stability along with their stability requirements. From this model, it is obtained eight equilibrium points, namely three population extinction points, two population extinction points, one population extinction point and three extinction points can coexist. Of the eight points, only two points are asymptotically stable if they meet certain conditions. Next, a numerical simulation will be performed to illustrate the model’s behavior. In this article, a numerical simulation was carried out using the RK-4 method. The simulation results obtained support the results of the dynamic analysis that has been done previously.[VM1] [VM1]To add a mathematical effect to the article. There can be added mathematical models produced in the study at the end of this section.

NUMERICAL SIMULATION OF DENGUE FEVER (SIR MODEL) USING DIFFERENTIAL TRANSFORM METHOD, MULTI-STEP DIFFERENTIAL TRANSFORM METHOD AND RK4 METHOD

This study investigates the application of the differential transformation method(DTM), multi-step differential transform method(MsDTM) with step-size and RK4 method (Mathematica) for finding the numerical solution of the SIR model of dengue fever in epidemiology. This model is a system of non-linear ordinary differential equations that have no analytic solution. Both the methods DTM and MsDTM are applied directly without any linearization, perturbation or discretization in the model equations to obtain semi-analytic solutions. The accuracy of the MSDTM is excellent and comparable to the RK4 method of Mathematica.

ADAPTIVE TIME-STEPPING FOR RUNGE-KUTTA METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS

Ordinary Differential Equations (ODEs) are usually used in numerous fields especially in solving the modelling problem. Numerical methods are one of the vital mathematical tools to solve the ODEs that appear in various modelling problems by determining the approximation solution close to the in exact solution if it exists. Runge-Kutta methods (RK) are the numerical methods used to integrate the ODEs by applying multistage methods at the midpoint of an interval which can efficiently produce a more accurate result or small magnitude of error. We proposed Runge-Kutta methods (RK) to solve the 1st_ order nonlinear stiff ODEs. The RK methods used in this research are known as the RK-2, RK-4, and RK-5 methods. We proved the existence and uniqueness of the ODEs before we solved it numerically. We also proved the absolute-stability of the RK methods to determine the overall stability of these methods. We found two suitable test cases which are the standard test problem and manufactured solution. We proved that by combining the adaptive step size with RK methods can result in more efficient computation. We implemented the 2nd_, 4th_ and 5th_ order of RK methods with step size adaptively algorithm to solve the test problem and manufactured solution via Octave programming language. The resulting numerical error and the stability of each method can be studied. We compared our results using several error plots versus the Central Processing Unit (CPU) time required to compute a given nonlinear 1st_ order stiff ODE problem. In a conclusion, RK methods which combine with the adaptive step size can result in more efficient computation and accuracy compare with the fixed step size RK methods.

Fast crack bounds in linear elastic fracture mechanics: a new approach to establish the upper bound

This work presents a way to improve the fast crack bounds method. There are several crack propagation models in linear elastic fracture mechanics. The fast crack bound (FCB) consists of an upper (UB) and lower bound (LB) to establish the crack size function. The FCB method was validated by comparing the UB and LB functions with the fourth-order Runge–Kutta (RK4) numerical method. The upper bound is based on $$a^*$$ parameter. In previous works, this parameter is established by numeric inspections. This work presents a new approach to determine the upper bound, without mathematical inspections necessity. Therefore, this work shows an alternative to define the upper bound without using the parameter $$a^*$$ , nor the RK4 method, despite the RK4 has been maintained as a study reference. Two numerical examples, involving Paris–Erdogan and Forman model, were used to analyze the performance of the new upper bound proposition.

Non-standard finite difference scheme and analysis of smoking model with reversion class

Smokers are at more risk to COVID-19 as the entertainment of smoking because their fingers are in touch with lips regularly during smoking that increases the probability of transmission of virus from hand to mouth. On other hand the smokers may have lung disease (or reduced lung capacity) which would greatly increase risk of serious illness especially COVID-19. For this esteem, in this research work, we first formulate a mathematical model contains the reversion class. Then, using different techniques for finding the local and global stability of the presented model related to equilibrium points that are free smoking and positive smoking equilibrium points. As the model consisting on the nonlinear equations, so we use the non-standard finite difference (NSFD) scheme, ODE45 and RK4 methods to find the numerical results. Finally, we show the graphs numerically through MATLAB.

A comparative epidemiological stability analysis of predictor corrector type non-standard finite difference scheme for the transmissibility of measles

In this article, a predictor-corrector type non-standard finite difference scheme has been formulated to avoid unnatural chaos, and numerical instabilities depend on a long time step, lead to a better strategy for overcoming the transmission of infectious diseases. The construction, development, and analysis of our proposed numerical scheme is done for the SEIR epidemiological model regarding the transmission dynamics of measles. Based on the mathematical study of the proposed scheme, the stability analyses are performed in detail. Further, the behavior of the scheme is accessed by the evaluation of the Eigenvalues of the Jacobian at a steady state. MATLAB algorithm is used for numerical simulations, and results demonstrated that the NSFD-PC scheme double refines the solution and gives physically realistic solutions even for large step sizes. The effectiveness of the scheme has been investigated by comparison with renowned numerical methods in literature like the RK4 and Euler method of a predictor-corrector type. It has been found that the presented scheme is dynamically compatible with the continuous system, unconditionally convergent, and satisfies the positivity of the state variables involved in the system of the SEIR model. Conclusively, the proposed numerical scheme preserves all essential control measuring features of the corresponding dynamical system and reduces additional cost when examined over long periods. Therefore, it is highly recommended.

An Efficient Numerical Simulation of a Reaction-Diffusion Malaria Infection Model using B-splines Collocation

• Reaction-diffusion malaria infection model • Neumann Boundary Conditions • Cubic B-splines collocation method • Runge-Kutta method Malaria is a potentially life-threatening disease caused by parasite. This disease is more common in countries with tropical climates. Due to chromosomal mutations, the dynamics of malaria parasites are quite complex to study as well as for any predictions. A reaction-diffusion model to characterize the dynamics of within-host malaria infection with adaptive immune responses is studied in this paper. A numerical scheme based on the collocation of cubic B-splines is proposed to approximate the solution of the considered reaction-diffusion model. Collocation forms of the partial differential equation results in a system of first order ordinary differential equations which in turn have been solved by RK4 method. The non-linearity of the model is being resolved without any transformation or linearization. The computed numerical results are in good agreement with those already available in the literature. Easy to apply and achieving accurate solutions in less CPU time are the strong points of the present method.