Runge-Kutta Formulas Research Articles

Jacobian-free Locally Linearized Runge–Kutta method of Dormand and Prince for large systems of differential equations

This paper introduces Jacobian-free Locally Linearized Runge–Kutta formulas of Dormand and Prince for integrating large systems of initial value problems. With these new Jacobian-free formulas, an adaptive time-stepping variable-order scheme with regulated Krylov dimension is constructed with new developments in the computation of the phi-function action over vectors through Jacobian-free Krylov–Padé approximations, and in the procedures for controlling the approximation errors and for estimating the dimension of the Krylov subspaces. At each integration step, the implemented scheme performs only one Krylov subspace decomposition and a few computations of low dimensional exponential matrices, which contrasts with the implementation of other exponential-type integrators of high order. Regarding to existing schemes, the performed numerical study demonstrates a higher effectiveness of the constructed scheme in the integration of a variety of test equations.

Derivation And Implementation of a Fifth Stage Fourth Order Explicit Runge-Kutta Formula using 𝑓(𝑥,𝑦) Functional Derivatives

This paper is aimed at using 𝒇(𝒙,𝒚) functional derivatives to derive a fifth stage fourth order Explicit Runge-Kutta formula for solving initial value problems in Ordinary Differential Equations. The 𝒇(𝒙,𝒚) functional derivatives from the general Runge-Kutta scheme will be compared with the 𝒇(𝒙,𝒚) functional derivatives from the Taylor series expansion to derive the method. The method will be implemented on some initial value problems, and results compared with results from the classical fourth order method. The results revealed that the method compared favorably well with the existing classical fourth order method.

Fuzzy adaptive recursive terminal sliding mode control for an agricultural omnidirectional mobile robot

Nowadays, agricultural robots play a vital role in the development of agriculture. This paper proposes a fuzzy adaptive recursive terminal sliding mode control strategy for a Mecanum-wheeled omnidirectional mobile robot to fulfill trajectory-tracking tasks in transporting agricultural products. First, a kinematic-and-dynamic model with four control inputs and three states is established for the mobile robot. Then, a fuzzy adaptive recursive terminal sliding mode controller is designed, and the control system’s stability is assured via detailed mathematical proof. Subsequently, a dead-reckoning scheme based on the Runge–Kutta formula is presented to reckon the robot’s position-and-posture information online. A conventional sliding mode controller and a nonsingular terminal sliding mode controller are designed for comparison. Lastly, experiments in two scenarios of circular and figure-eight trajectories with different loads are carried out. The experimental results reveal the superiority of the proposed control strategy compared with the benchmark controllers.

ROOTED TREE ANALYSIS OF AN EXPLICIT FOURTH-STAGE FOURTH ORDER RUNGE- KUTTA METHOD

This research paper is aimed at using Butcher’s rooted trees to separate the f(y) functional derivatives from the f(x,y) functional derivatives after applying Taylor series expansion on the general fourth stage fourth order Runge Kutta method. This approach revealed that the f(y) functional derivatives generated a set of linear/ nonlinear equations that gave birth to a fourth stage fourth order Runge Kutta formula. This idea is derivable from general graphs and combinatorics.

Effects of viscous variation, thermal radiation, and Arrhenius reaction: The case of MHD nanofluid flow containing gyrotactic microorganisms over a convectively heated surface

The numerical exploration of viscous variation, thermal radiation, and Arrhenius reaction modifications on an electrically conducting nanofluid flow through a convectively heated surface is scrutinized in this study. By means of well-suited similarity variables, the nonlinear coupled Ordinary Differential Equations obtained from the mathematical model governing the fluid flow are solved with MATLAB in-built bvp4c software package following shooting approach in conjunction with 4th order Runge–Kutta formula. In addition, a statistical tool called slope of linear regression through data point is introduced. A good agreement was established when the limiting case of the system of equations is compared with previous report in the literature. The results are presented in graphs and tables. It was discovered that enhancement in the viscous variation parameter boosts the velocity gradient and heat transfer rate but lowers the rate of mass transfer and density of motile microorganisms at different levels of R d and K r . More so, increase in chemical reaction lowers the concentration distribution but rises with thermal radiation.

Predictor Corrector Parallel Based on the Geometric Mean Runge-Kutta Formula for Solving Initial Value Problems

The purpose of this study is to present a new connection of the famous Runge-Kutta methods by using more than one old technique and obtain a new method, which has acceptable results for solving Initial Value problems.

Numerical Analysis of Laterally Loaded Long Piles in Cohesionless Soil

The capability of piles to withstand horizontal loads is a major design issue. The current research work aims to investigate numerically the responses of laterally loaded piles at working load employing the concept of a beam-on-Winkler-foundation model. The governing differential equation for a laterally loaded pile on elastic subgrade is derived. Based on Legendre-Galerkin method and Runge-Kutta formulas of order four and five, the flexural equation of long piles embedded in homogeneous sandy soils with modulus of subgrade reaction linearly variable with depth is solved for both free- and fixed-headed piles. Mathematica, as one of the world's leading computational software, was employed for the implementation of solutions. The proposed numerical techniques provide the responses for the entire pile length under the applied lateral load. The utilized numerical approaches are validated against experimental and analytical results of previously published works showing a more accurate estimation of the response of laterally loaded piles. Therefore, the proposed approaches can maintain both mathematical simplicity and comparable accuracy with the experimental 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.

Complex dynamics from heterogeneous coupling and electromagnetic effect on two neurons: Application in images encryption

This paper studies the effect of the electromagnetic flux on the dynamics of an introduced model of heterogeneous coupled neurons. Analytical investigation of the coupled neurons revealed that the obtained model is equilibrium free thus displays hidden firing activities. Based on the Helmholtz theorem, it is demonstrated that the coupled neurons possess a Hamilton energy, which enables to keep the electrical activity of the coupled neurons. Numerical simulations based on the fourth-order Runge-Kutta formula have enabled us to find a range of the electromagnetic induction strength where the model exhibits hysteretic dynamics. That hysteresis justifies the coexistence of two different firing activities for the same parameters captured. This latter behavior is further supported using bifurcation diagrams, the graph of the maximum Lyapunov exponent, phase portraits, time series, and attraction basins as arguments. Beside, the STM32F407ZE microcontroller development board is exploited for the digital implementation of the proposed model. The results of microcontroller implementation perfectly supported the results of the numerical simulation of bistability. Finally, a compressive sensing approach is used to compress and encrypt digital images based on the sequences of the above coupled-neurons model. The plain color image is decomposed into R, G, and B components. The DWT is applied to each component to obtain the corresponding sparse components. Confusion keys are obtained from the proposed coupled neurons to scramble each sparse component. The measurement matrixes obtained from the coupled neurons sequence are used to compress the confused sparse matrices corresponding to R, G, and B components. Each component is quantified, and a diffusion step is then applied to improve the randomness and consequently the information entropy.

Simple program for computing objective optical properties of magnetic lenses

The paper describes the basic features of a new program denoted Magnetic Electron Lens Optical Properties (MELOP), primarily intended for providing free and simple way to calculate the objective focal properties of rotationally-symmetric electron magnetic lenses with the presence of its axial magnetic field distribution. The calculation is done by solving the paraxial ray equation using the fourth-order Runge-Kutta formula. For specific beam voltage, the program computes the excitation parameter, the object or image plane, the objective principal plane, the objective focal length, the objective magnification, the spherical aberration coefficient, the chromatic aberration coefficient and the magnetic flux density at the object or image plane. These parameters are solved for zero, low, high or infinite magnification condition. The program can handle instantaneously plotting the variations of the calculated parameters relatively. In addition to that, the data can be transferred to xlsx or txt file format.

NURBS Modeling and Curve Interpolation Optimization of 3D Graphics

In order to solve the problem of complicated Non-Uniform Rational B-Splines (NURBS) modeling and improve the real-time performance of the high-order derivative of the curve interpolation process, the method of NURBS modeling based on the slicing and layering of triangular mesh is introduced. The research and design of NURBS curve interpolation are carried out from the two aspects of software algorithm and hardware structure. Based on the analysis of the characteristics of traditional computing methods with Taylor series expansion, the Adams formula and the Runge-Kutta formula are used in the NURBS curve interpolation process, and the process is then optimized according to the characteristics of NURBS interpolation. This can ensure accuracy, and avoid the calculation of higher-order derivatives. Furthermore, the hardware modules for the Adams and Runge-Kutta formulas are designed by using the parallel hardware construction technology of Field Programmable Gate Array (FPGA) chips. The parallel computing process using FPGA is compared with the traditional serial computing process using CPUs. Simulation and experimental results show that this scheme can improve the computational speed of the system and that the algorithm is feasible.

Locally Linearized Runge-Kutta method of Dormand and Prince for large systems of initial value problems

In this paper, new Locally Linearized Runge-Kutta formulas of Dormand and Prince are introduced with the purpose of integrating large systems of initial value problems. The new formulas are obtained by replacing the Padé approximation for matrix exponential by a Krylov-Padé approximation in the embedded formulas proposed in a previous work. Unlike other high order exponential integrators, these new formulas involve the approximation of just a single phi-function times vector, which makes them particularly attractive from a computational viewpoint. With these new formulas, adaptive schemes are constructed with novelties in the approximation of phi-functions times vectors by Krylov-Padé approximations, and in the strategies for controlling the approximation errors, for estimating the dimension of the Krylov subspaces, and for the reuse of the Jacobians. In addition, numerical experiments are performed to show the potential of the new embedded formulas and adaptive codes in the integration of known physical, biophysical, and physical-chemistry models.

Compact local integrated radial basis functions (Integrated RBF) method for solving system of non–linear advection-diffusion-reaction equations to prevent the groundwater contamination

The coupled advection-dominated diffusion-reaction equations which arise in the prevention of groundwater contamination problem are approximated by the compact local integrated radial basis function (CLIRBF) method. To efficiently solve the resulting nonlinear system of advection-diffusion equations, we use the integrated radial basis function (IRBF) for discretizing the spatial variables. Afterwards, the system of ordinary differential equations (ODEs) obtained is discretized by the method of lines (MOL). MOL is a general way of viewing a partial differential equation (PDE) as a system of ordinary differential equations (ODE). The efficient fourth-order exponential time differencing Runge-Kutta (ETD-RK4) formula is utilized for solving this system. The main aim of this paper is to show that the integrated radial basis method based on the local form can be exerted for solving the coupled non-linear advection-diffusion-reaction system. The numerical tests are provided to illustrate its validity and accuracy.

An improved version of the Continuous Newton’s method for efficiently solving the Power-Flow in Ill-conditioned systems

This paper tackles the efficient Power-Flow solution of ill-conditioned cases. In that sense, those methods based on the Continuous Newton’s philosophy look very promising, however, these methodologies still present some issues mainly related with the computational efficiency or the robustness properties. In order to overcome these drawbacks, we suggest several modifications about the standard structure of the Continuous Newton’s method. Thus, the standard Continuous Newton’s paradigm is firstly modified with a frozen Jacobian scheme for reducing its computational burden; secondly, it is extended for being used with High-order Newton-like method for achieving higher convergence rate and, finally, a regularization scheme is introduced for improving its robustness features. On the basis of the suggested improvements, a Power-Flow solution paradigm is developed. As example, a novel Power-Flow solver based on the introduced solution framework and the 4th order Runge-Kutta formula is developed. The novel technique is validated in several realistic large-scale ill-conditioned systems. Results show that the suggested modifications allow to overcome the drawbacks presented by those methodologies based on the Continuous Newton’s method. On the light of the results obtained it can be also claimed, that the developed solution paradigm constitutes a promising framework for developing robust and efficient Power-Flow solution techniques.

Lie group method for analyzing the generalized heat transfer mathematical model for Lake Tahoe

The one-dimensional time-dependent heat transfer equation in a vertical direction is introduced in terms of a general formula of density and thermal conductivity. One-parameter Lie symmetry group transformation is applied to determine the suitable forms of water density and thermal conductivity based on experimental measurements. The general equation is investigated again using Lie group analysis after substitution by the two possible forms of water density and thermal conductivity from the first part. The obtained partial differential equation is solved numerically using explicit 4th and 5th Runge–Kutta formula or analytically if it is possible assuming the physical parameters of Lake Tahoe in the Sierra Nevada of the United States. The temperature distribution across the lake depth from each case is illustrated graphically to indicate the thermal stratification phenomenon of lakes.

B-Spline Method of Lines for Simulation of Contaminant Transport in Groundwater

In this study, we propose a new numerical method, which can be effectively applied to the advection-dispersion equation, based on B-spline functions and method of lines approach. In the proposed approach, spatial derivatives are calculated using quintic B-spline functions. Thanks to the method of lines approach, the partial differential equation governing the contaminant transport in groundwater is converted into time-dependent ordinary differential equations. After this transformation, the time-integration of this system is realized by using an adaptive Runge–Kutta formula. In order to test the accuracy of the proposed method, four numerical examples were solved and the obtained results compared with various analytical and numerical solutions given in the literature. It is proven that the proposed method is faster and more reliable than other methods referenced herein and is a good alternative for simulation of contaminant transport problems as a result of these comparisons.

Modified Fractional Runge–Kutta Method for Solving Fuzzy Differential Equations of Fractional Order

The purpose of this research work is to solve fuzzy differential equations of fractional order under the Caputo-type. The modified fractional Runge–Kutta (R–K) method is proposed based on a generalized Runge–Kutta formula and modified trapezoidal rule. Moreover, the approach is followed to obtain a solution for fuzzy differential equations of fractional order in the sense of Caputo-type fuzzy fractional derivatives. Finally, illustrative example is provided to demonstrate the applicability, accuracy and efficiency of the proposed method.

Pressure Control of High Pressure Tubing

This article focuses on how to control the pressure in the high-pressure tubing. First, an overall analysis of the system is performed to determine the pressure balance conditions of the high-pressure fuel pipe; that is, the mass of fuel flowing out of the high-pressure fuel pipe is equal to the mass of fuel flowing in. This article uses Excel to fit and organize the data in all data processing. When solving differential equations of pressure and density, the fourth-order Runge-Kutta formula is used, and the numerical solution is obtained by MATLAB2017a.

Comparison of various robust and efficient load-flow techniques based on Runge–Kutta formulas

Based on Continuous Newton’s method, any well-assessed numerical scheme can be adapted for solving the Load-Flow (LF) problem. So far, LF techniques based on 4th order Runge–Kutta formula (RK4) and Adams–Bashfort’s methods (AB) have been proposed. However, there is a huge variety of numerical methods whose adequacy for solving the LF problem has not been explored yet. This paper tries to fill this gap by proposing several LF solvers based on Runge–Kutta (RK) formulas. Thus, several LF techniques based on Midpoint (MP), 3rd order Heun (H3), Simpson 3/8 (S3/8) and an accelerated 3rd order Runge–Kutta (ARK3) formulas are proposed. The performance of the proposed LF techniques is assessed using several medium and large-scale ill-conditioned power systems. The proposed techniques are compared with RK4, AB and other robust LF methods. In addition, their scalability and influence of the loading level are analyzed. The obtained results prove that the proposed LF techniques are faster than RK4 and robust enough to successfully tackle medium and large-scale ill-conditioned power systems. As main conclusion, it is proved that lower order methods might be as robust as higher order ones but more efficient. Therefore, its usage with respect higher order methods (e.g. 4th order ones), should be frequently preferable.

Runge–Kutta Ray-Tracing Technique for Radiative Transfer in a Three-Dimensional Graded-Index Medium

The Runge–Kutta ray-tracing method is combined with the Monte Carlo method to analyze the radiative transfer in a three-dimensional graded-index media. The analytical solution of the trajectory of a ray in a refractive index distribution of a Maxwell fisheye lens is obtained. The temperature field, heat flux, and apparent emissivity calculated by the analytical solution of the trajectory of a ray are treated as the standard solution to verify the results of the Runge–Kutta ray-tracing methods. The third-order fixed-step, fourth-order fixed-step, and third-order variable-step Runge–Kutta ray-tracing methods are discussed here. More accurate results can be obtained by using a higher-order Runge–Kutta algorithm; however, the third-order variable-step-size Runge–Kutta ray-tracing method has better precision than the fourth-order fixed-step Runge–Kutta formula for the temperature field. The maximum relative error of the temperature as compared to the standard solution is less than 0.19%, the integrated mean relative error is less than 0.06%, and the maximum relative error of the emissivity is less than 0.94% in the case of . The Runge–Kutta ray-tracing method can effectively solve the radiative transfer in a three-dimensional graded-index media.