Classical Fourth-order Runge-Kutta Method Research Articles

Transient and Optimization Analysis of Vibration Isolation Device Based on Geometric Nonlinear Theory

The vibration isolation device based on geometric nonlinear theory is a hotspot in the field of dynamics in recent years. In order to deeply explore the application of this technology in the field of vibration isolation for power grid equipment, we establish a transformer-casing coupling nonlinear dynamic model considering seismic wave excitation, and use the classical fourth-order Runge-Kutta method to calculate and analyse the transient response behaviour and characteristics of the coupled system under different conditions. The analysis results reveal that the influence of the three parameters, including stiffness, length and damping of springs in shock absorber, on the anti-seismic performance of this coupling structure, which lays a theoretical foundation for the application of geometric nonlinear theory in the field of large-scale transformation equipment vibration isolation.

SERIES SOLUTION OF TYPHOID FEVER MODEL USING DIFFERENTIAL TRANSFORM METHOD

This paper presents an analysis of PSIuIeTR type model, which are used to study the transmission dynamics of typhoid fever diseases in a population. Basic idea of typhoid fever disease transmission using compartmental modeling is discussed. Differential Transformation Method (DTM) is discussed in detail, which is used to compute the series solution of the non-linear system of differential equation governing the model equations. The validity of the (DTM) in solving the proposed model is established by classical fourth-order Runge-Kutta method which is implemented in Maple 18. Graphical results confirm that (DTM) is in good agreement with RK-4 and this produced correctly same behaviour of the model, thus validating the efficiency and accuracy of (DTM) in finding the series solution of an epidemic model.

Stability enhancement of high Prandtl number chaotic convection in an anisotropic porous layer with feedback control

The chaotic dynamical behaviour of thermal convection in an anisotropic porous layer subject to gravity, heated from below and cooled from above, is studied based on theory of dynamical system in the presence of feedback control. The extended Darcy model, which includes the time derivative has been employed in the momentum equation to derive a low dimensional Lorenz-like equation by using Galerkin-truncated approximation. The classical fourth-order Runge-Kutta method is used to obtain the numerical solution in order to exemplify the dynamics of the nonlinear autonomous system. The results show that stability enhancement of chaotic convection is feasible via feedback control.

A high-order accurate wavelet method for solving Schrödinger equations with general nonlinearity

A sampling approximation for a function defined on a bounded interval is proposed by combining the Coiflet-type wavelet expansion and the boundary extension technique. Based on such a wavelet approximation scheme, a Galerkin procedure is developed for the spatial discretization of the generalized nonlinear Schrodinger (NLS) equations, and a system of ordinary differential equations for the time dependent unknowns is obtained. Then, the classical fourth-order explicit Runge-Kutta method is used to solve this semi-discretization system. To justify the present method, several widely considered problems are solved as the test examples, and the results demonstrate that the proposed wavelet algorithm has much better accuracy and a faster convergence rate in space than many existing numerical methods.

Lie symmetry analysis, soliton and numerical solutions of boundary value problem for variable coefficients coupled KdV–Burgers equation

In this article, an initial and boundary value problem for variable coefficients coupled KdV–Burgers equation is considered. With the help of Lie group approach, initial and boundary value problem for variable coefficients coupled KdV–Burgers equation reduced to an initial value problem for nonlinear third-order ordinary differential equations (ODEs). Moreover, the systems of ODEs are solved to obtain soliton solutions. Further, classical fourth-order Runge–Kutta method is applied to systems of ODEs for constructing numerical solutions of coupled KdV–Burgers equation. Numerical solutions are computed, and accuracy of numerical scheme is assessed by applying the scheme half mesh principal to calculate maximum errors.

High Accurate NURBS Interpolation Algorithm Based on Runge-Kutta Method

High accurate NURBS interpolation algorithm based on Runge-Kutta method was proposed to solve the problem of low precision and efficiency during NURBS machining. The algorithm plans desired feed rate according to the constraint of the tolerant maximum chord error, and calculates the interpolation parameters through the classical fourth-order Runge-Kutta method during real-time interpolation process, which only needs the first derivative of NURBS curve. Then we can quickly get the high accurate interpolation parameters, which greatly reduces the error between the actual feed rate and desired feed rate. Namely, the feed rate fluctuation rate has been effectively controlled. The feasibility of the algorithm was verified by MATLAB simulation.

On the two-dimensional characteristics of laminar fluid transients in viscoelastic pipes

A two-dimensional numerical model is developed to study the behavior of laminar fluid transients in viscoelastic pipes. The present procedure couples a two-dimensional water hammer model with a single element Kelvin-Voigt model, where the latter is used to simulate the viscoelastic behavior of the pipe wall. The method of lines is implemented in the present study, where second-order accurate central difference expressions are used for the discretization of the spatial terms and the classical fourth-order accurate Runge-Kutta method is used for integrating the discretized equations in time. Through the present numerical results and dimensional analysis of the governing equations, a non-dimensional parameter is identified which governs the effect of pipe wall viscoelasticity on the fluid transient. The identified parameter shows that the effect of pipe wall viscoelasticity is more pronounced in the case of longer pipes with bigger diameters and larger wave speeds. Moreover, a parametric study is carried out to identify the effect of the non-dimensional parameter on the various characteristics of the fluid transient and on the pipe instantaneous and retarded strain components.

Lie group analysis and similarity solution for fractional Blasius flow

This paper presents an investigation for boundary layer flow of viscoelastic fluids past a flat plate. Fractional-order Blasius equation with spatial fractional Riemann–Liouville derivative is derived firstly by using Lie group transformation. The solution is obtained numerically by the generalized shooting method, employing the shifted Grünwald formula and classical fourth order Runge–Kutta method as the iterative scheme. The effects of the order of fractional derivative and the generalized Reynolds number on the velocity profiles are analyzed and discussed. Numerical results show that the smaller the value of the fractional order derivative leads to the faster velocity of viscoelastic fluids near the plate but not to hold near the outer flow. As the Reynolds number increases, the fluid is moving faster in the whole boundary layer consistently.

Eigenvalue Assignment for Control of Time-Delay Systems Via the Generalized Runge–Kutta Method

This paper presents an eigenvalue assignment method for the time-delay systems with feedback controllers. A new form of Runge–Kutta algorithm, generalized from the classical fourth-order Runge–Kutta method, is utilized to stabilize the linear delay differential equation (DDE) with a single delay. Pole placement of the DDEs is achieved by assigning the eigenvalue with maximal modulus of the Floquet transition matrix obtained via the generalized Runge–Kutta method (GRKM). The stabilization of the DDEs with feedback controllers is studied from the viewpoint of optimization, i.e., the DDEs are controlled through optimizing the feedback gain matrices with proper optimization techniques. Several numerical cases are provided to illustrate the feasibility of the proposed method for control of linear time-invariant delayed systems as well as periodic-coefficient ones. The proposed method is verified with high computational accuracy and efficiency through comparing with other methods such as the Lambert W function and the semidiscretization method (SDM).

The Multi-Step Differential Transform Method and Its Application to Determine the Solutions of Non-Linear Oscillators

In this paper, a reliable algorithm based on an adaptation of the standard differential transform method is presented, which is the multi-step differential transform method (MSDTM). The solutions of non-linear oscillators were obtained by MSDTM. Figurative comparisons between the MSDTM and the classical fourth-order Runge-Kutta method (RK4) reveal that the proposed technique is a promising tool to solve non-linear oscillators.

Exploration on initial structures of extrasolar protoplanets via new explicit RKAHeM(4,4) method

In this paper, a newly proposed embedded Runge–Kutta fourth order with four stages arithmetic Heronian mean method is employed and verified in determining the distribution of thermodynamic variables inside the extra-solar protoplanets formed through gravitational instability at their initial stages. In specific, the case of conduction–radiation is considered regarding the transference of heat inside the protoplanets. A general brief theoretical framework for the proposed numerical method is stated in addition to pseudo code followed by error estimation description. The results based on newly proposed explicit RKAHeM(4,4) method are found to be optimal and efficient in comparison with the ones obtained with classical fourth order Runge–Kutta method.

English

In this paper, we derive a new phase fitted Runge-Kutta method based on the existing classical Runge-Kutta method of order four to solve ordinary differential equations with oscillatory solutions. The new method has the property of zero phase-lag and zero dissipation. Phase-lag or dispersion error is the angle between the true and the approximated solution, whereas dissipation is the distance of the computed solution from the standard cyclic solution. A set of problems are tested upon over a large interval and the numerical results proved that the method is more accurate compared to the classical fourth order Runge-Kutta method.

A comparison of classical Runge-Kutta and Henon’s methods for capturing chaos and chaotic transients in an aeroelastic system with freeplay nonlinearity

A typical two-dimensional airfoil with freeplay nonlinearity in pitch undergoing subsonic flow is studied via numerical integration methods. Due to the existence of the discontinuous nonlinearity, the classical fourth-order Runge-Kutta (RK4) method cannot capture the aeroelastic response accurately. Particularly, it is because the RK4 method is incapable of detecting the discontinuous points of the freeplay that leads to the numerical instability and inaccuracy. To resolve this problem, the RK4 method is used with the aid of the Henon’s method (referred to as the RK4Henon method) to precisely predict the freeplay’s switching points. The comparison of the classical RK4 and the RK4Henon methods is carried out in the analyses of periodic motions, chaos, and long-lived chaotic transients. Numerical simulations demonstrate the advantages of the RK4Henon method over the classical RK4 method, especially for the analyses of chaos and chaotic transients. Another existing method to deal with the freeplay nonlinearity is to use an appropriate rational polynomial (RP) to approximate this discontinuous nonlinearity. Consequently, the discontinuity is removed. However, it is demonstrated that the RP approximation method is unable to capture the chaotic transients. In addition, an efficient tool for predicting the existence of chaotic transients is proposed by means of the evolution curve of the largest Lyapunov exponent. Finally, the effects of system parameters on the aeroelastic response are investigated.

A Fast Immersed Boundary Fourier Pseudo-spectral Method for Simulation of the Incompressible Flows

Abstract The present paper is devoted to implementation of the immersed boundary technique into the Fourier pseudo-spectral solution of the vorticity-velocity formulation of the two-dimensional incompressible Navier-Stokes equations. The immersed boundary conditions are implemented via direct modification of the convection and diffusion terms, and therefore, in contrast to some other similar methods, there is not an explicit external forcing function in the present formulation. At the beginning of each time step, the solenoidal velocities (also satisfying the desired immersed boundary conditions), are obtained and fed into a conventional pseudo-spectral solver, together with a modified vorticity. The classical explicit fourth-order Runge-Kutta method is used in time integration, and the boundary conditions are set at the beginning of each sub-step, in order to increase the time accuracy. The method is employed in simulation of some different test cases, including the flow behind impulsively started circular cylinder, oscillating circular cylinder in fluid at rest and insect-like flapping wing motion. The results show accuracy and efficiency of the method.

Effects of parallel strategies in the transitional flow investigation

The direct numerical simulation of transitional and turbulent incompressible flows is an area that is increasing with the advance in computational resources. Code parallelization has became a useful tool in these simulations. In the present paper the unsteady two dimensional Navier–Stokes equations are used as physical model. Tollmien–Schlichting waves propagating in a Poiseuille flow is adopted as test case. Three frequencies showing different behavior according to the linear stability theory (LST) are used: unstable, nearly neutral and stable waves. The parallelization is done via domain decomposition in the streamwise direction. The time derivative is integrated by a classical fourth-order Runge–Kutta method. High-order compact finite difference schemes are used for the spatial derivatives discretization. The Poisson equation is solved by multigrid methods. The present paper explores different parallelization techniques for solving the tridiagonal system, and the multigrid method. The results are compared with LST, and also between the different parallel strategies to show the advantages and disadvantages of each one.

Wavelet solutions of Burgers’ equation with high Reynolds numbers

A wavelet method is proposed to solve the Burgers’ equation. Following this method, this nonlinear partial differential equation is first transformed into a system of ordinary differential equations using the modified wavelet Galerkin method recently developed by the authors. Then, the classical fourth-order explicit Runge-Kutta method is employed to solve the resulting system of ordinary differential equations. Such a wavelet-based solution procedure has been justified by solving two test examples: results demonstrate that the proposed method has a much better accuracy and efficiency than many other existing numerical methods, and whose order of convergence can go up to 5. Most importantly, our results also indicate that the present wavelet method can readily deal with those fluid dynamics problems with high Reynolds numbers.

Implementation of method of lines to predict water levels due to a storm along the coastal region of Bangladesh

In this study, the method of lines (MOL) has been applied to solve two-dimensional vertically integrated shallow water equations in Cartesian coordinates for the prediction of water levels due to a storm surge along the coast of Bangladesh. In doing so, the partial derivatives with respect to the space variables were discretized by the finite difference (central) method to obtain a system of ordinary differential equations (ODEs) with time as independent variable. The classical fourth-order Runge–Kutta method was used to solve the obtained system of the ODEs. We used a nested finite difference scheme, where a high resolution fine grid model (FGM) capable of incorporating all major islands along the coastal region of Bangladesh was nested into a coarse grid model (CGM) covering up to 15°N latitude of the Bay of Bengal. The boundaries of the coast and islands were approximated through proper stair step. Appropriate tidal condition over the model domain was generated by forcing the sea level to be oscillatory with the constituent M 2 along the southern open boundary of the CGM omitting wind stress. Along the northeast corner of the FGM, the Meghna River discharge was taken into account. The developed model was applied to estimate water levels along the coast of Bangladesh due to the interaction of tide and surge associated with the April 1991 storm. We also computed our results employing the standard finite difference method (FDM). Simulated results show the MOL performs well in comparison with the FDM with regard to CPU time and stability, and ensures conformity with observations.

The multi-step homotopy analysis method for modified epidemiological model for computer viruses

In this paper, we consider the modified epidemiological model for computer viruses (SAIR) proposed by Piqueira and Araujo (Appl Math Comput 2(213):355–360, 2009). The multi-step homotopy analysis method (MHAM) is employed to compute an approximation to the solution of the model of fractional order. The fractional derivatives are described in the Caputo sense. Figurative comparisons between the MHAM and the classical fourth-order Runge-Kutta method reveal that this method is very effective. The solutions obtained are also presented graphically.

Enhanced Multistage Homotopy Perturbation Method: Approximate Solutions of Nonlinear Dynamic Systems

We introduce a new approach called the enhanced multistage homotopy perturbation method (EMHPM) that is based on the homotopy perturbation method (HPM) and the usage of time subintervals to find the approximate solution of differential equations with strong nonlinearities. We also study the convergence of our proposed EMHPM approach based on the value of the control parameterhby following the homotopy analysis method (HAM). At the end of the paper, we compare the derived EMHPM approximate solutions of some nonlinear physical systems with their corresponding numerical integration solutions obtained by using the classical fourth order Runge-Kutta method via the amplitude-time response curves.

Analytical Study of Fractional-Order Multiple Chaotic FitzHugh-Nagumo Neurons Model Using Multistep Generalized Differential Transform Method

The multistep generalized differential transform method is applied to solve the fractional-order multiple chaotic FitzHugh-Nagumo (FHN) neurons model. The algorithm is illustrated by studying the dynamics of three coupled chaotic FHN neurons equations with different gap junctions under external electrical stimulation. The fractional derivatives are described in the Caputo sense. Furthermore, we present figurative comparisons between the proposed scheme and the classical fourth-order Runge-Kutta method to demonstrate the accuracy and applicability of this method. The graphical results reveal that only few terms are required to deduce the approximate solutions which are found to be accurate and efficient.