Kutta Method Of Order Research Articles

Study of electrical circuits using Runge Kutta method of order 4

In this paper, Runge Kutta method of order 4 is used to study the electrical circuits designs through past, intermediate and present voltages. When integrating differential equations with Runge Kutta method of order 4, a constant step size (ℎ) is used until a testing procedure confirms that the discontinuity occurs in the present integration interval. This step size function calculations would take place at the end of the functional calculations, but before the dependent variables were updated. Runge Kutta methods along with convolution are given by array interpretation (Butcher matrix) representation, this leads to identify the equilibrium state. The input parameters indicate the voltage coefficient controlled by current sources and measures it a random periodic time. The output parameters provide stable independent values and calculated from past voltage and current values. Finally solutions are compared with exact values and RK method of order 4 along with Heun, Midpoint and Taylors’s method with various ℎ values.

Numerical Solutions of Nonlinear Fredholm Integro Differential Equations using Fifth Order Runge-kutta Method

This paper is considered with the numerical solution of a type of second-order nonlinear Fredholm integro-differential Equations (FIDs). The proposed scheme is based on Runge - kutta methods of order fifth. Moreover, two numerical examples are considered to show the capability and the accuracy of the proposed methods compared with other Runge-kutta methods of less order. Finally, we apply the least square errors (LSE) formula to make numerical comparisons between the numerical and the exact solutions. The obtained results show that the proposed method is superior and more accurate than Runge - kutta methods of orders two, three, and four.

Non-uniqueness of the solution of one mathematical model of an autocatalytic reaction with diffusion and the Showalter – Sidorov condition

The article is devoted to the numerical study of the phase space of one mathematical model of an autocatalytic reaction with diffusion, based on a degenerate system of equations for a distributed brusselator. We will obtain conditions for the existence, uniqueness or multiplicity of solutions to the Showalter – Sidorov problem and reveal the dependence of these conditions on the parameters of the system. The approach used in the numerical research of this problem is based on the reduction of the semilinear Sobolev-type equation to a system of algebraic-differential equations with the subsequent solution of this system using the Runge – Kutta method of order 4-5. The article also provides the result of a computational experiment which illustrates the operation of a program complex based on an algorithm for the numerical solution of the problem. The results of numerical simulation in the case of existence of two solutions of the investigated model are presented.

Optimized low-dispersion and low-dissipation two-derivative Runge–Kutta method for wave equations

The paper is devoted to the optimization of the explicit two-derivative sixth-order Runge–Kutta method in order to obtain low dissipation and dispersion errors. The method is dependent on two free parameters, used for the optimization. The optimized method demonstrates the lowest dispersion error in comparison with other widely-used high-order Runge–Kutta methods for hyperbolic problems. The efficiency of the method is demonstrated on the solutions of problems for five linear and nonlinear partial differential equations by the method-of-lines. Spatial derivatives are discretized by finite differences and Petrov–Galerkin approximations. The work-precision and error—CPU time plots, as the dependence of CPU time on space grid resolution, are considered. Also, the optimized method compared with other two-derivative methods. In most examples, the optimized method has better properties, especially for the cases of computations with adaptive stepsize control at high accuracy. The lowest CPU time takes place for the optimized method, especially in the cases of fine space grids.

Modelling and simulation of transmission lines in a biological neuron

In this paper, we present a new mathematical model that explains the transmission along a biological neuron. We also present a numerical scheme based on the four order β-method to simulate numerically the transmission. The idea is to couple the β-method of high order with Runge Kutta method in order to get high order schemes without oscillations. Furthermore, various numerical experiments are presented to show the power and efficiency of our proposed model.

Modelling and simulation of transmission lines in a biological neuron

In this paper, we present a new mathematical model that explains the transmission along a biological neuron. We also present a numerical scheme based on the four order β-method to simulate numerically the transmission. The idea is to couple the β-method of high order with Runge Kutta method in order to get high order schemes without oscillations. Furthermore, various numerical experiments are presented to show the power and efficiency of our proposed model.

Flow of a Jeffrey Fluid Between Two Long Vertical Thin Plates with Porous Medium

The present study investigates fully developed free - convection Jeffrey fluid flow between two vertical plates with porous medium. The vertical plates are moving with same velocity but in opposite directions. The coupled nonlinear governing equations are solved by using the linearization technique. The solutions for velocity distribution, temperature distribution, skin friction and rate of heat transfer is obtained in the presence of porous medium by Iterative procedure. Shooting technique with Runge - Kutta method of order four is proposed to compare the numerical results for velocity and temperature distribution. The numerical results obtained by both methods are compared and presented graphically. It is observed that an increase in the permeability parameter causes decrease in the fluid velocity and an increase in the Jeffrey fluid parameter causes an enhancement in the fluid velocity. The significance of various pertinent parameters like Grashof number, Prandtl number, Eckert number and the plate velocity are explained through graphs.

A Runge–Kutta Gegenbauer spectral method for nonlinear fractional differential equations with Riesz fractional derivatives

ABSTRACTA Runge–Kutta Gegenbauer spectral method is proposed to solve an initial-boundary value problem for a nonlinear two-dimensional fractional differential equation with variable coefficients. The solution to the problem at each time step is approximated by a bivariate polynomial based on shifted Gegenbauer polynomials and then the Runge–Kutta method of order 3 is applied to the problem. The convergence rate of the derived method is analysed. Numerical results are presented to verify the effectiveness of the method.

A new class of efficient one-step contractivity preserving high-order time discretization methods of order 5 to 14

A family of one-step, explicit, contractivity preserving, multi-stage, multi-derivative, Hermite–Birkhoff–Taylor methods of order p = 5,6,…,14, that we denote by CPHBTRK4(d,s,p), with nonnegative coefficients are constructed by casting s-stage Runge–Kutta methods of order 4 with Taylor methods of order d. The constructed CPHBTRK4 methods are implemented using efficient variable step control and are compared to other well-known methods on a variety of initial value problems. A selected method: CP 6-stages 9-derivative HBT method of order 12, denoted by CPHBTRK412, has larger region of absolute stability than Dormand–Prince DP(8,7)13M and Taylor method T(12) of order 12. It is superior to DP(8,7)13M and T(12) methods on the basis the number of steps, CPU time, and maximum global error on several problems often used to test higher-order ODE solvers. Also, we show that the contractivity preserving property of CPHBTRK412is very efficient in suppressing the effect of the propagation of discretization errors and the new method compares positively with explicit 17 stages Runge-Kutta-Nystrom pair of order 12 by Sharp et al. on a long-term integration of a standard N-body problem. The selected CPHBTRK412is listed in the Appendix.

Analysis of nano droplet dynamics with various sphericities using efficient computational techniques

Motion of a vertically falling nano droplet in incompressible Newtonian media with initial velocity is investigated. The instantaneous velocity and acceleration are carried out by using the variational iteration method (VIM) and homotopy perturbation method (HPM), which are analytical solution techniques. The obtained results are compared with Runge–Kutta method in order to verify the accuracy of the proposed methods. The results show that, the analytical solutions are in good agreement with each other and with the numerical solution. Also, the effects of sphericity (φ) on the velocity and acceleration profiles of the nano droplet are explained. Moreover, the results demonstrate that the VIM-Pade and HPM-Pade are very effective in generating analytical solutions for even highly nonlinear problems.

On variable step highly stable 4-stage Hermite-Birkhoff solvers for stiff ODEs

Variable-step (VS) \(4\)-stage \(k\)-step Hermite--Birkhoff (HB) methods of order \(p=(k+1)\), denoted by HB\((p)\), are constructed as a combination of linear \(k\)-step methods of order \((p-2)\) and a two-step diagonally implicit \(4\)-stage Runge--Kutta method of order 3 (TSDIRK3) for solving stiff ordinary differential equations. The main reason for considering this class of formulae is to obtain a set of \(k\)-step methods which are highly stable and are suitable for the integration of stiff differential systems whose Jacobians have some large eigenvalues lying close to the imaginary axis. The approach, described in the present paper, allows us to develop \(L\)-stable \(k\)-step methods of order up to 7 and \(L(\alpha)\)-stable methods of order up to 10 with \(\alpha > 64^\circ\). Fast algorithms are developed for solving confluent Vandermonde-type systems of the new methods in O\((p^2)\) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The step sizes of these methods are controlled by a local error estimator. Selected HB(\(p\)) of order \(p\), \(p=4,5,\ldots,9\), compare favorably with existing Cash modified extended backward differentiation formulae, MEBDF(\(p\)), \(p=4,5,\ldots,8\) in solving problems often used to test highly stable stiff ODE solvers on the basis of CPU time, number of steps and error at the endpoint of the integration interval.

Qualitatively stability of nonstandard 2-stage explicit Runge–Kutta methods of order two

When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Nonstandard finite differences (NSFDs) schemes can improve the accuracy and reduce computational costs of traditional finite difference schemes. In addition NSFDs produce numerical solutions which also exhibit essential properties of solution. In this paper, a class of nonstandard 2-stage Runge–Kutta methods of order two (we call it nonstandard RK2) is considered. The preservation of some qualitative properties by this class of methods are discussed. In order to illustrate our results, we provide some numerical examples.

Qualitatively stability of nonstandard 2-stage explicit Runge–Kutta methods of order two

Qualitatively stability of nonstandard 2-stage explicit Runge–Kutta methods of order two

On variable step Hermite–Birkhoff solvers combining multistep and 4-stage DIRK methods for stiff ODEs

Variable-step (VS) 4-stage k-step Hermite---Birkhoff (HB) methods of order p = (k + 2), p = 9, 10, denoted by HB (p), are constructed as a combination of linear k-step methods of order (p ? 2) and a diagonally implicit one-step 4-stage Runge---Kutta method of order 3 (DIRK3) for solving stiff ordinary differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge---Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems. This approach allows us to develop L(a)-stable methods of order up to 11 with a > 63°. Fast algorithms are developed for solving these systems in O (p2) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stepsizes of these methods are controlled by a local error estimator. HB(p) of order p = 9 and 10 compare favorably with existing Cash modified extended backward differentiation formulae of order 7 and 8, MEBDF(7-8) and Ebadi et al. hybrid backward differentiation formulae of order 10 and 12, HBDF(10-12) in solving problems often used to test higher order stiff ODE solvers on the basis of CPU time and error at the endpoint of the integration interval.

Limitations and improvements of standard spectral methods for pricing standard options

We discuss the issue of how to improve the efficiency of standard spectral methods for pricing options. As a prototype example, we consider American options which are frequently used in the market as they enjoyed the early exercise opportunities. Such options are often modeled by nonlinear partial differential equations, solutions of which are not obtainable in a closed form and thus necessitates robust numerical techniques. For the discretisation in space (asset) direction, we use a rational spectral methods. For time integration involved in the process, we looked at comparing the efficiency of the method by using contour integral method based on the inversion of the Laplace transform, and the Exponential Time Differencing Runge–Kutta method of order 4. It is known that when these spectral methods are applied to option pricing problems, they no longer retain their higher order convergence. This order reduction is attributed due to the non-smooth initial conditions. To overcome this, we propose a domain decomposition method based on our rational spectral method. For the problem under consideration, we divide the domain into two sub-domains, with the transition point as the strike price. To represent the solution in each sub-domains, we choose to approximate the solution by linear rational interpolants due to their improved stability properties as compared to the polynomial interpolant. Comparative results are also presented.

Solving Fuzzy Differential Equationsin Runge- Kutta Method of Order Three

Solving Fuzzy Differential Equationsin Runge- Kutta Method of Order Three

Three-stage Hermite–Birkhoff solver of order 8 and 9 with variable step size for stiff ODEs

Variable-step (VS) $$3$$3-stage Hermite---Birkhoff (HB) methods HB$$(p=k+1)$$(p=k+1) of order $$p=8$$p=8 and 9 are constructed as a combination of linear $$k$$k-step methods of order $$(p-2)$$(p-2) and a diagonally implicit one-step $$3$$3-stage Runge---Kutta method of order 3 (DIRK3) for solving stiff ordinary differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge---Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems. This approach allows us to develop A-stable methods of order up to 5 and A($$\alpha $$?)-stable methods of order up to 10. Fast algorithms are developed for solving these systems in O$$(p^2)$$(p2) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stepsize of these methods are controlled by a local error estimator. When programmed in C++, HB($$p$$p) of order $$p=8$$p=8 and 9 compare favorably with existing Cash modified extended backward differentiation formulae of order 7 and 8, MEBDF(7---8), in solving problems often used to test higher order stiff ODE solvers on the basis of CPU time and error at the endpoint of the integration interval.

Convection-radiation thermal analysis of triangular porous fins with temperature-dependent thermal conductivity by DTM

Increasing performance of porous fins is one of the common priorities nowadays. Hence attention here is given to the convection and radiation effects in the analysis of performance of a porous triangular fin with temperature-dependent thermal conductivity. Mathematical problem is solved by differential transformation method (DTM). The DTM results are compared with the numerical results obtained by fourth-order Runge–Kutta method in order to confirm the accuracy of the analytical solution. The dimensionless temperature distributions, fin efficiency and fin effectiveness are studied with respect to emerging parameters involved in the analysis. It is noted that there is an increase in fin performance in view of porous constraint.

A novel and developed approximation for motion of a spherical solid particle in plane coquette fluid flow

The present article solves the couple equations of a spherical solid particle’s motion in plane coquette fluid flow by using the HPM-Padé technique which is a combination of the Homotopy Perturbation Method (HPM) and Padé approximation. The series solutions of the couple equations are developed. Generally, the truncated series solution is adequately in a small region and to overcome this limitation, the Padé techniques which have the advantage of turning the polynomial approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. The current results compared with those derived from HPM and the established fourth order Runge–Kutta method in order to ascertain the accuracy of the proposed method. It is found that this method can achieve more suitable results in comparison to HPM.

Comparison of Soot Particle Movement based on Crank Angle

In a diesel engine, soot was produced due to incomplete fuel combustion in a combustion chamber. Some of this soot sticks to the cylinder wall and interferes with lubricant oil. This soot causes the lubricant oil to contaminate and this increases its viscosity. Contamination of lubricant oil is one of the major causes of engine wear. Therefore, the focus of this study is on soot movement in diesel engine that is the initial step to avoid contamination of lubricant oil. This work uses the data of the formation of soot particles from Kiva-3v obtained from previous investigation and then simulated it by a Matlab routine. Kiva-3v produced velocity vectors of the soot, fuel, temperature, pressure and others. Matlab routine uses trilinear interpolation and fourth order Runge Kutta method in order to calculate soot movement in a combustion chamber. In addition, the influence of drag force is considered in the calculation to achieve a higher accuracy. The objective of this study is to compare soot particle movement between 8° ATDC and 18° ATDC. Results show that 8° ATDC has a high risk to contaminate lubrication oil in certain location compare to 18° ATDC.