Classical Fourth-order Runge-Kutta Method Research Articles

Two-dimensional Coupled Asymmetric van der Pol Oscillator

A reliable algorithm based on an adaptation of the standard differential transform method (DTM) is presented, which is the multi-step differential transform method (Ms-DTM) since it may be difficult to directly apply differential transform method (DTM) to obtain the series solutions for the present two- dimensional coupled asymmetric van der Pol oscillator. The solutions of a two dimensional coupled asymmetric van der Pol oscillator were obtained by Ms-DTM. Figurative comparisons between the Ms-DTM and the classical fourth order Runge-Kutta method (RK4) are given. The obtained results reveal that the proposed technique is a promising tool to solve the considered van der Pol oscillator and yield same information on the phase portrait confirming the stability of the system, effectively. It can be said that the considered approach can be easily extended to other nonlinear van der Pol oscillator systems and therefore is widely applicable in engineering and other sciences.

The APPROXIMATE SOLUTIONS OF MALARIA DISEASE TRANSMISSION MODEL: USING MUTI-STEP DIFFERENTIAL TRANSFORMATION METHOD

In this paper, numerical solutions to the SPEIR-SEI Malaria disease model were obtained using the Multi-Step Differential Transformation Method (MS-DTM). MS-DTM is a semi-analytical method for solving a system of a non-linear differential equation where its exact solution is difficult to obtain. The analytical solution of MS-DTM was compared with the solution of Maple21’s in-built classical fourth-order Runge-Kutta method. The results demonstrate the reliability and efficiency of the method and the graphs show that the solutions from both methods agreed well with each other.

THE SOLUTION OF A MATHEMATICAL MODEL FOR COVID-19 TRANSMISSION AND VACCINATION IN NIGERIA BY USING A DIFFERENTIAL TRANSFORMATION METHOD

In this work, Differential Transform Method (DTM) was employed to obtain the series solution of the SIRV COVID-19 model in Nigeria. The validity of the DTM in solving the model was validated by Maple 21’s Classical fourth-order Runge-Kutta method. The comparison between DTM and Runge- Kutta (RK4) solutions was performed and there was a good correlation between the results obtained by the two methods. The result validates the accuracy and efficiency of the DTM to solve the model

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.

The Derivation of Various Arbitrary Parameters in the Formulation of Classical Fourth-Order Runge-Kutta Method

The Derivation of Various Arbitrary Parameters in the Formulation of Classical Fourth-Order Runge-Kutta Method

Chebyshev fifth-kind series approximation for generalized space fractional partial differential equations

<abstract><p>In this paper, we propose a numerical scheme to solve generalized space fractional partial differential equations (GFPDEs). The proposed scheme uses Shifted Chebyshev fifth-kind polynomials with the spectral collocation approach. Besides, the proposed GFPDEs represent a great generalization of significant types of fractional partial differential equations (FPDEs) and their applications, which contain many previous reports as a special case. The fractional differential derivatives are expressed in terms of the Caputo sense. Moreover, the Chebyshev collocation method together with the finite difference method is used to reduce these types of differential equations to a system of differential equations which can be solved numerically. In addition, the classical fourth-order Runge-Kutta method is also used to treat the differential system with the collocation method which obtains a great accuracy. Numerical approximations performed by the proposed method are presented and compared with the results obtained by other numerical methods. The introduced numerical experiments are fractional-order mathematical physics models, as advection-dispersion equation (FADE) and diffusion equation (FDE). The results reveal that our method is a simple, easy to implement and effective numerical method.</p></abstract>

A strong-form local meshless approach based on radial basis function-finite difference (RBF-FD) method for solving multi-dimensional coupled damped Schrödinger system appearing in Bose–Einstein condensates

In this work, one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) coupled damped Schrödinger system is solved numerically. A strong-form local meshless approach established on radial basis function-finite difference (RBF-FD) method for spatial approximation is developed. Polyharmonic splines are used as radial basis function with augmented polynomials. The use of the polyharmonic splines saves us from choosing an optimum shape parameter which is not a simple task for infinitely smooth RBFs such as multiquadrics or Gaussians. For time discretization classical fourth-order Runge Kutta method is utilized. L∞ error norm and conserved quantities are computed to indicate performance of the proposed method. Stability of the proposed method is examined numerically. Some computer codes are devised in Julia programming language for obtaining numerical results. Acquired numerical results and their comparison with other studies available in literature such as cubic B-spline Galerkin method and direct meshless local Petrov–Galerkin (DMLPG) method endorse the performance and reliability of the proposed method.

Characterization of hydrodynamic properties from free vibration tests of a large-scale bridge model

To accurately predict the dynamic response of a structure subjected to fluid induced loading, a thorough understanding of the dynamic properties (mass, stiffness, and damping) and associated interactions is required. Limited data are available to characterize dynamic fluid–structure interactions. Data are particularly limited for large-scale and flexible structural models. This article reports the results of the first free vibration tests of a dynamic large-scale laboratory highway bridge superstructure model. The dynamic response characteristics of the model were extracted and analyzed from free vibration tests under varying levels of water submersion and for different horizontal substructure flexibilities. The nature of the damping response was identified based on the empirically measured logarithmic decrements of the model’s free vibration displacement amplitudes, and a suitable equation of motion (EOM) was subsequently developed. Using the classical fourth-order Runge–Kutta method, the EOM was solved for the different test trials and the dynamic properties of the model were obtained through an optimization approach. The concept of added mass was introduced to explain the observed decrease in natural frequency with increasing levels of water submersion. Finally, added mass factor was computed for the case where the water level was even with the top of the bridge superstructure model. This study provides a suitable EOM needed for numerical simulations of this and similar models that study fluid–structure interaction and also provides a methodology for establishing the structural dynamic properties of generalized hydrodynamic analytical models.

Application of SPD-RBF method of lines for solving nonlinear advection–diffusion–reaction equation with variable coefficients

PurposeThe purpose of this study is to use the method of lines to solve the two-dimensional nonlinear advection–diffusion–reaction equation with variable coefficients.Design/methodology/approachThe strictly positive definite radial basis functions collocation method together with the decomposition of the interpolation matrix is used to turn the problem into a system of nonlinear first-order differential equations. Then a numerical solution of this system is computed by changing in the classical fourth-order Runge–Kutta method as well.FindingsSeveral test problems are provided to confirm the validity and efficiently of the proposed method.Originality/valueFor the first time, some famous examples are solved by using the proposed high-order technique.

A general solution procedure for the scaled boundary finite element method via shooting technique

The scaled boundary finite element method (SBFEM) is known for its inherent ability to simulate unbounded domains and singular fields, and its flexibility in the meshing procedure. Keeping the analytical form of the field variables along one coordinate intact, it transforms the governing partial differential equations of the problem into a system of one-dimensional (initial–)boundary value problems. However, closed-form solution of the said system is not available for most cases (e.g. transient heat transfer, acoustics, ultrasonics, etc.) since the system cannot be diagonalized in general. This paper aims to establish a numerical tool within the context of the shooting technique to evaluate the coefficient matrices of the subdomains without a priori knowledge of the analytical solution of the semi-discretized system. With proper choice of boundary conditions, the technique uses the strong form of the scaled boundary finite element equations to pass the required information and with the desired accuracy from one boundary to another. Due to generality of the technique, its procedure can be adjusted for any field equations. Since this technique is presented here for the first time, linear elastostatics, for which the closed-form solution is well-established, is formulated to provide valid comparisons. In addition, any direct solution method can be used for integrating the scaled boundary equations. Thus, without loss of generality, a Nyström extension of the classical fourth-order Runge–Kutta method is employed. A quantitative sensitivity analysis is also conducted, and efficiency of the classical and proposed solution techniques is compared in terms of computational time. Finally, some numerical examples, including bounded and unbounded domains, as well as singular stress fields are simulated based on the classical and proposed solution techniques.

The kinematics behaviour of coupled pendulum using differential transformation method

The topic of coupled oscillations is rich in physical content which is both interesting and complex. In this research work we aimed to study the kinematics behaviour of an important physical system known as coupled pendulum, in which two identical pendulums were coupled together by a spring. The Lagrangian of the system was first constructed, then the equations of motion have been derived. Analytical solutions were obtained for these equations, and in addition we apply a reliable algorithm based on an adaptation of the standard differential transformation method is presented, which is the multi-step differential transformation method to obtain numerical solutions for the considered system for some selected initial conditions. The solutions of the equations of motion were obtained by the multi-step differential transform method. Of course, this method is needed if it predicts the motion of the considered system. For this reason figurative comparisons between the multi-step differential transformation method, the standard differential transformation method and the classical fourth-order Runge–Kutta method are given. The results reveal that the proposed technique is a promising tool to predict the behaviours of the considered system in long time interval. Moreover, the motion of curves represents the initial conditions that sensitive depend and how much oscillatory motion. In conclusion, the proposed technique is a reliable method to solve the considered double pendulum problem in long time interval. We believe that the system considered in this work along with the method used is interesting for physicists, mathematicians and engineers.

A Numerical Model of Carbon Dioxide Concentration Measurement in a Room with an Opened Ventilation System

A vast number of patients visit the facility every day, causing a major air pollution issue that may pose a risk of exposure of respiratory infectious diseases in outpatient rooms and harm human health. TB, COVID-19, MERS, and SARS are dangerous communicable diseases that transmit from person to person through the air or aerosol in a variety of forms, such as coughing, spitting, sneezing, speaking, or through wounds. COVID-19, TB, MERS and SARS are risks and the chances of success toward lethal infection make more patients ill in the hospital. We should also be notified of the care and control of these diseases. As a result, effective air quality monitoring is needed to monitor and reduce the potential for infected air, such as carbon dioxide (CO₂) concentrations. Measuring and controlling carbon dioxide in a hospital with a ventilation system where the number of patients in each room varies in time is challenging. In this research, the numerical model of carbon dioxide concentration measurement in a space with an opened ventilation system is proposed. The model sets the concentration of carbon dioxide at any point when the number of people and the rate of ventilation varies. The classical fourth-order Runge-Kutta method is employed to approximate the model solution. There are many cases of scenarios for improving air quality in the proposed simulations. In the air quality management process, the proposed model provides a balance between the number of persons allowed to stay in the room and the capacity of the air ventilation system.

A novel iterative method for solving chemical kinetics system

The purpose of this research is to impose a semi-analytical method called the iterative method to the chemical kinetics system, which appears in the form of a system of ordinary differential equations. To test the accuracy of the standard iterative method, we have applied the classical fourth-order Runge–Kutta method and the iterative method to the chemical kinetics system. It is significantly notable that approximate analytical precisions of standard iterative method made a high agreement with those obtained from the fourth-order Runge–Kutta technique. Numerical outputs and solution procedures indicate that iterative method can be easily applicable to a large class of scientific numeric applications with high accuracy.

Numerical Solution of First Order Ordinary Differential Equation by Using Runge-Kutta Method

In this paper, the classical fourth-order Runge-Kutta methodis presented for solving the first-order ordinary differential equation. First, the given solution domain is discretizedby using a uniform discretization grid point. Next by applyingthe forward difference method, we discretized the given ordinary differential equation. And formulating a difference equation. Then using this difference equation, the given first-order ordinary differential equation is solved by using the classicalfourth-order Runge-Kutta method at each specified grid point. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supportedthe theoretical and mathematical statementsand the accuracy of the solution is obtained. The accuracy of the present methodhas been shown in the sense ofmaximumabsolute error and the local behavior of the solution is captured exactly. Numerical and exact solutions have been presented in tables and graphs and the corresponding maximumabsolute errorisalso presented in tables and graphs. The present method approximates the exact solution very well and it is quite efficient and practically well suitedfor solving first-order ordinary differential equations. The numerical result presented in tables and graphsindicates that the approximate solution is in good agreement with the exact solution. Hence the proposed method is accruable to solve ordinary differential equations.

Solution of typhoid fever model by Adomian decomposition method

In this paper, we presents a deterministic mathematical model on the dynamics of typhoid fever disease. The Adomian Decomposition Method (ADM) is used to solve the model equations. In solving the model, the validity of the ADM is established by the classical fourth-order Runge-Kutta method implemented in Maple 18. In other to confirm the accuracy of the method, a comparison was carried out between the ADM solution and Runge-Kutta(RK4). The findings obtained confirm the precision and accuracy of the ADM to cope with the study of morden epidemics.

The Effect of Modulation on Heat Transport by a Weakly Nonlinear Thermal Instability in the Presence of Applied Magnetic Field and Internal Heating

The present paper deals with a weakly nonlinear stability problem under an imposed time-periodic thermal modulation. The temperature has two parts: a constant part and an externally imposed time-dependent part. We focus on stationary convection using the slow time scale and quantify convective amplitude through the real Ginzburg-Landau equation (GLE). We have used the classical fourth order Runge-Kutta method to solve the real Ginzburg-Landau equation. The effect of various parameters on heat transport is discussed through GLE. It is found that heat transport analysis is controlled by suitably adjusting the frequency and amplitude of modulation. The applied magnetic field (effect of Ha) is to diminish the heat transfer in the system. Three different types of modulations thermal, gravity, and magnetic field have been compared. It is concluded that thermal modulation is more effective than gravity and magnetic modulation. The magnetic modulation stabilizes more and gravity modulation stabilizes partially than thermal modulation.

Application of Adomian Decomposition Method on a Mathematical Model of Malaria

In this paper, we consider a deterministic model of malaria transmission. Adomian decomposition method (ADM) is used to calculate an approximation to the solution of the non-linear couple of differential equations governing the model. Classical fourth-order Runge-Kutta method implemented in Maple18 confirms the validity of the ADM in solving the problem. Graphical results show that ADM agrees with R-K 4. In order words, these produced the same behaviour, validating ADM's efficiency and accuracy of ADM in finding the malaria model solution.

An Intial-Value Technique for Self-Adjoint Singularly Perturbed Two-Point Boundary Value Problems

In this paper, we present an initial value technique for solving self-adjoint singularly perturbed linear boundary value problems. The original problem is reduced to its normal form and the reduced problem is converted to first order initial value problems. This replacement is significant from the computational point of view. The classical fourth order Runge-Kutta method is used to solve these initial value problems. This approach to solve singularly perturbed boundary-value problems is numerically very appealing. To demonstrate the applicability of this method, we have applied it on several linear examples with left-end boundary layer and right-end layer. From the numerical results, the method seems accurate and solutions to problems with extremely thin boundary layers are obtained.

Comparison of Numerical Methods for System of First Order Ordinary Differential Equations

In this paper three numerical methods are discussed to find the approximate solutions of a systems of first order ordinary differential equations. Those are Classical Runge-Kutta method, Modified Euler method and Euler method. For each methods formulas are developed for n systems of ordinary differential equations. The formulas explained by these methods are demonstrated by examples to identify the most accurate numerical methods. By comparing the analytical solution of the dependent variables with the approximate solution, absolute errors are calculated. The resulting value indicates that classical fourth order Runge-Kutta method offers most closet values with the computed analytical values. Finally from the results the classical fourth order is more efficient method to find the approximate solutions of the systems of ordinary differential equations.

Semi analytical method for solving lymphatic filariasis epidemic model

In this paper, we present a deterministic model on the transmission dynamics of Lymphatic Filariasis. Non-Standard Finite Difference Method (NSFDM) is employed to attempt the solution of the model. The validity of the NSFDM in solving the model is established by using the computer in-built classical fourth-order Runge-Kutta method. The comparism between Non-Standard Finite Difference Method solution and Runge-Kutta (RK4) were performed which were found to be efficient, accurate and rapidly convergence. Keywords: Lymphatic Filariasis, Non-Standard Finite Difference, Runge-Kutta; Mathematical model