Numerical Treatment of Initial Value Problems of Nonlinear Ordinary Differential Equations by Duan-Rach-Wazwaz Modified Adomian Decomposition Method

We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.

Multi-scale Runge-Kutta_Galerkin method for solving one-dimensional Kd V and Burgers equations

In this paper, the multi-scale Runge-Kutta_Galerkin method is developed for solving the evolution equations, with the spatial variables of the equations being discretized by the multi-scale Galerkin method based on the multi-scale orthogonal bases in Ho （a, b） and then the classical fourth order explicit Runge-Kutta method being applied to solve the resulting initial problem of the ordinary differential equations for the coefficients of the approximate solution. The proposed numerical scheme is validated by applications to the Burgers equation （nonlinear convection-diffusion problem）, the KdV equation （single solitary and 2-solitary wave problems） and the KdV-Burgers equation, where analytical solutions are available for estimating the errors. Numerical results show that using the algorithm we can solve these equations stably without the need for extra stabilization processes and obtain accurate solutions that agree very well with the corresponding exact solutions in all cases.

Influence of viscous force on the dynamic process of micro-sphere in optical tweezers

With the advantages of noncontact,high accuracy,and high flexibility,optical tweezers hold huge potential for micro-manipulation and force measurement.However,the majority of previous research focused on the state of the motion of particles in the optical trap,but paid little attention to the early dynamic process between the initial state of the particles and the optical trap.Note that the viscous forces can greatly affect the motion of micro-spheres.In this paper,based on the equations of Newtonian mechanics,we investigate the dynamics of laser-trapped micro-spheres in the surrounding environment with different viscosity coefficients.Through the calculations,over time the particle trajectory clearly reveals the subtle details of the optical capture process,including acceleration,deceleration,turning,and reciprocating oscillation.The time to equilibrium mainly depends on the corresponding damping coefficient of the surrounding environment and the oscillation frequency of the optical tweezers.These studies are essential for understanding various mechanisms to engineer the mechanical motion behavior of molecules or microparticles in liquid or air.

A-stable Explicit Nonlinear Runge-Kutta Methods

Nonlinear methods are combined with Runge Kutta methods to develop A stable explicit nonlinear Runge Kutta methods for solving stiff differential equations and a class of the third order formulae are constructed.It avoids solving the nonlinear equations which implicit methods must solve. Implementation is very simple and the computation cost for each step is small.This paper uses a shift transformation to avoid the order reduction of nonlinear methods at y = 0 . Thus the method is very practicable. Numerical tests show that the method is more efficient than explicit methods or implicit methods of the same order.

Simulation and comparison of several numerical algorithms for solving ballistic differential equations

Several numerical methods of differential equations and their applications in ballistic calculation are discussed for the purpose of simplification of the dynamic differential equations of projectile trajectory.Program simulations of Euler method,Heun method,lassic fourth-order Runge Kutta（RK4）method,ABM method and Hamming method are achieved based on Matlab.In addtion,the approximate solutions,local truncation errors and calculation time of the dynamic differential equations are obtained.By analyzing the simultaion results,the advantages and disadvantages of these methods are compared,which provides a basis for choice of ballistic calculation methods.

HIGH-RESOLUTION NUMERICAL MODEL FOR SHALLOW WATER FLOWS AND POLLUTANT DIFFUSIONS

A finite-volume high-resolution numerical model for coupling the shallow water flows and pollutant diffusions was presented based on using a hybrid TVD scheme in space discretization and a Runge-Kutta method in time discretization. Numerical simulations for modelling dam-break, enlarging open channel flow and pollutant dispersion were implemented and compared with experimental data or other published computations. The validation of this method shows that it can not only deal with the problem involving discontinuities and unsteady flows, but also solve the general shallow water flows and pollutant diffusions.

Error Analysis in Frequency Domain for Linear Multipass Algorithms

Error analysis methods in frequency domain are developed in this paper for determining the characteristic root and transfer function errors when the linear multipass algorithms are used to solve linear differential equations. The relation between the local truncation error in time domain and the error in frequency domain is established, which is the basis for developing the error estimation methods. The error estimation methods for the digital simulation model constructed by using the Runge-Kutta algorithms and the linear multistep predictor-corrector algorithms are also given.

Effects of heat and mass transfer on nonlinear MHD boundary layer flow over a shrinking sheet in the presence of suction

This work is concerned with Magnetohydrodynamic viscous flow due to a shrinking sheet in the presence of suction. The cases of two dimensional and axisymmetric shrinking are discussed. The governing boundary layer equations are written into a dimensionless form by similarity transformations. The transformed coupled nonlinear ordinary differential equations are numerically solved by using an advanced numeric technique. Favorability comparisons with previously published work are presented. Numerical results for the dimensionless velocity, temperature and concentration profiles as well as for the skin friction, heat and mass transfer and deposition rate are obtained and displayed graphically for pertinent parameters to show interesting aspects of the solution.

EULER EQUATION SIMULATION FOR THE FLOWFIELD AROUND PROPELLER

An Euler solver is developed for calculating the flowfield around the propeller. The Euler equations are recast in the blade attached rotating coordinate system with absolute flow variables.The finite volume explicit Runge Kutta time stepping scheme is employed for solving the equations.The presented method is applied for a two bladed propeller NACA10 (3) (066) 033.An algebraic O O grid is generated. Comparison of the mumerical results shows good agreement with the experimental data.

Longtime Convergence of Numerical Approximations for Semilinear Parabolic Equations (Ⅰ)

The numerical approximations of the dynamical systems governed by semilinear parabolic equations are considered. An abstract framework for long time error estimates is established. When applied to reaction diffusion equation, Navier Stokes equations and Chan Hilliard equation, approximated by Galerkin and nonlinear Galerkin methods in space and by Runge Kutta method in time, our framework yields error estimates uniform in time.

SEIRS model for malaria transmission dynamics incorporating seasonality and awareness campaign

Malaria,a devastating disease caused by the Plasmodium parasite and transmitted through the bites of female Anopheles mosquitoes,remains a significant public health concern,claiming over 600,000 lives annually,predominantly among children.Novel tools,including the application of Wolbachia,are being developed to combat malaria-transmitting mosquitoes.This study presents a modified susceptible-exposed-infectious-recovered-susceptible(SEIRS)compartmental mathematical model to evaluate the impact of awareness-based control measures on malaria transmission dynamics,incorporating mosquito interactions and seasonality.Employing the next-generation matrix approach,we calculated a basic reproduction number(R0)of 2.4537,indicating that without robust control measures,the disease will persist in the human population.The model equations were solved numerically using fourth and fifth-order Runge-Kutta methods.The model was fitted to malaria incidence data from Kenya spanning 2000 to 2021 using least squares curve fitting.The fitting algorithm yielded a mean absolute error(MAE)of 2.6463 when comparing the actual data points to the simulated values of infectious human population(Ih).This finding indicates that the proposed mathematical model closely aligns with the recorded malaria incidence data.The optimal values of the model parameters were estimated from the fitting algorithm,and future malaria dynamics were projected for the next decade.The research findings suggest that social media-based awareness campaigns,coupled with specific optimization control measures and effective management methods,offer the most cost-effective approach to managing malaria.

New transfer matrix method for long-period fiber gratings with coupled multiple cladding modes

A new transfer matrix method for long-period fiber gratings with coupled multiple cladding modes is proposed and numerically characterized. The transmission spectra of uniform and non-uniform longperiod fiber gratings are numerically characterized. The theoretical results excellently agree with the experimental measurements, Compared with commonly used methods, such as using the fourth-order adaptive step size control of the Runge-Kutta algorithm in solving the coupled mode equation, the new transfer matrix method exhibits a faster calculation speed.

Study on the spectral response of fiber Bragg grating sensor under non-uniform strain distribution in structural health monitoring

Many theoretical studies have been developed to study the spectral response of a fiber Bragg grating (FBG) under non-uniform strain distribution along the length of FBG in recent years. However, almost no experiments were designed to obtain the evolution of the spectrum when a FBG is subjected to non-uniform strain. In this paper, the spectral responses of a FBG under non-uniform strain distributions are given and a numerical simulation based on the Runge-Kutta method is introduced to investigate the responses of the FBG under some typical non-uniform transverse strain fields, including both linear strain gradient and quadratic strain field. Experiment is carried out by using loads applied at different locations near the FBG. Good agreements between experimental results and numerical simulations are obtained.

Stagnation temperature effect on the conical shock with application for air

The aim of this work is to realize a new numerical program based on the development of a mathematical model allowing determining the parameters of the supersonic flow through a conical shock under hypothesis at high temperature, in the context of correcting the perfect gas model. In this case, the specific heat at constant pressure does not remain constant and varies with the increase of temperature. The stagnation temperature becomes an important parameter in the calculation.The mathematical model is presented by the numerical resolution of a system of first-order nonlinear differential equations with three coupled unknowns for initial conditions. The numerical resolution is made by adapting the higher order Runge Kutta method. The parameters through the conical shock can be determined by considering a new model of an oblique shock at high temperature. All isentropic parameters of after the shock flow depend on the deviation of the flow from the transverse direction. The comparison of the results is done with the perfect gas model for low stagnation temperatures, upstream Mach number and cone deviation angle. A calculation of the error is made between our high temperature model and the perfect gas model. The application is made for air.

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.

High-order implicit discontinuous Galerkin schemes for unsteady compressible Navier–Stokes equations

Efficient solution techniques for high-order temporal and spatial discontinuous Galerkin（DG） discretizations of the unsteady Navier–Stokes equations are developed. A fourth-order implicit Runge–Kutta（IRK） scheme is applied for the time integration and a multigrid preconditioned GMRES solver is extended to solve the nonlinear system arising from each IRK stage. Several modifications to the implicit solver have been considered to achieve the efficiency enhancement and meantime to reduce the memory requirement. A variety of time-accurate viscous flow simulations are performed to assess the resulting high-order implicit DG methods. The designed order of accuracy for temporal discretization scheme is validate and the present implicit solver shows the superior performance by allowing quite large time step to be used in solving time-implicit systems. Numerical results are in good agreement with the published data and demonstrate the potential advantages of the high-order scheme in gaining both the high accuracy and the high efficiency.