Numerical Scheme Based on the Implicit Runge-Kutta Method and Spectral Method for Calculating Nonlinear Hyperbolic Evolution Equations

A numerical scheme for nonlinear hyperbolic evolution equations is made based on the implicit Runge-Kutta method and the Fourier spectral method. The detailed discretization processes are discussed in the case of one-dimensional Klein-Gordon equations. In conclusion, a numerical scheme with third-order accuracy is presented. The order of total calculation cost is O(Nlog2N). As a benchmark, the relations between numerical accuracy and discretization unit size and that between the stability of calculation and discretization unit size are demonstrated for both linear and nonlinear cases.

Lotka-Volterra Model of Wastewater Treatment in Bioreactor System using 4th Order Runge-Kutta Method

Wastewater treatment is essential to preserve the ecosystem and to ensure water resources are uncontaminated. This paper presents the Lotka-Volterra model of nonlinear ordinary differential equations of the interaction between predator-prey and substrate. The dimensionless ordinary differential equations of the model are solved using the 4th Order Runge-Kutta method (RK4) in MATLAB®. This study discusses the behaviour parameters of predators, prey and substrate. The results are shown graphically for different values of each parameter. Hence, the biological reaction of clean water from the interaction of predator-prey and substrate in wastewater treatment is identified. The higher the concentration of prey, the faster the concentration of substrate reaches 0 with and without the natural death of prey. The clean water will be produced whenever the concentration of prey and the concentration of predator are in balance regardless of the natural death rate. Stability analysis using the Jacobian matrix at the equilibrium point is also performed to determine the stability of the system.

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.

Modeling of constrained motion

This paper presents an investigation of modeling and solving of differential equations in the study of mechanical systems with holonomic constraints. The 2D and 3D mathematical models of constrained motion are made. The structure of the models consists of nonlinear first or second order differential equations. Cases of free movement and movement with resistance are investigated. Solutions of the Cauchy problem of obtained differential equations were obtained by Runge–Kutta method.

Numerical simulation of a class of space fractional bistable systems based on the Fourier spectral method

Nonlinear vibration arises everywhere in a bistable system. The bistable system has been widely applied in physics, biology, and chemistry. In this article, in order to numerically simulate a class of space fractional-order bistable system, we introduce a numerical approach based on the modified Fourier spectral method and fourth-order Runge-Kutta method. The fourth-order Runge-Kutta method is used in time, and the Fourier spectrum is used in space to approximate the solution of the space fractional-order bistable system. Numerical experiments are given to illustrate the effectiveness of this method.

Modeling the Effects of Chemotherapy and Immunotherapy on Tumor Growth

Mathematical modeling has been used to simulate the interaction of chemotherapy and immunotherapy drugs intervention with the dynamics of tumor cells growth. This work studies the interaction of cells in the immune system, such as the natural killer, dendritic, and cytotoxic CD8+ T cells, with chemotherapy. Four different cases were considered in the simulation: no drug intervention, independent interventions (either chemotherapy or immunotherapy), and combined interventions of chemotherapy and immunotherapy. The system of ordinary differential equations was initially solved using the Runge-Kutta method and compared with two additional methods: the Explicit Euler and Heun’s methods. Results showed that the combined intervention is more effective compared to the other cases. In addition, when compared with Runge-Kutta, the Heun’s method presented a better accuracy than the Explicit Euler technique. The proposed mathematical model can be used as a tool to improve cancer treatments and targeted therapy.

PREDICTOR-CORRECTOR TECHNIQUE FOR IMPLEMENTING AN SIXTH ORDER IMPLICIT RUNGE-KUTTA METHOD

Bài báo này đưa ra hướng tiếp cận mới đối với bài toán xây dựng trình thực thi cho một lớp các phương pháp Runge-Kutta dạng ẩn. Phương pháp Runge-Kutta dạng ẩn được nghiên cứu cụ thể ở đây được phát triển dựa trên các đa thức Gauss-Legendre, phương pháp xuất hiện đầu tiên trong bài báo của J. C. Butcher (2009). Sự cải tiến mà hướng tiếp cận mới mang lại là rất hữu ích. Điều này có được do những lợi thế của phương pháp một bước dạng ẩn chỉ có ba bước, đặc biệt phù hợp với các bài toán stiff, với khối lượng tính toán nhỏ mà độ chính xác cao của một phương pháp bậc sáu, một bậc tương đối cao của sự hội tụ mà vẫn đảm bảo điều kiện bền vững. Chứng minh cho sự hội tụ của phương pháp này được ra. Hướng tiếp cận này cũng có thể áp dụng cho một phương pháp Runge-Kutta dạng ẩn khác được đưa ra với bậc thấp hơn được xây dựng dựa trên các đa thức Gauss-Legendre. Sự kết hợp giữa hướng tiếp cận mới và phương pháp sai phân dạng khối Off-step bậc sáu có thể mang đến sự hợp lý trong việc xấp xỉ các bài toán stiff. Phương pháp này cũng được nghiên cứu trong bài báo. Sau cùng, các so sánh thực nghiệm đưa ra nhằm minh họa cho sự ưu việt của hướng tiếp cận đạt được.

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.

Uniqueness of Solutions, Stability and Simulations for a Differential Problem Involving Convergent Series and Time Variable Singularities

We focus on a new type of nonlinear integro-differential equations with nonlocal integral conditions. The considered problem has one nonlinearity with time variable singularity. It involves also some convergent series combined to Riemann-Liouville integrals. We prove a uniqueness of solutions for the proposed problem, then, we provide some examples to illustrate this result. Also, we discuss the Ulam-Hyers stability for the problem. Some numerical simulations, using Rung Kutta method, are discussed too. At the end, a conclusion follows.