Vol. 30 No. 4 (2023)

Open Access

Articles

Article ID: 2417

NON-CLASSICAL OPTIMAL CONTROL PROBLEM: A CASE STUDY FOR CONTINUOUS APPROXIMATION OF FOUR-STEPWISE FUNCTION

by Wan Noor Afifah Wan Ahmad, Suliadi Firdaus Sufahani, Mahmod Abd Hakim Mohamad, Mohd SaifullahRusiman, Mohd Zulariffin Md Maarof, Muhamad Ali lmran Kamarudin

Advances in Differential Equations and Control Processes, Vol.30, No.4, 2023;

The numerical properties of a contemporary optimal control problem (OCP) within the realm of financial aspects deviate from the conventional OCP framework. In our specific scenario, the final state condition is unknown, while the integrand exhibits a piecewise capacity that aligns with the unknown terminal state value. Since this is not a classical OCP, it cannot be solved using Pontryagin’s maximum approach under the expected end limit conditions. A free final state in the non-classical issue results in a critical limit condition of the final shadow value not being equal to zero. The new fundamental condition must be comparable to a particular necessary condition because the integrand is a part of the unidentified final state value. By employing the hyperbolic tangent (tanh) function, we showcase a continuous approximation of the piecewise constant integrand function. Furthermore, we tackle a specific scenario utilizing the shooting method in C++ programming language. This is by combining the Newton and Golden Section Search methods in the shooting technique to calculate the limiting free final state value in an external circle emphasis. Discretization methods such as Euler and Runge-Kutta approximations were used in the validation procedure. The program was constructed in AMPL programming language with MINOS solver.

Open Access

Articles

Article ID: 2418

NUMERICAL SIMULATION OF ROSENAU-KORTEWEG-DE VRIES REGULARIZED LONG WAVE EQUATION WITH FLUX LIMITERS METHOD

by OUEDRAOGO Mamadou, LAMIEN Kassiénou, SOMA Mifiamba, SO Ousséni

Advances in Differential Equations and Control Processes, Vol.30, No.4, 2023;

In this paper, the flux limiter technique based on the method of lines is designed to simulate the nonlinear Rosenau-Korteweg-de Vries-regularized long wave equation. In order to illustrate the efficiency, accuracy and essentially non-oscillatory property of the present method, the error norms, discrete mass, momentum and energy conservative properties have been calculated. These calculations give good agreement for the exact solutions and numerical solutions of solitary and shock wave.

Open Access

Articles

Article ID: 2419

PICARD’S METHOD OF SUCCESSIVE APPROXIMATION FOR FRACTIONAL ORDER INITIAL VALUE PROBLEM

by Jag Mohan, Anju Sood

Advances in Differential Equations and Control Processes, Vol.30, No.4, 2023;

In this paper, we derived Picard’s successive approximation technique for fractional differential systems in which the derivative has been taken in the Riemann-Liouville sense. We investigated the existence and uniqueness results of the present method. Two numerical examples are given to show the efficiency of the presented method.

Open Access

Articles

Article ID: 2420

A STOCHASTIC MODEL FOR THE SPREAD OF HUMAN PLASMODIA

by Abdoul KarimDRABO, Frédéric BERE, Sibiri NarcisseDOLEMWEOGO, S. P. Clovis NITIEMA

Advances in Differential Equations and Control Processes, Vol.30, No.4, 2023;

We use stochastic differential equations (SDEs) to model the spread of the malaria parasite through a compartmental model of the type $\left(S_h L_h I_h R_h S_h-I_v\right)$. Considering the transmission rates $\bar{\beta}(t)$ and $\bar{v}(t)$ by introducing standard Brownian motion in order to render the ordinary differential equation into SDE, we obtain the existence and uniqueness of the solution.

Open Access

Articles

Article ID: 2421

SIMULATION OF TWO-STEP ORDER 2 IMPLICIT STRONG METHOD FOR APPROXIMATING STRATONOVICH STOCHASTIC DIFFERENTIAL EQUATIONS

by Yazid Alhojilan

Advances in Differential Equations and Control Processes, Vol.30, No.4, 2023;

This paper introduces a novel two-step order strong scheme to numerically solve Stratonovich Stochastic Differential Equations (SDEs) of order 2. The approach involves a unique technique that replaces stochastic integrals $J_\alpha$ with random variables, eliminating the need for their explicit calculation. The methodology combines the Stratonovich-Taylor expansion with the Runge-Kutta method to obtain approximate solutions with the desired order of accuracy. To validate the method’s effectiveness, the paper includes experimental results that assess the approximation quality and quantify the associated errors.

Open Access

Articles

Article ID: 2422

HOPF BIFURCATION OF A DIFFUSIVE PREDATOR-PREY SYSTEM WITH NONLOCAL INTRASPECIFIC COMPETITION

by Haoming Wu, Zhaoyan Shi, Ming Liu

Advances in Differential Equations and Control Processes, Vol.30, No.4, 2023;

The study of predator-prey models and reaction-diffusion equations helps us to more comprehensively and accurately explain the changes in population density in the natural world, and is an important aspect of biological and mathematical research. The study of Hopf bifurcations is a significant topic of research on reaction-diffusion equations, and it is of great importance for our understanding of population behavior. Firstly, we modify a predator-prey system with local effects studied by Geng et al. [1] and conduct further research based on this system. Secondly, we investigate the existence of the positive equilibrium in the system. We find that the positive equilibrium exists only under certain conditions, and we provide criteria through the study of the properties of a cubic function. Thirdly, we present the characteristic equations for two different systems under the scenarios of $n = 0$ and $n \neq 0$. Since the model includes two integral terms, we categorize into two different Hopf bifurcation scenarios based on the magnitude of the value of certain parameters. We provide conditions under which the system undergoes Hopf bifurcation for each case.

Open Access

Articles

Article ID: 2423

AN INTEGRO-DIFFERENTIAL EQUATION IN COMPOUND POISSON RISK MODEL WITH VARIABLE THRESHOLD DIVIDEND PAYMENT STRATEGY TO SHAREHOLDERS AND TAIL DEPENDENCE BETWEEN CLAIMS AMOUNTS AND INTER-CLAIM TIME

by Kiswendsida Mahamoudou OUEDRAOGO, Delwendé Abdoul-Kabir KAFANDO, Francois Xavier OUEDRAOGO, Lassané SAW ADOGO, Pierre Clovis NITIEMA

Advances in Differential Equations and Control Processes, Vol.30, No.4, 2023;

This article is an extension of the compound Poisson risk model with variable threshold dividend payment strategy to shareholders and a dependence between claims amounts and inter-claim times via Spearman copula. We find the integro-differential equation associated to this risk model.