Vol. 30 No. 1 (2023)

Open Access

Articles

Article ID: 2398

TWO-STEP ORDER 3/2 STRONG METHOD FOR APPROXIMATING STOCHASTIC DIFFERENTIAL EQUATIONS

by Yazid Alhojilan

Advances in Differential Equations and Control Processes, Vol.30, No.1, 2023;

In this paper, we consider two-step order strong scheme for getting numerical solutions of stochastic differential equations (SDEs) of order 3/2. It follows a new technique based on replacing stochastic integrals $I_\alpha$ by random variables. Thus we do not need to calculate $I_\alpha$. We employ Itô-Taylor expansion and Runge-Kutta method to get the approximate solutions of the desired order. The experimental results of the approximation method and its error are provided to confirm the validity of the method.

Open Access

Articles

Article ID: 2399

CHEBYSHEV WAVELETS BASED TECHNIQUE FOR NUMERICAL DIFFERENTIATION

by Inderdeep Singh, Preeti

Advances in Differential Equations and Control Processes, Vol.30, No.1, 2023;

Numerical differentiation plays a significant role in numerical analysis. In this research paper, Chebyshev wavelets based efficient scheme has been developed to find the numerical differentiation problems arising in numerical analysis. Proposed technique based on the expansion of unknown function into a series of Chebyshev wavelets. Some numerical examples have been performed to find the accuracy of the proposed technique.

Open Access

Articles

Article ID: 2400

AN EFFECTIVE SHOOTING PIECEWISE ANALYTICAL INTEGRATION METHOD FOR SINGULAR PERTURBATION TWO-POINT BOUNDARY VALUE PROBLEMS

by Haifa S. Al-Juaydi, Essam R. El-Zahar

Advances in Differential Equations and Control Processes, Vol.30, No.1, 2023;

In this paper, an effective semi-analytical-numerical method is proposed for solving singular perturbation two-point boundary value problems (SPBVPs). Firstly, the original problem is replaced by an equivalent singular perturbation initial value problems (SPIVPs) of first-order with an unknown initial condition that can be determined iteratively using shooting method. Then, an adaptive one-step explicit piecewise analytical integration scheme over a special non-uniform mesh is presented to integrate these SPIVPs. The accuracy and stability properties of the scheme are investigated and shown to yield at least second-order of accuracy and L-stability property. A good estimation of the missed initial condition is obtained and suggested as a starting initial guess to ensure accelerated convergence of the shooting method. To demonstrate the applicability of the method, we have applied it to linear and nonlinear test problems at different values of the perturbation parameter. The method can be extended to higher-order SPBVPs. We have applied it to the well-known third-order Blasius’ viscous flow problem for a large suction case. The results indicate that the method approximates the solution very well not only over the boundary layer region but also overall the problem domain. Moreover, the method is more accurate and has a higher computational efficiency compared to other methods in the literature.

Open Access

Articles

Article ID: 2401

EXISTENCE AND OPTIMAL CONTROL ANALYSIS OF ACID-MEDIATED TUMOR INVASION MODEL

by P. T. Sowndarrajan

Advances in Differential Equations and Control Processes, Vol.30, No.1, 2023;

The distributed optimal control problem of a highly nonlinear coupled system of reaction-diffusion equations is investigated in the study. Normal cell density, tumor cell density, excess H+ ion concentration, and chemotherapy drug concentration are all represented by partial differential equations (PDEs) in the coupled system of acid-mediated tumor invasion model. It is a usual factor to formulate an optimal control problem by introducing control interventions while considering the tumor invasion model with drug chemotherapy. However, in our model, we consider a constant drug injection rate as a control variable based on biological motivation. The major goal of our optimal control problem is to reduce the overall amount of medicine supplied while minimizing cancer cell proliferation. First, we prove the existence of solutions to the direct problem using the Faedo-Galerkin approximation method, deriving a priori estimates, and then passing to the limit in the approximate solutions using monotonicity and compactness arguments. We introduce a functional to minimize and to establish the existence of optimal control for the proposed optimal control problem. Using the Lagrangian framework, we derive the adjoint problem and necessary optimality condition associated with our problem. Finally, we prove the existence of weak solutions to the adjoint system.

Open Access

Articles

Article ID: 2402

SOLITON SOLUTIONS OF 10th ORDER 2-D BOUSSINESQ EQUATION

by K. Bharatha, R. Rangarajan

Advances in Differential Equations and Control Processes, Vol.30, No.1, 2023;

The 2-D Boussinesq equation of 10th order is derived from its bilinear form. Its soliton solutions are studied in detail using the Hirota’s bilinear method. Since the 2-D Boussinesq equation is not completely integrable, we only obtain its 1-soliton and 2-soliton solutions. The equation is solved by the tanh method to reconstruct the 1-soliton solution obtained by the Hirota’s method.