Vol. 29 (2022)

Open Access

Articles

Article ID: 2559

SOLUTION OF SOME LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

by Anarkul Urdaletova, Syrgak Kydyraliev, Elena Burova

Advances in Differential Equations and Control Processes, Vol.29, No., 2022;

The problem of integrability of ordinary differential equations to find their exact solutions is a celebrated problem in the theory of differential equations which attracted attention of several workers in the area. This is due to the fact that: (a) differential equations are the most widely used continuous models of dynamic systems in physics, medicine, economics, biology and other sciences that study the surrounding reality, for which the explicit trajectory of the dynamic system’s behavior is important as the explicit solution contains in itself the maximum information about the behavior of the system; (b) an explicit solution of the equation is necessary to confirm the mathematical and physical intuition, to compare the solutions obtained by various approximate methods and to compare these methods. It is also worth noting that in the presence of various methods for obtaining an explicit form of solving differential equations, the advantage is given to simpler algorithms. This paper presents a method for finding an explicit form of the solution of one class of systems of linear ordinary differential equations of the first order with variable coefficients. Examples are given for illustration. This method includes elements of the well-known classical methods of the theory of integration of ordinary differential equations: the Leonard Euler method, based on the roots of the characteristic equation, and the Jean Leron D’Alembert method of integrable combinations.

Open Access

Articles

Article ID: 2560

CHOICE OF A BASIS TO SOLVE THE LANE-EMDEN EQUATION

by Obaid J.Algahtani

Advances in Differential Equations and Control Processes, Vol.29, No., 2022;

If the set of basis functions is chosen by overlooking physics of a problem, then the results can be misleading. It is shown that for the Lane-Emden equation, a set of functions with semi-infinite domain sometimes fails to produce results of desired accuracy. A qualitative analysis of the problem shows that the solution is bounded when $m$ is an odd integer but is unbounded when $m$ is even. Solution of the Lane-Emden equation with rational Legendre functions, as basis, is poorer in accuracy when $m = 2$ as compared with the one when $m = 3$ with the same basis. Since the physically important region is contained in a finite interval, a set of scaled Legendre polynomials, as basis, produces results which are much more accurate on the interval of interest.

Open Access

Articles

Article ID: 2561

2D-WAVELETS BASED EFFICIENT SCHEME FOR SOLVING SOME PDEs

by Inderdeep Singh, Manbir Kaur

Advances in Differential Equations and Control Processes, Vol.29, No., 2022;

We propose two-dimensional Hermite wavelet method for solving some applications of partial differential equations. Kronecker tensor product has been utilized to resolve and control huge matrices operations and calculations. Proposed method is based on the approximation of largest mixed derivatives of the given partial differential equation into a series of two-dimensional Hermite wavelet basis functions. To validate the efficiency and accuracy of the proposed technique, some numerical examples are presented.

Open Access

Articles

Article ID: 2562

CLOSED FORM EXPRESSIONS OBTAINED FROM THE SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS OF THE PROBABILITY DENSITY FUNCTION OF THE BETA DISTRIBUTION

by Olasunmbo O.Agboola, Hilary I.Okagbue, Adedayo F.Adedotun, Paulinus O.Ugwoke

Advances in Differential Equations and Control Processes, Vol.29, No., 2022;

In this paper, some closed form expressions for selected parameters for the probability density function (PDF) of the beta distribution are obtained. The closed form expressions are recovered from the solution of the ordinary differential equations (ODEs), obtained from the differentiation of the PDF of the distribution. The paper shows that the shape of the distributions also determines the nature of the resulting ODE which has shown how distributions related to the beta distribution can be traced via the solutions of the ODEs. Numerical methods are unnecessary because the closed form expressions are the same with the values obtained from the standard statistical software.

Open Access

Articles

Article ID: 2563

SOLUTIONS OF FRACTIONAL HYBRID DIFFERENTIAL EQUATIONS VIA FIXED POINT THEOREMS AND PICARD APPROXIMATIONS

by Sahar Abusalim

Advances in Differential Equations and Control Processes, Vol.29, No., 2022;

We investigate the following fractional hybrid differential equation: $$ \left\{\begin{array}{l} D_{t_0+}^\alpha\left[x(t)-f_1(t, x(t))\right]=f_2(t, x(t)) \text { a.e } t \in J, \\ x\left(t_0\right)=x_0 \in \mathbb{R}, \end{array}\right. $$ where $D_{t_0+}^\alpha$ is the Riemann-Liouville differential operator order of $\alpha>0, J=\left[t_0, t_0+a\right]$, for some $t_0 \in \mathbb{R}, a>0$, $f_1 \in C(J \times \mathbb{R}, \mathbb{R}), \quad f_2 \in \mathcal{L}_p^\alpha(J \times \mathbb{R}, \mathbb{R}), \quad p \geq 1$ and satisfies certain conditions. We investigate such equations in two cases: $\alpha \in(0,1)$ and $\alpha \geq 1$. In the first case, we prove the existence and uniqueness of a solution which extends the main result of [1]. Moreover, we show that the Picard iteration associated to an operator $T: C(J \times \mathbb{R}) \rightarrow C(J \times \mathbb{R})$ converges to the unique solution of (1.0) for any initial guess $x \in C(J \times \mathbb{R})$. In particular, the rate of convergence is $n^{-1}$. In the second case, we investigate this equation in the space of $k$ times differentiable functions. Naturally, the initial condition $x\left(t_0\right)=x_0$ is replaced by $x^{(k)}\left(t_0\right)=x_0, 0 \leq k \leq n_{\alpha, p}-1$ and the existence and uniqueness of a solution of (1.0) is established. Moreover, the convergence of the Picard iterations to the unique solution of (1.0) is shown. In particular, the rate of convergence is $n^{-1}$. Finally, we provide some examples to show the applicability of the abstract results. These examples cannot be solved by the methods demonstrated in [1]

Open Access

Articles

Article ID: 2564

ON THE ENERGY EQUALITY FOR WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS

by N. V.Giang, D. Q.Khai, N. M.Tri

Advances in Differential Equations and Control Processes, Vol.29, No., 2022;

In this paper, we first introduce the concept of absolutely continuous functions of order $s(0