Fractal–Fractional Michaelis–Menten Enzymatic Reaction Model via Different Kernels

In this paper, three new models of fractal–fractional Michaelis–Menten enzymatic reaction (FFMMER) are studied. We present these models based on three different kernels, namely, power law, exponential decay, and Mittag-Leffler kernels. We construct three schema of successive approximations according to the theory of fractional calculus and with the help of Lagrange polynomials. The approximate solutions are compared with the resulting numerical solutions using the finite difference method (FDM). Because the approximate solutions in the classical case of the three models are very close to each other and almost matches, it is sufficient to compare one model, and the results were good. We investigate the effects of the fractal order and fractional order for all models. All calculations were performed using Mathematica software.

Development of the Approximating Functions Method for Problems in a Planar Waveguide with Constant Polarization

Article introduces an extension of the approximating functions method, a particular case of the finite element method (FEM) with interpolating functions in the form of Lagrange polynomials of a special form, to solve electrodynamics problems in a planar waveguide with constant polarization in the spatial-temporal domain using the Volterra integral equation method. The main goal of the article is to expand the area of ​​applicability of this method to three-dimensional problems in a planar waveguide with constant polarization, as well as to obtain general interpolation expressions in analytical form, which will be used to construct a system of nonlinear equations for solving specific problems.

Parameterized Trajectory Optimization and Tracking Control of High Altitude Parafoil Generation

Parafoil trajectory directly affects the power generation of a high-altitude wind power generation (HAWPG) device. Therefore, it is particularly important to optimize the parafoil trajectory and then to track it effectively. In this paper, the trajectory of the parafoil at high altitudes is optimized and tracked in a comprehensively parameterized manner. Both the complex dynamic characteristics of the parafoil and the dexterous demand of the high-altitude controller are considered. Firstly, the trajectory variables and control signals are parameterized as Lagrange polynomials in terms of the corresponding values at the selected nodes. Then, the Radau pseudospectral method (PSM) is employed to reformulate the original dynamic trajectory optimization problem into a static nonlinear programming (NLP) problem. By doing so, the parameterized optimal trajectory, which has the maximum net power generation, can be obtained. To attenuate the strong nonlinear, multivariable and coupling characteristics of the flexible parafoil, a bandwidth parameterized linear extended state observer (ESO) is used to estimate and reject these dynamics explicitly in a unified way. Finally, the simulation results demonstrate the effectiveness of the proposed parameterized trajectory optimization and control strategies. The main contribution of this study is that complicated nonlinear parafoil dynamics with a complex trajectory can be well regulated by a PID-type linear time-invariant controller, which is appealing for practitioners.

ON SOLUTIONS OF LINEAR FUNCTIONAL INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS VIA LAGRANGE POLYNOMIALS

In this study, a matrix-collocation method is developed numerically to solve the linear Fredholm-Volterra-type functional integral and integro-differential equations. The linear functional integro-differential equations are considered under initial conditions. The mentioned type problems often appear in various branches of science and engineering such as physics, biology, mechanics, electronics. The method essentially is a collocation method based on the Lagrange polynomials and matrix operations. By using presented method, the problem is reduced to a system of linear algebraic equations. The solution of this system gives the coefficients of assumed solution. An error analysis based on the residual function is studied. Some examples are solved to demonstrate the accuracy and efficiency of the method.

Arbitrary-Order Bernstein–Bézier Functions for DGFEM Transport on 3D Polygonal Grids

In this paper, we present an arbitrary-order discontinuous Galerkin finite element discretization of the SN transport equation on 3D extruded polygonal prisms. Basis functions are formed by the tensor product of 2D polygonal Bernstein–Bézier functions and 1D Lagrange polynomials. For a polynomial degree p, these functions span {xayb}(a+b)≤p⊗{zc}c∈(0,p) with a dimension of np(p+1)+(p+1)(p−1)(p−2)/2 on an extruded n-gon. Numerical tests confirm that the functions capture exactly monomial solutions, achieve expected convergence rates, and provide full resolution in the thick diffusion limit.

On Properties of a Regular Simplex Inscribed into a Ball

Let $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of continuos functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions upon ${\mathbb R}^n$. The interpolation projector $P:C(B)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in B$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$, $j=1,\ldots, n+1$.The norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the formula $\|P\|_B=\max_{x\in B}\sum |\lambda_j(x)|.$ Here $\lambda_j$ are the basic Lagrange polynomials corresponding to the $n$-dimensional nondegenerate simplex $S$ with the vertices $x^{(j)}$. Let $P^\prime$ be a projector having the nodes in the vertices of a regular simplex inscribed into the ball. We describe the points $y\in B$ with the property $\|P^\prime\|_B=\sum |\lambda_j(y)|$. Also we formulate some geometric conjecture which implies that $\|P^\prime\|_B$ is equal to the minimal norm of an interpolation projector with nodes in $B$. We prove that this conjecture holds true at least for $n=1,2,3,4$.

Buckling Analysis of Laminated Composite Beams by Using an Improved First Order Formulation

In this work, a finite element model based on an improved first-order formulation (IFSDT) is developed to analyze buckling phenomenon in laminated composite beams. The formulation has five independent variables and takes into account thickness stretching. Three-dimensional constitutive equations are employed to define the material properties. The Trefftz criterion is used for the stability analysis. The finite element model is derived from the principle of virtual work with high-order Lagrange polynomials to interpolate the field variables and to prevent shear locking. Numerical results are compared and validated with those available in literature. Furthermore, a parametric study is presented.