A new Model for Solving Three Mixed Integral Equations with Continuous and Discontinuous Kernels

In this paper, we discuss a new model to obtain the answer to the following question: how can we establish the different types of mixed integral equations from the Fredholm integral equation? For this, we consider three types of mixed integral equations (MIEs), under certain conditions. The existence of a unique solution of such equations is guaranteed. Using analytic and numerical methods, the three MIEs formulas yield the same Fredholm integral equation (FIE) formula of the second kind. For continuous kernel, the solution of these three MIEs, via the FIEs, is discussed analytically. In addition, for a discontinuous kernel, the Toeplitz matrix method (TMM) and Product Nyström method (PNM) are used to obtain, in each method, a linear algebraic system (LAS). Then, the numerical results are obtained, the error is computed in each case, and compared as well.

How to Tackle Underdeterminacy in Metabolic Flux Analysis? A Tutorial and Critical Review

Metabolic flux analysis is often (not to say almost always) faced with system underdeterminacy. Indeed, the linear algebraic system formed by the steady-state mass balance equations around the intracellular metabolites and the equality constraints related to the measurements of extracellular fluxes do not define a unique solution for the distribution of intracellular fluxes, but instead a set of solutions belonging to a convex polytope. Various methods have been proposed to tackle this underdeterminacy, including flux pathway analysis, flux balance analysis, flux variability analysis and sampling. These approaches are reviewed in this article and a toy example supports the discussion with illustrative numerical results.

Modeling the trajectory of motion of a linear dynamic system with multi-point conditions

<abstract><p>The motion of the linear dynamic system with given properties is modeled; conditions for system state at various arbitrarily points in time are given. Simulated movement carried out due to the calculated input vector function. The method of undefined coefficients is used to construct the input vector function and the corresponding trajectory. The proposed method consists in the formation of the state vector function, the trajectory of motion and the input vector function in exponential-polynomial form, that is, in the form of linear combinations of the powers of the time parameter with vector coefficients. This linear combination is complemented by a scalar exponential function with an additional parameter in the exponent to change the type of trajectory. To find the introduced coefficients, formulas and a linear algebraic system are formed. To find the introduced coefficients, the formed linear combinations are substituted directly into the equations describing the dynamic system and into the given multipoint conditions for finding the entered coefficients. All this leads to obtaining algebraic formulas and linear algebraic systems. Only the matrices included in the system that describe the dynamics of the model (and similar matrices with higher exponents) are the coefficients for the unknown parameters of the resulting algebraic system. It is proved that the fulfillment of the condition Kalman is sufficient for the solvability of the resulting system. To substantiate the solvability of the system, the properties of finite-dimensional mappings are used: decomposition of spaces into subspaces, projectors on subspaces, semi-inverse operators. But for the practical use of the proposed method, it is sufficient to solve the obtained linear algebraic system and use the obtained linear formulas. The correctness of the obtained model is investigated. Due to the non-uniqueness of the solution to the problem posed, the trajectory of motion can be unstable. It is revealed which components of the desired coefficients are arbitrary. It is showed which ones to choose, to make the movement steady, that is, so that small changes in the given multi-point values, as well as a small change parameters of the dynamic system corresponded to a small change in the trajectory of motion. An example is given of constructing trajectories of a material point in a vertical plane under the action of a reactive force in order to hit a given point with a given speed.</p></abstract>

Solution of Integral Equation in Two-Dimensional using Spectral Relationships

This paper concerned using spectral relationships in the solution of the integral equation (IE) in two-dimensional. To discuss that, the (IE) in two-dimensional under certain conditions was considered. The existence of at least one solution of the (IE) was discussed by proving the continuity and compactness of an integral operators. Chebyshev polynomials of the first kind were used to transform the (IE) to a linear algebraic system. Many numerical results and estimating errors were calculated and plotted by the Maple program in different cases.

Метод точного расчёта длины цилиндрической оболочки

Как известно, для решения конкретных задач расчета прочности и устойчивости оболочек вращения используется теория расчетов осесимметричных тел вращения произвольной формы, основанная на гипотезах Кирхгофа и предположениях об однородности и изотропности материалов изготовления. В общей теории тонких оболочек данная задача сводится к решению системы уравнений равновесия в частных производных восьмого порядка. Для цилиндрических оболочек ввиду принятых допущений система уравнений равновесия в перемещениях преобразуется в линейную алгебраическую систему. Из данной системы на основе дополнительных допущений получают простое уравнение, из которого и определяется величина критического давления устойчивости по заданной длине оболочки. Однако из основной системы уравнений возможно решение обратной задачи: по заданной величине критического давления определять точное значение длины цилиндрической оболочки. При этом задача имеет точное решение без каких либо дополнительных допущений и упрощений системы уравнений.As is known, to solve specific problems of calculating the strength and stability of shells of revolution, the theory of calculations of axisymmetric bodies of revolution of arbitrary shape is used, based on Kirchhoff hypotheses and assumptions about the homogeneity and isotropy of manufacturing materials. In the general theory of thin shells, this problem reduces to solving a system of eighth-order partial differential equilibrium equations. For cylindrical shells, in view of the accepted assumptions, the system of equations of equilibrium in displacements is transformed into a linear algebraic system. From this system, on the basis of additional assumptions, a simple equation is obtained, from which the critical pressure of stability for a given shell length is determined. However, it is possible to solve the inverse problem from the main system of equations: determine the exact value of the length of a cylindrical shell for a given critical pressure. Moreover, the problem has an exact solution without any additional assumptions and simplifications of the system of equations.

On a method for solving the inverse Sturm–Liouville problem

A method for solving the inverse Sturm–Liouville problem on a finite interval is proposed. It is based on a Fourier–Legendre series representation of the integral transmutation kernel. Substitution of the representation into the Gel’fand–Levitan equation leads to a linear algebraic system of equations and consequently to a simple algorithm for recovering the potential. Numerical illustrations are presented.

Perturbation Bounds and Condition Numbers for a Complex Indefinite Linear Algebraic System

We consider perturbation bounds and condition numbers for a complex indefinite linear algebraic system, which is of interest in science and engineering. Some existing results are improved, and illustrative numerical examples are provided.