Algebraic Equations Research Articles

An analytic method to investigate hemodynamics of the cardiovascular system: Biventricular system.

Previously, we found analytic solutions for single ventricular system based on the lumped parameter model (LPM). In this study, we generalized the method to biventricular system and derived its analytic solutions. LPM is just a set of differential equations, but it is difficult to solve due to time-varying ventricular elastance and high order. Mathematically, there exist no elementary solutions for time-varying equations. It turns out that instead of differential equations, according to volume conservation, a set of algebraic equations can be carried out. The solutions of the set of equations are just physiological states at end of systolic and diastolic phases such as end systolic/diastolic pressure/volume of left ventricle. As a preliminary application, the method is utilized to deduce the hemodynamic effects of VA ECMO. Left ventricular (LV) distension, a serious complication of VA ECMO, is usually attributed to factors such as increased afterload, inadequate LV unloading, reduced myocardial contractility or aortic valve regurgitation (AR), bronchial and Thebesian return in the absence of aortic valve (AoV) opening. Among these, reduced contractility and AR are strongly associated with LV distension. However, in the absence of reduced contractility or AR, it is less clear whether increased afterload or inadequate LV unloading alone can cause LV distension. This leads to the critical question: under what conditions does LV distension occur in the absence of reduced contractility or AR? The analytic formulas derived in this study give conditions for LV distension. Furthermore, the results show that the analytic hemodynamics are coincident with simulated results.

Four extremal solutions of discrete-time algebraic Riccati equations: existence theorems and computation

Four extremal solutions of discrete-time algebraic Riccati equations: existence theorems and computation

Modeling a poiseuille flow through a rectangular pipe in a floor slab (PSD)

This study was devoted to the modeling of a laminar fluid flow in a pipe of the rectangular section located in a floor slab. This theme was raised because of its relevance in the large business sector as well as in the areas of energy saving and storage at the residential scale. The equations of motion quantity and heat transport are the classical equations of forced convection. However, to describe the movement amount within the heating slab pipeline, Prandtl simplified model was most appropriate. In addition, this model retains the most important terms for a laminar regime flow. In addition, the modeling of fluid flow conservation in the pipeline was taken into account in the equation system. To achieve the above-mentioned objective, a numerical resolution was made for the system resolution of the Navier-Stokes equations that govern this fluid flow. These equations were discretized by an implicit finite difference method. The system of algebraic equations as well as obtained was solved by the algorithm of Gauss and the algorithm of Thomas. The results obtained such as: the evolution of the fluid velocity profile within the slab according to pipeline to the Reynolds number variation, the fluid temperature have clearly demonstrated the Poiseuille effect on fluid flow. Moreover, the stabilization of the Gauss algorithm and the Thomas algorithm in this numerical modeling was well verified against the tricks that were taken into consideration. The result of this study was that Thomas algorithm and Gauss algorithm are the best in this type of numerical modeling.

Symbolic computation in algebra, geometry, and differential equations

In this survey article we describe how symbolic computation in algebra and geometry leads to symbolic, i.e., formula solutions of algebraic differential equations. Symbolic solutions of algebraic differential equations can be derived from parametrizations of corresponding algebraic varieties. Such parametrizations in turn can be computed by elimination methods, i.e., methods for solving systems of polynomial equations.

The numerical solution of Fredholm integral equation of the first kind using the Sinc-Galerkin method

Sinc-Galerkin scheme is one of the new techniques used in solving numerical problems involving integral equations. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. In this article, a numerical solution of Fredholm integral equation of the first kind which arise in a large number of problems in applied mathematics will be discussed, for this result, we choose Sinc functions as basis functions and Galerkin method to estimate a solution for an unknown function in this equation. So, in this paper, some properties of the Sinc-Galerkin method required for our subsequent development are given and are utilized to reduce integral equation of the first kind to some linear algebraic equations. Then convergence with exponential rate is proved by a theorem to guarantee applicability of numerical technique. Finally, numerical examples are included to confirm applicability and justify rapid convergence of our method.

A numerical study for non-linear multi-term fractional order differential equations

In this article, a numerical method is presented for solving non-linear multi-order fractional differential equations (FDEs) using orthonormal Boubaker polynomials as basis functions. The operational matrix is constructed for the product of basis functions. The fractional derivative is considered in the Caputo sense. Using this technique, non-linear multi-order FDEs converted into a system of non-linear algebraic equations, which can then be solved numerically. The proposed method is implemented using MATLAB for numerical analysis. Using numerical results, validity, and accuracy of the proposed method is given and compared the results with existing literature

Numerical solution of different kinds of fractional‐order optimal control problems using generalized Lucas wavelets and the least squares method

AbstractGeneralized Lucas wavelets (GLWs) have two more parameters ( and ), comparing with some existing classical wavelet functions. In this manner, we have different types of wavelet functions (orthogonal and non‐orthogonal) by choosing various values of parameters and . Due to the impressive feature of the GLWs, we design a new computational method for the solution of fractional optimal control problems and fractional pantograph optimal control problems. This technique uses the GLWs and least squares method. The scheme includes expanding the required functions using GLW elements. We present new Riemann–Liouville and pantograph operational matrices for GLWs. Applying the operational matrices and least squares method, the considered problems lead to systems of algebraic equations, which can be solved numerically. A brief discussion of the error of the estimate used is investigated. Finally, some numerical experiments are exhibited to demonstrate the validity and applicability of the suggested scheme. The proposed algorithm is easy to implement and presents very accurate results.

Solution analysis of Solow Growth Model for financial practices and applications

Solow Growth Model (SGM) is an economic model that is exogenous in nature and examines the relationship between the output and input levels in an economy over a period of time. It projects long-term economic growth in relation to labour (population growth), savings rate, and technological development. However, traditional approaches to solving the Solow growth model may rely on complex mathematical techniques that might not give an accurate representation of real-world economic dynamics. Thus, this paper applies the Natural Decomposition Method (NDM) to the Solow growth financial model. The NDM is a numerical technique that combines the Natural Transform (NT) and the ADM-Adomian decomposition method. The NDM simplifies problem-solving by converting the original differential equations into algebraic equations as regards limitations associated with nonlinear models. From the results obtained by applying the NDM to the Solow growth financial model, researchers and policymakers can better understand the interplay between financial variables, such as savings rates, investment, and capital allocation, and their impact on economic growth dynamics, as a systematic approach to capturing the complex relationship between finance and economic development within the Solow framework is ensured. Further research and application of the NDM can contribute to advancing the knowledge of economic dynamics and support evidence-based decision-making in economic policy.

An optimal midcourse guidance method for dual pulse air-to-air missiles using linear Gauss pseudospectral model predictive control method

This paper proposes an optimal midcourse guidance method for dual pulse air-to-air missiles, which is based on the framework of the linear Gauss pseudospectral model predictive control method. Firstly, a multistage optimal control problem with unspecified terminal time is formulated. Secondly, the control and terminal time update formulas are derived analytically. In contrast to previous work, the derivation process fully considers the Hamiltonian function corresponding to the unspecified terminal time, which is coupled with control, state, and costate. On the assumption of small perturbation, a special algebraic equation is provided to represent the equivalent optimal condition for the terminal time. Also, using Gauss pseudospectral collocation, error propagation dynamical equations involving the first-order correction term of the terminal time are transformed into a set of algebraic equations. Furthermore, analytical modification formulas can be derived by associating those equations and optimal conditions to eliminate terminal error and approach nonlinear optimal control. Even with their mathematical complexity, these formulas produce more accurate control and terminal time corrections and remove reliance on task-related parameters. Finally, several numerical simulations, comparisons with typical methods, and Monte Carlo simulations have been done to verify its optimality, high convergence rate, great stability and robustness.

An approximate solution of the Blasius problem using spectral method

This paper aims at finding the numerical approximation of a classical Blasius flat plate problem using spectral collocation method. This technique is based on Chebyshev pseudospectral approach that involves the solution is approximated using Chebyshev polynomials, which are orthogonal polynomials defined on the interval [−1, 1]. The Chebyshev pseudospectral method employs Chebyshev- Gauss- Lobatto points, the extrema of the Chebyshev polynomials. The differential equation is approximated as a sum of Chebyshev polynomials. A differentiation matrix, based on these polynomials and their derivatives at the collocation points, transforms the differential equation into a system of algebraic equations. By evaluating the differential equation at these points and applying boundary conditions, the original boundary value problem reduced the solution to the solution of a system of algebraic equations. Solving for the coefficients of the polynomials yields the numerical approximation of the solution. The implementation of this method is carried out in Mathematica and its validity is ensured by comparing it with a built in MATLAB numerical routine called bvp4c.

Additional damping control for multi-terminal AC-DC hybrid system based on reachability analysis

Abstract The application proportion of multi-terminal AC-DC hybrid power distribution systems is constantly increasing. In order to describe the operation status and trajectory of the system and judge whether the system can maintain stable and safe operation, in this paper, the differential algebraic equation is firstly established by mathematical and physical methods for a typical multi-terminal AC-DC hybrid power distribution system. Then, the reachable set in the operation process is analyzed according to the reachability analysis calculation method to judge the stable operation status. An additional damping control method is proposed, which can effectively restrain the negative impedance characteristics of the DC micro-grid and improve the operation stability of the system. Finally, the improvement performance of the control method is verified by combining the reachability analysis results.

Stagnant rings and uniform film analysis of Phan-Thien Tanner fluid film flow on a vertically upward moving tube

Analyzing Phan-Thien Tanner fluid film behavior on a vertically upward moving tube can help predictive models in engineering, notably in coating and lubrication operations. This paper provides a theoretical analysis of the dynamics of stagnant rings and uniform Phan-Thien Tanner fluid film adhered to a vertically upward moving tube. Formulated ordinary differential equations are solved to get exact analytic expressions for velocity, flow rate, average velocity, shear stress components, and stagnant rings. Highly nonlinear algebraic equations are solved using Newton's method in MAPLE to find the linear and exponential Phan-Thien Tanner film thicknesses. The uniform film thickness widens with increasing constant tube velocity, while it decreases with increasing Deborah number, Stokes number, and elongation parameter. The analysis delineates that stagnant rings shrink around the tube as the Stokes number, Deborah number, and elongation parameter increase. The minimal Stokes condition for the existence of a realistic stagnant ring is also determined. It has been established that whenever the Stokes number is less than the minimal Stokes condition, stagnant rings do not form. A comparison between the exponential Phan-Thien Tanner, linear Phan-Thien Tanner, upper convected Maxwell, and Newtonian fluids is also provided for stagnant rings and fluid film thickness. Following the application of an efficient approximation on tube geometry, plate geometry is approximated, and the outcomes are consistent with existing literature. The results of this research are significant for a wide range of biofluid applications, including agrochemical uses, paint and surface coating flow behavior, thin films on the cornea and lungs, and chemical and nuclear reactor design.

Global well-posedness for two-phase fluid motion in the Oberbeck–Boussinesq approximation

This paper focuses on the global well-posedness of the Oberbeck–Boussinesq approximation for the unsteady motion of a drop in another bounded fluid separated by a closed interface with surface tension. We assume that the initial state of the drop is close to a ball BR with the same volume as the drop, and that the boundary of the drop is a small perturbation of the boundary of BR. To begin, we introduce the Hanzawa transformation with an added barycenter point to obtain the linearized Oberbeck–Boussinesq approximation in a fixed domain. From there, we establish time-weighted estimates of solutions for the shifted equation using maximal Lp–Lq regularities for the two-phase fluid motion of the linearized system, as obtained by Hao and Zhang [J. Differ. Equations 322, 101–134 (2022)]. Using time decay estimates of the semigroup, we then obtain decay time-weighted estimates of solutions for the linearized problem. Additionally, we prove that these estimates are less than the sum of the initial value and its own square and cube by estimating the corresponding non-linear terms. Finally, the existence and uniqueness of solutions in the finite time interval (0, T) was proven by Hao and Zhang [Commun. Pure Appl. Anal. 22(7), 2099–2131 (2023)]. After that, we demonstrate that the solutions can be extended beyond T by analyzing the properties of the roots of algebraic equations.

A New Numerical Simulation for Modified Camassa-Holm and Degasperis-Procesi Equations via Trigonometric Quintic B-spline

In this study, the soliton solutions of the modified Camassa-Holm (mCH) and Degasperis-Procesi (mDP) equations, called modified b-equations with important physical properties, were obtained. The soliton waves' movement and positions formed by solving the mCH and mDP equations were calculated. Ordinary differential equation systems were obtained using trigonometric quintic B-spline bases for position and time direction derivatives in the equations to obtain numerical solutions. An algebraic equation system was then created by writing Crank-Nicolson type approximations for time and position-dependent terms. The stability analysis of this system was examined using the von-Neumann Fourier series method. L₂, L_{∞} and absolute error norms were used to measure the numerical results' convergence to the real solution. The numerical results calculated were compared with the exact solution and some studies in the literature.

Relativistic Effects in Ligand Field Theory (I): Optical Properties of d1 Atoms in Oh' Symmetry.

Ligand field theory, which explains the splitting of degenerate nd atomic orbitals due to static electric fields from point-charge ligands, is rederived using Dirac orbitals instead of Schrödinger orbitals, specifically using the nd3/2 and nd5/2 spinors. This formalism is, to some extent, equivalent to incorporating the spin-orbit interaction either in the nd atomic orbitals or in the ligand field orbitals (e.g., the t2g and eg orbitals arising from Oh symmetry). The spin-orbit interaction is of fundamental importance in the description of the magnetic and optical properties of the 4d and 5d transition metal complexes. Algebraic equations for the relativistic energy levels of d1 octahedral complexes as functions of the spin-orbit coupling constant ξnd and the ligand field parameters Dq and Dp are derived. It is demonstrated that these parameters allow a direct link between the ligand field theory and ab initio relativistic calculations, consistent with the emerging ab initio ligand field theory. The spin-orbit coupling constant and ligand field parameters of ReF6 obtained from optical absorption spectra are carefuly in the light of the new theory.

Interaction of short-crested waves with cylinder and concentric segmented arc breakwaters

Using the outer breakwater of Palm Jumeirah in Dubai as a physical geometric model, the diffraction problem of short-crested waves interacting with concentric segmented arc structures was theoretically investigated using linear water wave theory. In this study, the watershed was partitioned into two regions, and a unique coordinate system was established to describe the velocity potential function within each region. The interface between the two regions connects the velocity potentials across them using the matching eigenfunction expansion technique. By accounting for the boundary conditions, the integral equation was streamlined into a series of algebraic equations, which were subsequently resolved to determine an unidentified function. The precision of the model was validated through a comparison with findings from the literature. Numerical investigations have elucidated the correlation between changes in wave force, cylinder run-up, and parameters such as the spacing between adjacent arcs, opening angle, and porous coefficient. The numerical results indicated that the impermeable segmented arcs structure provides better protection against short-crested and plane waves, whereas the permeable segmented arc structure provides better protection against standing waves. Interestingly, for a segmented three-arc structure, when the segments are moderately spaced, the protection of the interior region is superior to that of an unsegmented arc structure.

Supersymmetric Expansion Algorithm and Complete Analytical Solution for the Hulthén and Anharmonic Potentials

Abstract An algorithm for providing analytical solutions to Schrödinger’s equation with non-exactly solvable potentials is elaborated. It represents a symbiosis between the logarithmic expansion method and the techniques of the supersymmetric quantum mechanics as extended toward non shape invariant potentials. The complete solution to a given Hamiltonian H0 is obtained from the nodeless states of the Hamiltonian H0 and of a set of supersymmetric partners H1, H2, …, Hr. The nodeless states (dubbed “edge” states) are unique and in general can be ground or excited states. They are solved using the logarithmic expansion which yields an infinite system of coupled first order hierarchical differential equations, converted later into algebraic equations with recurrence relations which can be solved order by order. We formulate the aforementioned scheme, termed to as “Supersymmetric Expansion Algorithm” step by step and apply it to obtain for the first time the complete analytical solutions of the three dimensional Hulthén–, and the one-dimensional anharmonic oscillator potentials.

Diffraction of P1, SV, or Rayleigh waves by an arbitrary-shaped canyon in an unsaturated half-plane

The diffraction of P1, SV, or Rayleigh waves by an arbitrary-shaped canyon in an unsaturated half-plane is investigated in this paper. The scattered wave potentials with unknown coefficients are solved by Fredlund’s theory and the Helmholtz’s decomposition. The general expressions of displacements, stresses, pore pressures, and seepage are obtained using complex variable method. The half-surface and the canyon surface are mapped onto semi-circles in the image plane with Möbius transformation and Schwarz–Christoffel mapping. The boundary-valued problems yield an infinite series of linear algebraic equations with unknown coefficients. The truncating numbers are selected to assure the convergence and accuracy of the solution. A parametric study is performed to investigate the site responses of the canyons by incident P1 waves in an unsaturated half-plane. Numerical results show that the aspect ratio, saturation degree, circular frequency, and angle of incident wave significantly affect the site response of the canyon. The effects of saturation degree on the ground motions are significant owing to the complex tri-interaction of the three phases in the unsaturated medium. The dynamic concentrations occur at the corner region of the canyon. The shielding effect of the canyon to obliquely incident P1 waves is obvious.

Extinction and stationary distribution of a novel SIRS epidemic model with general incidence rate and Ornstein–Uhlenbeck process

We propose, in this paper, a novel stochastic SIRS epidemic model to characterize the effect of uncertainty on the distribution of infectious disease, where the general incidence rate and Ornstein–Uhlenbeck process are also introduced to describe the complexity of disease transmission. First, the existence and uniqueness of the global nonnegative solution of our model is obtained, which is the basis for the discussion of the dynamical behavior of the model. And then, we derive a sufficient condition for exponential extinction of infectious diseases. Furthermore, through constructing a Lyapunov function and using Fatou’s lemma, we obtain a sufficient criterion for the existence and ergodicity of a stationary distribution, which implies the persistence of the disease. In addition, the specific form of the density function of the model near the quasiendemic equilibrium is proposed by solving the corresponding Fokker–Planck equation and using some relevant algebraic equation theory. Finally, we explain the above theoretical results through some numerical simulations.

The Effects of Concrete and Virtual Manipulatives on Solving Algebraic Equations in Students With Disabilities

The Effects of Concrete and Virtual Manipulatives on Solving Algebraic Equations in Students With Disabilities