Terms Of Algebraic Equations Research Articles

Theoretical analysis of colonic crypt and colorectal cancer model through Caputo–Fabrizio fractional derivative

This study aims to analyze the solution of a system of differential equations that describes the mathematical modeling of cell population dynamics in colonic crypt and colorectal cancer. The Caputo–Fabrizio fractional order derivatives are used to fractionalize the model. The corresponding mathematical model is solved by the Laplace transform, which helps transform differential equations into terms of algebraic equations. The Partial fraction technique is used to find the inverse Laplace of the governing equations. To assess the credibility of the results, graphical simulation has been investigated by manipulating certain parameters. There is a special scenario, namely when [Formula: see text], where the solutions obtained align with those already documented in the literature. This alignment ensures that the initial conditions are met and confirms the accuracy of our solutions.

A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series Solution

In this paper, a new method based on an accurate analytical series solution for free vibration of triangular isotropic and orthotropic plates is presented. The proposed solution is expressed in terms of undetermined arbitrary coefficients, which are exactly satisfied by the governing differential equation in free vibration. The approach used is based on an innovative extension of the superposition method through the application of a modified system of trigonometric functions. The boundary conditions for bending displacements and bending rotations on the sides of the triangular plate led to an infinite system of linear algebraic equations in terms of the undetermined coefficients. Following this development, the paper then presents an algorithm to solve the boundary value problem for isotropic and orthotropic triangular plates for any kinematic boundary conditions. Of course, the boundary conditions with zero displacements and zero rotations on all sides correspond to the case when the plate is fully clamped all around. The convergence of the proposed method is examined by numerical simulation applying stringent accuracy requirements to fulfill the prescribed boundary conditions. Some of the computed numerical results are compared with published results and finally, the paper draws significant conclusions.

The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic Equations

This work proposes the Enhanced Fixed Point Method (EFPM) as a straightforward modification to the problem of finding an exact or approximate solution for a linear or nonlinear algebraic equation. The proposal consists of providing a versatile method that is easy to employ and systematic. Therefore, it is expected that this work contributes to breaking the paradigm that an effective modification for a known method has to be necessarily long and complicated. As a matter of fact, the method expresses an algebraic equation in terms of the same equation but multiplied for an adequate factor, which most of the times is just a simple numeric factor. The main idea is modifying the original equation, slightly changing it for others in such a way that both have the same solution. Next, the modified equation is expressed as a fixed point problem and the proposed parameters are employed to accelerate the convergence of the fixed point problem for the original equation. Since the Newton method results from a possible fixed point problem of an algebraic equation, we will see that it is relatively easy to get modified versions of the Newton method with orders of convergence major than two. We will see in this work the convenience of this procedure.

Anomalous Nonlinear Dynamics Behavior of Fractional Viscoelastic Beams.

Fractional models and their parameters are sensitive to intrinsic microstructural changes in anomalous materials. We investigate how such physics-informed models propagate the evolving anomalous rheology to the nonlinear dynamics of mechanical systems. In particular, we study the vibration of a fractional, geometrically nonlinear viscoelastic cantilever beam, under base excitation and free vibration, where the viscoelasticity is described by a distributed-order fractional model. We employ Hamilton's principle to obtain the equation of motion with the choice of specific material distribution functions that recover a fractional Kelvin-Voigt viscoelastic model of order α. Through spectral decomposition in space, the resulting time-fractional partial differential equation reduces to a nonlinear time-fractional ordinary differential equation, where the linear counterpart is numerically integrated through a direct L1-difference scheme. We further develop a semi-analytical scheme to solve the nonlinear system through a method of multiple scales, yielding a cubic algebraic equation in terms of the frequency. Our numerical results suggest a set of α-dependent anomalous dynamic qualities, such as far-from-equilibrium power-law decay rates, amplitude super-sensitivity at free vibration, and bifurcation in steady-state amplitude at primary resonance.

Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group H3

PurposeIn the Lorentz Heisenberg space H3 endowed with flat metric g3, a translation surface is parametrized by r(x, y) = γ1(x)*γ2(y), where γ1 and γ2 are two planar curves lying in planes, which are not orthogonal. In this article, we classify translation surfaces in H3, which satisfy some algebraic equations in terms of the coordinate functions and the Laplacian operator with respect to the first fundamental form of the surface.Design/methodology/approachIn this paper, we classify some type of space-like translation surfaces of H3 endowed with flat metric g3 under the conditionΔri = λiri. We will develop the system which describes surfaces of type finite in H3. For solve the system thus obtained, we will use the calculation variational. Finally, we will try to give performances geometric surfaces that meet the condition imposed.FindingsClassification of six types of translation surfaces of finite type in the three-dimensional Lorentz Heisenberg group H3.Originality/valueThe subject of this paper lies at the border of geometry differential and spectral analysis on manifolds. Historically, the first research on the study of sub-finite type varieties began around the 1970 by B.Y.Chen. The idea was to find a better estimate of the mean total curvature of a compact subvariety of a Euclidean space. In fact, the notion of finite type subvariety is a natural extension of the notion of a minimal subvariety or surface, a notion directly linked to the calculation of variations. The goal of this work is the classification of surfaces in H3, in other words the surfaces which satisfy the condition/Delta (ri) = /Lambda (ri), such that the Laplacian is associated with the first, fundamental form.

Exact and approximate analytical solutions for nonlocal nanoplates of arbitrary shapes in bending using the line element-less method

In this study, an innovative procedure is presented for the analysis of the static behavior of plates at the micro and nano scale, with arbitrary shape and various boundary conditions. In this regard, the well-known Eringen’s nonlocal elasticity theory is used to appropriately model small length scale effects. The proposed mesh-free procedure, namely the Line Element-Less Method (LEM), only requires the evaluation of simple line integrals along the plate boundary parametric equation. Further, variations of appropriately introduced functionals eventually lead to a linear system of algebraic equations in terms of the expansion coefficients of the deflection function. Notably, the proposed procedure yields approximate analytical solutions for general shapes and boundary conditions, and even exact solutions for some plate geometries. In addition, several applications are discussed to show the simplicity and applicability of the procedure, and comparison with pertinent data in the literature assesses the accuracy of the proposed approach.

Application of Gyarmati’s principle to study active boundary layer control of ionic fluid past a Riga plate

Similarity solutions of laminar electromagnetohydrodynamic boundary layer flows in forced convection are investigated aiming at the evaluation of irreversible momentum, heat and mass transport balance equations by a variational principle. The transportation fields inside the boundary layer are approximated as trial polynomial functions, and the functionals of variational principle are formulated. The Euler–Lagrange equations of variational principle are constructed as algebraic equations in terms of momentum, thermal and concentration boundary layer thicknesses. Electrically conductive fluid flow over a semi-infinite flat plate controlled by electromagnetic force (Z) is considered. The effects of viscous dissipation (Ec), thermophoretic particle deposition (Nt) and thermal diffusion (Sr) in the presence of heat source/sink (Q) on skin friction, heat and mass transfer rates have been examined in this work. To demonstrate the efficiency of the present technique, some specific obtained results are compared with the known numerical results in literature and the accuracy is ensured. Further, the regression analysis is performed to state the dependency of these parameters on heat and mass transfer rates.

Convergence analysis on time scales for HOBAM neural networks in the Stepanov-like weighted pseudo almost automorphic space

We start by presenting the concept of Stepanov-like weighted pseudo almost automorphic on time-space scales. Besides, we introduce a novel model of high-order BAM neural networks with mixed delays. To the best of our knowledge, this is the first time to study the convergence analysis for one system modeling the recurrent neural networks. Since it is a $$\Delta $$ -dynamic system on time-space scales, the results obtained in our work are new and attractive. The main difficulty in our theoretical work is the construction of Lyapunov–Krasovskii functional and the application of the Banach’s fixed-point theorem. By fabricating an appropriate Lyapunov–Krasovskii Functional (LKF), some new sufficient conditions are obtained in terms of linear algebraic equations to guarantee the convergence to the Stepanov-like WPAA on time-space scales solution for the labeled neural networks solutions. The obtained conditions are expressed in terms of algebraic equations whose feasibility can be checked easily by a simple calculus. Furthermore, we have collated our effort with foregoing one in the available literatures and showed that it is less conserved. Finally, two numerical examples show the feasibility of our theoretical outcomes.

A Dynamic Model with Friction for Comprehensive Tubular-Stress Analysis

Summary A new model technique is described for comprehensive dynamic stress and displacement analysis of wellbore-completion tubulars, including friction loads with history. A dynamic model of tubing forces is necessary to predict local pipe velocity, which in turn determines the magnitude and direction of localized frictional contact. By tracking dynamic changes in axial force starting from the initial running state, a complete load history can be simulated for the installed casing and tubing through the service life of the well. The dynamic friction model subdivides the casing or tubing string joint by joint and uses an elastic pipe-momentum balance. Pipe velocity is related to axial force by the elasticity equation. Dynamically determined velocity is necessary to predict the magnitude and orientation of local node-friction vectors. Damping for the dynamic analysis is provided by annular fluid viscosity. The elastic equations are solved as a set of algebraic equations in terms of past and future values of pipe axial force and velocity. Key model inputs such as pressure, temperature, fluid, and wellbore-friction coefficients can be changed at each successive load step. Running loads and packer setting with slackoff or pickup loads determine the initial tubing-stress configuration. Given the initial configuration, each subsequent load case is calculated starting from the prior load and resultant friction state, allowing for full history dependence. The surface velocity profile of running individual stands is a key input. Unexpected magnitudes of downhole transfer of surface load are demonstrated. A change in the operation-load sequence is shown to produce significant differences in tubular axial loads, indicating that special attention to load history should be considered when performing a tubular-stress analysis. For slackoff, overpull, or packer-setting events, the model can track dynamic load response at downhole points, such as a packer or cement top. An example well with a deviated profile and a planned sequence of life-cycle operations including stimulation, production, and shut-in was simulated for a variety of load sequences. The model has been validated against field data using the actual hookload plot during installation of a single-trip, multizone intelligent completion in an offshore highly deviated extended-reach-drilling (ERD) well. Example calculations are given for a high-pressure/high-temperature (HP/HT) subsea well and a horizontal unconventional well. The dynamic friction model allows for the seamless integration of running loads with friction into a fully sequential stress analysis of subsequent well life-cycle loads for landed completion strings. Although dynamic analysis has been extensively applied to complex drilling phenomena such as drillstring vibration or bottomhole-assembly design, current industry models for completion tubulars such as casing and tubing separate the installation state from the in-service life envelope or attempt to solve the problem with a static analysis. This represents a critical deficiency in the current industry state of the art for completion tubulars, which the present work proposed herein strives to address. From a comparison with appropriate static analytic solutions and industry-standard drag-and stress-models, dynamics were found to affect friction-force directions and magnitudes, suggesting that tubular dynamics cannot be neglected.

Hybrid functions for numerical solution of fractional Fredholm‐Volterra functional integro‐differential equations with proportional delays

AbstractIn this work, we propose an efficient and accurate computational approach based on the Genocchi hybrid functions (GHFs) for solving a class of fractional Fredholm‐Volterra functional integro‐differential equations (FVFIDEs) with proportional delays. First, the Genocchi hybrid functions are constructed. Then, the operational matrix of the Caputo fractional derivative with some properties of the GHFs is employed to reduce the fractional FVFIDEs with proportional delays to systems of algebraic equations in terms of the unknown coefficients. Ultimately, the upper bound of error and convergence of approximate solution to exact solution are discussed. Moreover, we consider some examples to demonstrate the validity and applicability of our method.

Bistability criterion for electrostatically actuated initially curved micro plates

The criterion defining the geometric parameters guaranteeing bistable behavior of electrostatically actuated curved axisymmetric circular plate is established. The usage of Berger’s approximation for von-Kármán nonlinear plates, combined with single degree of freedom (DOF) reduced order (RO) modeling, allowed derivation of a simple semi-analytical bistability criterion, obtained in the form of an implicit algebraic equation in terms of critical deflection and plate geometric parameters. The criterion is verified by direct numerical solutions, combined with the arc-length method. Case studies are presented, illustrating the implementation of the suggested criterion as a useful tool for the early design stage for MEMS/NEMS devices.

Vibration of antisymmetric angle-ply composite annular plates of variable thickness

Based on first order shear deformation theory, free vibrational behaviour of antisymmetric angle-ply composite annular plate is investigated. The thickness variations are assumed to be linear, exponential and sinusoidal along the radial direction. The equilibrium equations are obtained from which the coupled differential equations are derived in terms of displacement and rotation functions. These functions are approximated using splines to obtain the system of algebraic equations in terms of spline coefficients where the equation becomes a generalized eigenvalue problem by combining the boundary conditions. This problem is solved for finding eigenfrequencies and corresponding eigenvectors. This study is conducted for clamped-clamped and clamped-free boundary conditions. Detailed parametric investigation is carried out to determine the influences of radii ratio, number of layers, ply orientations and materials on the frequencies of plates. The primary results of this study are compared with those available in the literatures.

Helicoidal Surfaces in the three dimensional simply isotropic space I₃¹

In this paper, we classify helicoidal surfaces in the three dimensional simply isotropic space I₃¹ satisfying some algebraic equations in terms of the coordinate functions and the Laplacian operators with respect to the first, the second and the third fundamental form of the surface. We also give explicit forms of these surfaces.

TRANSLATION SURFACES OF TYPE 2 IN THE THREE DIMENSIONAL SIMPLY ISOTROPIC SPACE I13

In this paper, we classify translation surfaces of Type 2 in the three dimensional simply isotropic space <TEX>${\mathbb{I}}_3^1$</TEX> satisfying some algebraic equations in terms of the coordinate functions and the Laplacian operators with respect to the first, the second and the third fundamental form of the surface. We also give explicit forms of these surfaces.

A new numerical scheme for solving system of Volterra integro-differential equation

In this article, we apply Genocchi polynomials to solve numerically a system of Volterra integro-differential equations. This is done by approximating functions using Genocchi polynomials and derivatives using Genocchi polynomials operational matrix of integer order derivative. Combining approximation with collocation method, the problem is reduced to a system of algebraic equations in terms of Genocchi coefficients of the unknown functions. By solving the Genocchi coefficients, we obtain good approximate functions of the exact solutions of the system. A few numerical examples show that our proposed Genocchi polynomials method achieves better accuracy compared to some other existing methods.

Forced Vibration of Surface Foundation on Viscoelastic Isotropic Multi-Layered Stratum

A numerical approach is presented for analyzing the forced vibration of a rigid surface foundation. In the analysis, the foundation is discretized into a number of sub square-elements. The dynamic response within each sub-element is described by the Green’s function, which is obtained by the Fourier–Bessel transform and the precise integration method (PIM). Then, a system of linear algebraic equation in terms of the contact forces within each sub-element is derived, which leads to the desired dynamic impedance functions of the foundation. Numerical results are obtained for the foundation not only with a simple geometry, such as circular one, but also with irregular shapes. Comparisons between the results obtained by the proposed approach and the thin layered method are made, for which good agreement is achieved. Also, parametric studies are performed on the dynamic response of the foundation, considering the effects of the material damping, stratum depth, Poisson’s ratio and the contact condition of the soil–foundation interface. Several conclusions are drawn concerning the significance of each parameter. Further application of the method can be easily extended to the analysis of a foundation on a viscoelastic anisotropic multi-layered stratum because no further complexity is introduced except the constitutive matrix needs to be modified.

Dynamic Interaction Between Rigid Surface Foundations on Multi-Layered Half Space

A numerical approach is presented to calculate the dynamic response of a group of rigid surface foundations. The formulation is unconditionally stable and has the computational simplicity with only the algebraic calculations involved. It imposes no limit to the foundation shape, foundation separations, thickness of the layered medium and magnitude of frequency. In the analysis, the foundation–ground interface is discretized into a number of sub square-regions. The Green’s function, which is obtained by the Fourier–Bessel transform and precise integration method, is employed to calculate the dynamic response of each sub-region. Finally, a system of linear algebraic equation in terms of the contact forces within each sub-region is observed, which leads to the desired dynamic impedance functions of the foundations. Comparison is carried out between the proposed method and the solutions available in the literature. Parametric studies on the dynamic interaction between adjacent foundations are also described. Addressed in this study are the effects of the distance and direction of foundation alignment. Several conclusions are drawn the significance of each factor. Illustrative results for a case of several closely-spaced foundations are also presented. Although the dynamic interaction analysis of foundations is concerned here, further applications of this approach can be extended to the interaction analysis of structures.

The Application of the Laplace Transform for Modeling of Gas Flow Using Maple¯

AbstractThe Laplace transform is powerful method for solving differential equations. This paper presents the application of Laplace transform to solve the mathematical model of gas flow through the measuring system. The basis of the mathematical model used to describe and simulate the analyzed process is a system of ordinary differential equations. As a result is obtained a single algebraic equation in terms of the complex variable s. In order to study the dynamics of the system these equation should be reverted to the time domain by performing an inverse Laplace transform. Calculations were performed in the environment of Maple?. Determining of mixing in a u-shape vessel is aim of presented calculations.

The Discreet Charm of Abstraction: Hyperspace Worlds and Victorian Geometry

Analytic or Cartesian-coordinate geometry, which describes space in terms of algebraic equations, was conceived by most Victorians as a translation of the “real-world” truths of Euclidean geometry. However, other commentators were concerned that once Euclid’s axioms were translated into a purely symbolic system, there would be no guarantee that that system corresponded to reality. The epistemological crisis intensified with mathematical advances in the later decades of the century, as algebraic equations began to yield problems that seemed to require additional spatial dimensions for their resolution. Charles Howard Hinton, an important popularizer of hyperspace philosophy, argued that imperceptible higher-dimensional worlds must actually exist, since they are conceivable mathematically. The algebraic system is no longer merely meant to represent an a priori truth (of Euclidean space); it also reveals something about that space that is not accessible through perceptual means. This paper examines the anxiety attendant upon the threat of an abstract hyperreal in the work of three imaginative hyperspace writers: Gustav Theodor Fechner, Charles Hinton, and H. G. Wells.

Influence of interfaces on effective properties of nanomaterials with stochastically distributed spherical inclusions

The method of conditional moments is generalized to include evaluation of the effective elastic properties of particulate nanomaterials and to investigate the size effect in those materials. Determining the effective constants necessitates finding a stochastically averaged solution to the fundamental equations of linear elasticity coupled with surface/interface conditions (Gurtin–Murdoch model). To obtain such a solution the system of governing stochastic differential equations is first transformed to an equivalent system of stochastic integral equations. Using statistical averaging, the boundary-value problem is then converted to an infinite system of linear algebraic equations. A two-point approximation is considered and the stress fluctuations within the inclusions are neglected in order to obtain a finite system of algebraic equations in terms of component-average strains. Closed-form expressions are derived for the effective moduli of a composite consisting of a matrix and randomly distributed spherical inhomogeneities. As a numerical example a nanoporous material is investigated assuming a model in which the interface effects influence only the bulk modulus of the material. In that model the resulting shear modulus is the same as for the material without surface effects. Dependence of the bulk moduli on the radius of nanopores and on the pore volume fraction is analyzed. The results are compared to, and discussed in the context of other theoretical predictions.