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In this research article, two standard models of liquid chromatograophy, namely the dispersive equilibrium model (DEM) and the kinetic lumped model (KLM) are approximated numerically. We studied the transport of multi components in a single column of chromatography considering non linear adsorption thermodynamics. The models are analyzed for standard Bi-Langmuir and generalized Bi-Langmuir types adsorption equilibrium isotherms using Danckwert (Robin) boundary conditions. Mathematically, the model equations form a non linear system of partial differential equations accounting for the phenomena of advection and diffusion, paired with an algebraic equation or a differential equation for adsorption isotherm. An extended semi-discrete high-resolution finite volume scheme is employed to obtain the approximate solutions of the governing model equations. The method has second to third order accuracy. Several test case studies are conducted to examine the influence of various critical parameters on the process performance. The contemplated case studies incorporate the elution process of liquid chromatography with an increasing number of components. In particular, single component, two component and three component mixtures are considered for the assessment of process performance. The formulated numerical algorithm provide an efficacious mechanism for investigating the retention behavior and the influence of mass transfer kinetics on the shapes of elution profiles.

In this research, Genocchi wavelets method, a quite new type of wavelet-like basis, is adopted to obtain a numerical solution for the classical and time-fractional Rosenau-Hyman or K(n, n) equation arising in the formation of patterns in liquid drops. The considered partial differential equation can be transformed into a system of non-linear algebraic equations by utilizing the wavelets method including an integral operational matrix and then discretizing the equation at the collocation points. The system can be simply solved by several traditional methods. Finally, the algorithm is implemented for some numerical examples and the numerical solutions are compared with the exact solutions using MAPLE. The obtained results are demonstrated using figures and tables. When the results are compared, it is evinced that the algorithm is quite effective and advantageous due to its easily computable algorithm, high accuracy, and less process time. (C) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.

This article is devoted to the study of discrete Klein-Gordon type equations that describe the dynamics of an infinite chain of linearly coupled nonlinear oscillators. These equations represent a countable system of ordinary differential equations. Such systems are infinite-dimensional Hamiltonian systems. Equations of the Klein-Gordon type with power nonlinearities of odd degree are considered. When substituting the ansatz in the form of a standing wave, a system of algebraic equations for the amplitude of the standing wave is obtained. Further, we consider a system with a more general operator L of linear interaction of oscillators, which is bounded and self-adjoint in the Hilbert space of real two-sided sequences l2. The problem of the existence of periodic and localized (converging to zero at infinity) solutions for such systems is considered. The main conditions for the existence of these solutions are the spatial periodicity of the coefficients of the linear interaction operator of the oscillators and the belonging of the standing wave frequency to the spectral interval of the operator L. If the right end of the spectral interval is finite, then the system has nontrivial solutions. This article shows that periodic and localized solutions of this system can be constructed as critical points of the corresponding functionals Jk and J. The existence of periodic solutions was established using the linking theorem. In particular, it is shown that the functional Jk satisfies the so-called Palais-Smale condition and the linking geometry, and therefore has nontrivial critical points, which are the periodic solutions of the system. In the case of localized solutions, the linking theorem cannot be used, since the Palais-Smale condition does not hold for the functional J. Therefore, in this case, the method of periodic approximations is used, that is, the critical points of the functional J are constructed using the passage to the limit as k→∞ at the critical points of the functional Jk. By virtue of the well-known properties of the discrete Laplace operator, a corollary is obtained in which conditions for the existence of localized solutions for the original system are established.

Nonlinear dynamics perspective is an interesting approach to describe COVID-19 epidemics, providing information to support strategic decisions. This paper proposes a dynamical map to describe COVID-19 epidemics based on the classical susceptible-exposed-infected-recovered (SEIR) differential model, incorporating vaccinated population. On this basis, the novel map represents COVID-19 discrete-time dynamics by adopting three populations: infected, cumulative infected and vaccinated. The map promotes a dynamical description based on algebraic equations with a reduced number of variables and, due to its simplicity, it is easier to perform parameter adjustments. In addition, the map description allows analytical calculations of useful information to evaluate the epidemic scenario, being important to support strategic decisions. In this regard, it should be pointed out the estimation of the number death cases, infectious rate and the herd immunization point. Numerical simulations show the model capability to describe COVID-19 dynamics, capturing the main features of the epidemic evolution. Reported data from Germany, Italy and Brazil are of concern showing the map ability to describe different scenario patterns that include multi-wave and plateaus behaviors. The effect of vaccination is analyzed considering different campaign strategies, showing its importance to control the epidemics.

This work adopts to the time-fractional Klein–Gordon equation (FKGE) in the Caputo sense. We present a new technique using the clique polynomial as basis function for the operational matrices to obtain solution of time-FKGE. The key advantage of this technique is converting the time-FKGE to algebraic equations, which can be simply solved the problem under study. For the approximation of a bivariate function using the clique polynomial, an error bound is given. Numerical results derived using the proposed technique are compared with the exact solution. The results show that the proposed technique is very user friendly for solving the time-FKGE and accurate.

Although all advantages of a standard approach to teaching students new skills, we are increasingly faced with problems such as the inability to pay an equal amount of attention to many students, to work through and unambiguously highlight all possible problems and mistakes, to close knowledge gaps. Also, all these difficulties are becoming even more urgent given the current state of affairs in the world and the global transition to an online learning format. As a possible solution to the problem, one can consider the creation of independent intelligent systems capable of taking on a part of the load of teachers and automatically participating in the process of teaching students. The subject of research in this article is the process of analyzing the steps for solving algebraic equations using the Lobachevsky-Graeffe-Dandelen method. The goal is to model the process of solving algebraic equations and to identify all possible steps, difficulties and problems in solving such problems. Objective: development of a system capable of monitoring the execution of all necessary steps for a given solution, identifying and classifying possible student mistakes in the process of mastering the skill and work them out. In the process of the task, the following results were obtained: one possible solution for learning to solve an n-degree algebraic equation using the Lobachevsky-Greffe-Dandelen method has been described. On the basis of the signal-parametric approach to diagnostics of faults in dynamic systems the mathematical diagnostic models are created which allow detecting classes of errors by comparing the results of Student's calculations and the results of system calculations. The features and possible difficulties of application of the proposed diagnostic models are presented. An intelligent self-contained tutor system was developed and integrated into the work at practical classes on "Theory of Automatic Control" by 3rd year students of the National Aerospace University.

This article describes an algorithm for obtaining a non-negative basic solution of a system of linear algebraic equations. This problem, which undoubtedly has an independent interest, in particular, is the most time-consuming part of the famous simplex method for solving linear programming problems.Unlike the artificial basis Orden’s method used in the classical simplex method, the proposed algorithm does not attract artificial variables and economically consumes computational resources.The algorithm consists of two stages, each of which is based on Gaussian exceptions. The first stage coincides with the main part of the Gaussian complete exclusion method, in which the matrix of the system is reduced to the form with an identity submatrix. The second stage is an iterative cycle, at each of the iterations of which, according to some rules, a resolving element is selected, and then a Gaussian elimination step is performed, preserving the matrix structure obtained at the first stage. The cycle ends either when the absence of non-negative solutions is established, or when one of them is found.Two rules for choosing a resolving element are given. The more primitive of them allows for ambiguity of choice and does not exclude looping (but in very rare cases). Use of the second rule ensures that there is no looping.

This paper proposes two numerical approaches for solving the coupled nonlinear time-fractional Burgers’ equations with initial or boundary conditions on the interval $[0, L]$ [ 0 , L ] . The first method is the non-polynomial B-spline method based on L1-approximation and the finite difference approximations for spatial derivatives. The method has been shown to be unconditionally stable by using the Von-Neumann technique. The second method is the shifted Jacobi spectral collocation method based on an operational matrix of fractional derivatives. The proposed algorithms’ main feature is that when solving the original problem it is converted into a nonlinear system of algebraic equations. The efficiency of these methods is demonstrated by applying several examples in time-fractional coupled Burgers equations. The error norms and figures show the effectiveness and reasonable accuracy of the proposed methods.

The purpose of this present work is to investigate the geometrical non-linearity in free and forced vibration of a shallow arch elastically restrained at the ends. The non-linear governing equilibrium equation of the shallow arch is obtained after the Euler Bernoulli theory and the Von Karman geometrical non-linearity assumptions. After applying the ends conditions, the eigenvalues problem of the generalized trancendant equation have been determined iteratively using the Newton-Raphson algorithm. The kinetic and total strain energy have been discretized into a series of a finite spatial functions which are a combination of linear modes and basic function contribution coefficients. Using Hamilton’s principle energy and spectral analysis, the problem is reduced into a set of non-linear algebraic equations that solved numerically using an approximate explicit method developed previously the so-called second formulation. Considering a multimode approach, the effect of initial rise and concentrated force on non-linear behaviour of system has been illustrated in the backbone curves giving the non-linear amplitude-frequency dependence. The corresponding non-linear deflections and curvatures have been plotted for various vibration amplitudes.

In this paper, a new type of wavelet method to solve fractional differential equations (linear or nonlinear) is proposed. The proposed method is based on the generalized Gegenbauer–Humbert polynomial. First, we derived the operational matrices for integer and fractional order derivatives. Then, using these operational matrices with the proposed method, we transformed the given problem into a system of algebraic equations. Then, some linear and nonlinear examples were considered and discussed to confirm the efficiency and accuracy of the proposed method. 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/ 4.0/).

This paper presents a new computational method for solving Abel integral equation (both first kind and second kind). The numerical scheme is based on approximations in Daubechies wavelet basis. The properties of Daubechies scale functions are employed to reduce an integral equation to the solution of a system of algebraic equations. The error analysis associated with the method is given. The method is illustrated with some examples and the present method works nicely for low resolution.

This study proposed an efficient numerical technique for coupled differential equations (CDEs) using the clique polynomials of the Complete graph. Recently, Graph theory has dragged the attention of many mathematicians due to its wide applications. Here, some problems have been considered to examine the proposed scheme proficiency. Some theorems on convergence are discussed. Here, the CDEs are rehabilitated into an algebraic equation system using the clique polynomials and collocation technique. The proposed scheme results are compared with the undetermined coefficients (UDCM) and variation of parameters method (VPM) solutions through tables and graphs. Obtained results reveal that the current approach is more accurate.

This paper contributes to develop a highly accurate numerical method for solving two-dimensional mass transfer equations during convective air drying of apple slices. The numerical scheme is based on the interpolating the solution of the mentioned equations over the roots of the orthogonal Jacobi polynomials (i.e., the Jacobi-Gauss-Lobatto points) in a nodal form. Moreover, for speeding up the procedure of numerical technique, operational matrices of differentiation are applied to discretize the derivatives of both spatial and temporal variables. After implementing the proposed technique, two-dimensional mass transfer equations would be transferred into the associated systems of linear algebraic equations which can be solved by appropriate iterative solvers such as robust Krylov subspace iterative methods. Some constructed artificial examples are provided to show the efficiency and applicability of the Jacobi pseudo-spectral method for solving two-dimensional mass transfer equations. Finally, a real example is considered and numerical results are validated by the experimental data which confirm the accuracy of the presented numerical approach.

Time-dependent viscous fluid flow due to a stretchable rotating disk is investigated. Magnetic field is applied in vertical direction to the disk. Temperature equation is assisted with Joule heating effect. Governing system of PDE's is transformed to dimensionless form by suitable variables. One of the numerical techniques known as finite difference scheme is adopted to tackle the given dimensionless partial differential system. This method results in a system of simple algebraic equations. The unknown function is analyzed inside domain of interest. In this technique of solution, a system is subdivided into many smaller parts called finite elements. The obtained simpler algebraic equations are then assembled to form a system of equations which governs the original problem. Variational method is used to get approximate solution by reducing the error function. Behaviors of pertinent variables on surface drag force, temperature, velocity and heat transfer rate are shown graphically. The obtained outcomes guarantee that velocity decreases for Hartmann number while it enhances with Reynolds number. Temperature increases for higher Prandtl, Eckert and Hartmann numbers. Skin friction boosts for larger values of Hartmann number. Nusselt number enhances with Hartmann number.

This article presents a novel numerical method for seeking the numerical solutions of fractional order differential equations using hybrid functions consisting of block-pulse functions and Taylor polynomials. The fractional integrals operational matrix of the hybrid function is conducted through projecting the hybrid functions onto block-pulse functions. Then, the fractional order differential equations are converted to a set of algebraic equations via the derived operational matrix. Then, the numerical solutions are obtained via solving the algebraic equations. Moreover, we perform error analysis of the algorithm and gives the upper bound of absolute error. Finally, numerical examples are given to show the effectiveness of the proposed method.

In this work, a technique for finding approximate solutions for ordinary fraction differential equations (OFDEs) of any order has been proposed. The method is a hybrid between Galerkin and collocation methods. Also, this method can be extended to approximate fractional integro-differential equations (FIDEs) and fractional optimal control problems (FOCPs). The spatial approximations with their derivatives are based on shifted ultraspherical polynomials (SUPs). Modified Galerkin spectral method has been used to create direct approximate solutions of linear/nonlinear ordinary fractional differential equations, a system of ordinary fraction differential equations, fractional integro-differential equations, or fractional optimal control problems. The aim is to transform those problems into a system of algebraic equations. That system will be efficiently solved by any solver. Three spaces of collocation nodes have been used through that transformation. Finally, numerical examples show the accuracy and efficiency of the investigated method.

We propose a polynomial-based numerical scheme for solving some important nonlinear partial differential equations (PDEs). In the proposed technique, the temporal part is discretized by finite difference method together withθ-weighted scheme. Then, for the approximation of spatial part of unknown function and its spatial derivatives, we use a mixed approach based on Lucas and Fibonacci polynomials. With the help of these approximations, we transform the nonlinear partial differential equation to a system of algebraic equations, which can be easily handled. We test the performance of the method on the generalized Burgers–Huxley and Burgers–Fisher equations, and one- and two-dimensional coupled Burgers equations. To compare the efficiency and accuracy of the proposed scheme, we computed$L_{\infty }$L∞,$L_{2}$L2, and root mean square (RMS) error norms. Computations validate that the proposed method produces better results than other numerical methods. We also discussed and confirmed the stability of the technique.

The convection-diffusion process in porous electrodes depends greatly upon the porous structure. A fractal model for porous catalyst in a thin-zone bed reactor is established using He?s fractal derivative, and a variational principle is also established in a fractal space, and an approximate solution is obtained. Additionally an ancient Chinese algorithm is adopted to solve an algebraic equation.

This paper proposes a unified procedure for evaluating the resonant frequencies and the corresponding resonant modes for an arbitrary source region, based on the general expression for the difference between the stored magnetic and electric field energies in terms of the source current distribution. It is shown that the resonant frequencies and the corresponding resonant modes are governed by a homogeneous integral equation. By following a standard procedure of the method of moments (MoM), the integral equation can be discretized into a real homogenous algebraic equation, from which the resonant frequencies and the corresponding resonant modes can be determined by requiring that a non-trivial solution exists for the real homogeneous algebraic equation. Different from other modal theories that expand the fields by a linear combination of fundamental field patterns that are derived from the boundary conditions, the proposed theory of resonant modes is derived from the difference of stored field energies for an arbitrary source region. As an application of the theory of the resonant modes, a crossed-dipole antenna, a dual-band bowtie antenna, and a dual-band circular polarization antenna are designed through the appropriate excitations of the resonant modes on the three selected source regions, and they are validated by independent simulations and experiments.

This article introduces a category of variable-order fractional optimization problems subject to a partial differential equation. The Taylor’s series expansion method is developed as a suitable high-precision method for solving such problems. The proposed approach through the consecutive formulation describes how an optimization problem can be transformed into a system of algebraic equations using the Taylor’s series expansion. This achievement is significantly advantageous since the obtained algebraic system straightforwardly can be solved. The accuracy of the presented method is analyzed by solving a number of numerical examples.

In this article, a fractional-order mathematical physics model, advection–dispersion equation (FADE), will be solved numerically through a new approximative technique. Shifted Vieta–Lucas orthogonal polynomials will be considered as the main base for the desired numerical solution. These polynomials are used for transforming the FADE into an ordinary differential equations system (ODES). The nonstandard finite difference method coincidence with the spectral collocation method will be used for converting the ODES into an equivalence system of algebraic equations that can be solved numerically. The Caputo fractional derivative will be used. Moreover, the error analysis and the upper bound of the derived formula error will be investigated. Lastly, the accuracy and efficiency of the proposed method will be demonstrated through some numerical applications.

In this study, an efficient numerical scheme based on shifted Chebyshev polynomials is established to obtain numerical solutions of the Bagley–Torvik equation. We first derive the shifted Chebyshev operational matrix of fractional derivative. Then, by the use of these operational matrices, we reduce the corresponding fractional order differential equation to a system of algebraic equations, which can be solved numerically by Newton’s method. Furthermore, the maximum absolute error is obtained through error analysis. Finally, numerical examples are presented to validate our theoretical analysis.

In this paper, numerical solution of the linear second kind Fredholm integral equations is studied. These integral equations are widely used for solving many problems in mathematical physics and engineering. A new hybrid Bernstein and Improved Block-Pulse Functions ( HBIBP ) method is introduced and utilized to convert linear (nonlinear) second kind Fredholm integral equations into an algebraic equation. The new methodology is a combination of Bernstein polynomials (BPs) and improved block-pulse functions (IBPFs) on the interval [0, 1). Convergence analysis and numerical examples that illustrate the pertinent features of the method are presented.

In this paper, we employ some algebraic equations due to Hardy and Littlewood to establish some conditions on weights in dynamic inequalities of Hardy and Copson type. For illustrations, we derive some dynamic inequalities of Wirtinger, Copson and Hardy types and formulate the classical integral and discrete inequalities with sharp constants as particular cases. The results improve some results obtained in the literature.

In this study, a numerical technique is applied to investigate solutions of linear variable order differential equations(VODEs) in fluid mechanics. We investigate variable order(VO) in the Caputo sense. Our aim is to apply Bernstein polynomials(Bps) as the basis functions and operational matrices. We calculate two types of operational matrices of Bps. Using these operational matrices the main equation can be transformed into a system of algebraic equations, which must then be solved to get approximate solutions. Some examples are provided to indicate the accuracy of the presented method.

This paper is concerned with the Lane–Emden boundary value problems arising in many real-life problems. Here, we discuss two numerical schemes based on Jacobi and Bernoulli wavelets for the solution of the governing equation of electrohydrodynamic flow in a circular cylindrical conduit, nonlinear heat conduction model in the human head, and non-isothermal reaction–diffusion model equations in a spherical catalyst and a spherical biocatalyst. These methods convert each problem into a system of nonlinear algebraic equations, and on solving them by Newton’s method, we get the approximate analytical solution. We also provide the error bounds of our schemes. Furthermore, we also compare our results with the results in the literature. Numerical experiments show the accuracy and reliability of the proposed methods.

This is the Accepted Manuscript version of an article accepted for publication in Nonlinearity. IOP Publishing Ltd are not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1361-6544/ab9498., We construct a nonlinear lattice that has a particular symmetry in its potential function consisting of long-range pairwise interactions. The symmetry enhances smooth propagation of discrete breathers, and it is defined by an invariance of the potential function with respect to a map acting on the complex normal mode coordinates. Condition of the symmetry is given by a set of algebraic equations with respect to coefficients of the pairwise interactions. We prove that the set of algebraic equations has a unique solution, and moreover we solve it explicitly. We present an explicit Hamiltonian for the symmetric lattice, which has coefficients given by the solution. We demonstrate that the present symmetric lattice is useful for numerically computing traveling discrete breathers in various lattices. We propose an algorithm using it.

This article investigates effects of axial load distribution on buckling loads and their modes of functionally graded (FG) beams including a shear effect for the first time, since all previous studies focused on constant axial load. The parabolic higher order shear theory of Reddy is proposed to consider the shear effect of both thick and thin beams. The gradation of the material is distributed continuously through thickness direction with generalized power-law function. The in-plane axial compressive load is simulated by continuous functions through the axial direction, which are constant, linear, and parabolic functions. The equilibrium equations are developed by Hamilton’s principles, which yields variable-coefficients differential equations. Modified differential quadrature method (DQM) is employed to discretize and mesh the spatial domain and covert the differential equations and boundary conditions to algebraic equations. Algebraic set of equations is formed as a generalized matrix eigenvalue problem, then solved to obtain buckling loads (eigenvalues) and mode-shapes (eigenvectors). Numerical results are presented to illustrate effects of gradation parameter, load types, slenderness ratio and boundary conditions on buckling loads and mode-shapes of functionally graded beams structures. This model can be applied in designing of space-shuttle, spacecraft, nuclear structure, and naval structure subjected to distributed in-plane axial load.

This paper presents new sufficient conditions for convergence and asymptotic or exponential stability of a stochastic discrete-time system, under which the constructed Lyapunov function always decreases in expectation along the system's solutions after a finite number of steps, but without necessarily strict decrease at every step, in contrast to the classical stochastic Lyapunov theory. As the first application of this new Lyapunov criterion, we look at the product of any random sequence of stochastic matrices, including those with zero diagonal entries, and obtain sufficient conditions to ensure the product almost surely converges to a matrix with identical rows; we also show that the rate of convergence can be exponential under additional conditions. As the second application, we study a distributed network algorithm for solving linear algebraic equations. We relax existing conditions on the network structures, while still guaranteeing the equations are solved asymptotically., 14 pages, 1 figure

In this article, the authors report the Chebyshev pseudospectral method for solving twodimensional nonlinear Schrodinger equation with fractional order derivative in time and space both. The modified Riemann-Liouville fractional derivatives are used to define the new fractional derivatives matrix at CGL points. Using the Chebyshev fractional derivatives matrices, the given problem is reduced to a diagonally block system of nonlinear algebraic equations, which will be solved using Newton's Raphson method. The proposed methods have shown error analysis without any dependency on time and space step restrictions. Some model examples of the equations, defined on a convex and rectangular domain, have tested with various values of fractional order α and β. Moreover, numerical solutions are demonstrated to justify the theoretical results.

In this paper, we approximate the solution of fractional Bagley–Torvik equation by using the exponential spline function and the shifted Grünwald difference operator. The proposed methods reduce to the system of algebraic equations. The convergence analysis of the methods has been discussed. The numerical examples are presented to illustrate the applications of the methods and to compare the computed results with the other methods.

A numerical treatment to a system of Caputo fractional order differential- algebraic equations (SFDAEs) is presented throughout this article. The suggested method based upon the shifted Chebyshev pesedu- spectral method (SCPSM). The shifted Chebyshev polynomials (SCPs) are handled to reduce the SFDAEs into the solution of linear/ nonlinear systems of algebraic equations. By using some tested applications, the effectiveness and the accuracy of the suggested approach are demonstrated graphically. Also numerical comparisons between the proposed technique with other numerical methods in the existing literature are held. The numerical results show that the proposed technique is computationally efficient, accurate and easy to implement.

The indirect illumination system with diffuse reflection light-emitting diode (LED) is widely applied in various fields that require high illumination uniformity. On the basis of the Lambertian characteristic of LEDs and the ideal diffuse surface, a design method that features a multiple diffuse reflectance freeform surface is proposed and its related mathematical model is established in this paper. A nonlinear algebraic equation is solved to redistribute the energy emitted by the LED to a uniformity illuminated scene. The outline of the diffuse surface is obtained by solving equations numerically. With the use of the ray tracing software TracePro, simulation results show that the effect is generally the best and the uniformity and efficiency are significantly improved compared with other methods when the proportion between the radius of the target surface and the distance between the LED light source and the target surface is 1:4.