Finite Element Orthogonal Collocation Approach for Time Fractional Telegraph Equation with Mamadu-Njoseh Polynomials

Finite element method (FEM) is an efficient numerical tool for the solution of partial differential equations (PDEs). It is one of the most general methods when compared to other numerical techniques. PDEs posed in a variational form over a given space, say a Hilbert space, are better numerically handled with the FEM. The FEM algorithm is used in various applications which includes fluid flow, heat transfer, acoustics, structural mechanics and dynamics, electric and magnetic field, etc. Thus, in this paper, the Finite Element Orthogonal Collocation Approach (FEOCA) is established for the approximate solution of Time Fractional Telegraph Equation (TFTE) with Mamadu-Njoseh polynomials as grid points corresponding to new basis functions constructed in the finite element space. The FEOCA is an elegant mixture of the Finite Element Method (FEM) and the Orthogonal Collocation Method (OCM). Two numerical examples are experimented on to verify the accuracy and rate of convergence of the method as compared with the theoretical results, and other methods in literature.

Simulation of Cefoselis hydrochloride adsorption on macroporous resin in a fixed-bed column using orthogonal collocation

Adsorption operation is of great importance for separation and purification of semi-synthetic cephalosporin compounds in pharmaceutical industry. The adsorption dynamics of Cefoselis hydrochloride(CFH) on XR 920 C adsorbent in fixed bed was predicted by the model of modified film-pore diffusion(MFPD). The intraparticle diffusion equation and mass balance equation in fixed bed are discretized into two ordinary differential equations(ODEs) using the method of orthogonal collocation which largely improves the calculation accuracy. The MFPD model parameters including the pore diffusion coefficient(Dp), external mass-transfer coefficient(kf), and the axial dispersion(DL) were estimated. The kfvalue was calculated by the Carberry equation, in which the effective diffusion coefficient Dewas fitted based on Crank Model through experimental data. Moreover, three key operating parameters(i.e., initial adsorbate concentration; flow rate of import feed, and bed height of adsorbent) and the corresponded breakthrough curves were systematically studied and optimized. Therefore,this research not only provides valuable experimental data, but also a successfully mathematical model for designing the continuous chromatographic adsorption process of CFH.

SNAP－THROUGH OF LAMINATED ORTHOTROPIC SHALLOW SPHERICAL IMPERFECT SHELLS WITH TRANSVERSE SHEAR

The equations of large deformations of laminated orthotropic spherical shellsare derived. The effects of transverse shear deformation and initial imperfection are considered. on this basis. the semi-analytical solution of the axisymrnetric snap-throughbuckling of laminated orthotropic shallow spherical shells under uniform pressure is obtained using orthogonal collocation method. The effects of material parameters, structuralparameters, initial imperfection and transverse shear deformation are discussed.

Real-Time Optimization Model for Continuous Reforming Regenerator

An approach for the simulation and optimization of continuous catalyst-regenerative process of reforming is proposed in this paper.Compared to traditional method such as finite difference method,the orthogonal collocation method is less time-consuming and more accurate,which can meet the requirement of real-time optimization(RTO).In this paper,the equation-oriented method combined with the orthogonal collocation method and the finite difference method is adopted to build the RTO model for catalytic reforming regenerator.The orthogonal collocation method was adopted to discretize the differential equations and sequential quadratic programming(SQP)algorithm was used to solve the algebraic equations.The rate constants,active energy and reaction order were estimated,with the sum of relative errors between actual value and simulated value serving as optimization objective function.The model can quickly predict the fields of component concentration,temperature and pressure inside the regenerator under different conditions,as well as the real-time optimized conditions for industrial reforming regenerator.

ALTERNATING DIRECTION IMPLICIT OSC SCHEME FOR THE TWO-DIMENSIONAL FRACTIONAL EVOLUTION EQUATION WITH A WEAKLY SINGULAR KERNEL

In this paper, a new kind of alternating direction implicit （ADI） Crank-Nicolson-type orthogonal spline collocation （OSC） method is formulated for the two-dimensional frac-tional evolution equation with a weakly singular kernel arising in the theory of linear viscoelas-ticity. The novel OSC method is used for the spatial discretization, and ADI Crank-Nicolson-type method combined with the second order fractional quadrature rule are considered for thetemporal component. The stability of proposed scheme is rigourously established, and nearlyoptimal order error estimate is also derived. Numerical experiments are conducted to supportthe predicted convergence rates and also exhibit expected super-convergence phenomena.

NONLINEAR DYNAMIC RESPONSE AND DYNAMIC BUCKLING OF SHALLOW SPHERICAL SHELLS WITH CIRCULAR HOLE

In this paper, the nonlinear equations of motion for shallow spherical shells with axisymmetric deformation including transverse shear are derived. The nonlinear static and dynamic response and dynamic buckling of shallow spherical shells with circular hole on elastically restrained edge are investigated. By using the orthogonal point collocation method for space and Newmarh-β scheme for time, the displacement functions are separated and the nonlinear differential equations are replaced by linear algebraic equations to seek solutions. The numerical results are presented for different cases and compared with available data.

The Crank-Nicolson Hermite Cubic Orthogonal Spline Collocation Method for the Heat Equation with Nonlocal Boundary Conditions

We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval.Using an extension of the analysis of Douglas and Dupont[23]for Dirichlet boundary conditions,we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space.We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROWfor solving almost block diagonal linear systems.We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.

Alternating Direction Implicit Orthogonal Spline Collocation on Non-Rectangular Regions

The alternating direction implicit(ADI)method is a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems on rectangles.When the ADI technique is coupled with orthogonal spline collocation(OSC)for discretization in space we not only obtain the global solution efficiently but the discretization error with respect to space variables can be of an arbitrarily high order.In[2],we used a Crank Nicolson ADI OSC method for solving general nonlinear parabolic problems with Robin’s boundary conditions on rectangular polygons and demonstrated numerically the accuracy in various norms.A natural question that arises is:Does this method have an extension to non-rectangular regions?In this paper,we present a simple idea of how the ADI OSC technique can be extended to some such regions.Our approach depends on the transfer of Dirichlet boundary conditions in the solution of a two-point boundary value problem(TPBVP).We illustrate our idea for the solution of the heat equation on the unit disc using piecewise Hermite cubics.

Numerical Solutions of Coupled Nonlinear Schrödinger Equations by Orthogonal Spline Collocation Method

In this paper,we present the use of the orthogonal spline collocation method for the semi-discretization scheme of the one-dimensional coupled nonlinear Schrödinger equations.This method uses the Hermite basis functions,by which physical quantities are approximatedwith their values and derivatives associatedwith Gaussian points.The convergence rate with order O(h4+t2)and the stability of the scheme are proved.Conservation properties are shown in both theory and practice.Extensive numerical experiments are presented to validate the numerical study under consideration.