Numerical Collocation Method Research Articles

Optimal control for nonlinear time-fractional Schrödinger equation: an application to quantum optics

In this research article, a fractional optimal control problem (FOCP) is applied to a nonlinear time-fractional Schrödinger equation (NTFSE) incorporating a trapping potential. The NTFSE is an innovative mathematical advancement in the field of quantum optics, bridging fractional calculus with nonlinear quantum mechanics and addressing the intricacies of systems involving memory and nonlinearity. This exploration helps with potential technological advancements in quantum optics and related domains. Examining the FOCP within this system allows one to design quantum optical systems with enhanced performance, improved precision stability, and robustness against disturbances. In this work, the performance index for the problem is constructed, and then it is reformulated using the fractional variational principle and the Lagrange multiplier method. Additionally, the Jacobi collocation numerical method is employed to solve the FOCP and numerical simulations are demonstrated across various parameters which offer valuable insights into the implemented methodology.

On flexural vibrations of orthotropic strips of variable thickness taken into account of transverse effects with elasticaly built in edges in the presence of axial force

Abstract The problem of bending vibrations of orthotropic strips of variable thickness is considered, taking into account transverse effects elastically built in edges in the presence of axial force. The problem was solved using the numerical collocation method. A numerical analysis of the problem was carried out. The problem of Euler stability of orthotropic plate-strips of variable thickness under plane strain conditions is considered, taking into account transverse shear and rotational inertia. The resulting equations make it possible to determine both the frequencies of natural vibrations and the critical values of the axial force at which the plate-strip loses stability. The problems are solved using a modified collocation method under various boundary conditions at the ends and values of geometric and physical parameters characterizing a plate-strip of variable thickness.

A Collocation Numerical Method for Highly Oscillatory Algebraic Singular Volterra Integral Equations

The highly oscillatory algebraic singular Volterra integral equations cannot be solved directly. A collocation numerical method is proposed to overcome the difficulty created by the highly oscillatory algebraic singular kernel. This paper is composed primarily of two methods—the piecewise constant collocation method and the piecewise linear collocation method—in which uniformly distributed nodes serve as collocation points. For the efficient computation of highly oscillatory and algebraic singular integrals, the steepest descent method as well as the Gauss–Laguerre and generalized Gauss–Laguerre quadrature rules are employed. Consequently, the resulting linear system is solved for the unknown function approximated by the Lagrange interpolation polynomial. Detailed theoretical analysis is carried out and numerical experiments showing high accuracy are also presented to confirm our analysis.

Magnetic Field Effect and Heat Transfer of Nanofluids within Waveform Microchannel

In this research, a numerical study of mixed convection of non-Newtonian fluid and magnetic field effect along a vertical wavy surface was investigated. A simple coordinate transformation to transform wavy surface to a flat surface is employed. A cubic spline collocation numerical method is employed to analyze transformed equations. The effect of various parameters such as Reynolds number, volume fraction 0-, Hartmann number, and amplitude of wave length was evaluated in improving the performance of a wavy microchannel. According to the presented results, the sinusoidal shape of the microchannel has a direct impact on heat transfer. By increasing the microchannel wave amplitude, the Nusselt number has risen. On the other hand, increasing the heat transfer in the higher wavelength ratio corrugated channel is seen as an effective method of increasing the heat transfer, especially at higher Reynolds numbers. The results showed that with increasing Hartmann numbers, the flow line near the wall becomes more regular and, according to the temperature gradient created, the Nusselt number growth.

Elastic wave scattering by flat-bottomed indentations on a plate

• Method for multiple scattering by scatterers with unmatched midplanes is proposed. • The method can analyze coupled flexural and in-plane extensional motions. • Mode conversions between different elastic waves are observed. • Stop bands of shear and flexural waves are captured simultaneously. • Broadband scattering polarization of flexural waves is achieved by multiple scattering. Out-of-plane flexural and in-plane extensional motions are independently analyzed in available studies of multiple scattering due to the assumption of the consistent mid-plane between the scatterer and host medium. When mid-planes of the scatterer and the host medium are inconsistent, such as flat-bottomed indentations, the coupled out-of-plane flexural and in-plane extensional motions should be considered. This paper aims to investigate a general numerical procedure to solve the three-dimensional multiple scattering problem of elastic waves on a plate with cylindrically flat-bottomed indentations. Wave expansion functions derived from the Poisson and Mindlin plate theories are used to express three-dimensional displacements of the host plate and flat-bottomed indentations. A linear matrix representing analytical relationships between wave expansion coefficients of exciting and scattered fields is deduced by the continuity conditions of flat-bottomed indentations. A set of linear algebraic equations is then developed by the fact that the exciting fields around a scatterer are the sum of scattered fields coming from all other scatterers. The numerical collocation method is used to solve the preceding algebraic equations. Simulation results verify mode conversions between compressional, shear, and flexural waves. Details of stop-band formations of shear and flexural waves are captured simultaneously. Furthermore, far-field scattering polarizations of shear and flexural waves are found in the case of multiple scattering.

Solution of the Second Order of the Linear Hyperbolic Equation Using Cubic B-Spline Collocation Numerical Method

Wave equation is one of the second order of the linear hyperbolic equation. Telegraph equation as a special case of wave equation has interesting point to investigate in the numerical point of view. In this paper, we consider the numerical methods for one dimensional telegraph equation by using cubic B-spline collocation method. Collocation method is one method to solve the partial differential equation model problem. Cubic spline interpolation is an interpolation to a third order polynomial. This polynomial interpolate four point. B-Spline is one of spline function which related to smoothness of the partition. For every spline function with given order can be written as linear combination of those B-spline. As we known that the result of the numerical technique has difference with the exact result which we called as, so that we have an error. The numerical results are compared with the interpolating scaling function method which investigated by Lakestani and Saray in 2010. This numerical methods compared to exact solution by using RMSE (root mean square error), L2 norm error and L_∞ norm error . The error of the solution showed that with the certain function, the cubic collocation of numerical method can be used as an alternative methods to find the solution of the linear hyperbolic of the PDE. The advantages of this study, we can choose the best model of the numerical method for solving the hyperbolic type of PDE. This cubic B-spline collocation method is more efficiently if the error is relatively small and closes to zero. This accuration verified by test of example 1 and example 2 which applied to the model problem.

Novel Analytical and Numerical Approximations to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Method

In this work, some novel approximate analytical and numerical solutions to the forced damped driven nonlinear (FDDN) pendulum equation and some relation equations of motion on the pivot vertically for arbitrary angles are obtained. The analytical approximation is derived in terms of the Jacobi elliptic functions with arbitrary elliptic modulus. For the numerical approximations, the Chebyshev collocation numerical method is introduced for analyzing the equation of motion. Moreover, the analytical approximation and numerical approximation using the Chebyshev collocation numerical method and the MATHEMATICA command Fit are compared with the Runge–Kutta (RK) numerical solution. Also, the maximum distance error to all obtained approximations is estimated with respect to the RK numerical solution. The obtained results help many authors to understand the mechanism of many phenomena related to the plasma physics, classical mechanics, quantum mechanics, optical fiber, and electronic circuits.

LS-based and GL-based thermoelasticity in two dimensional bounded media: A Chebyshev collocation analysis

In this paper, the Chebyshev collocation numerical method is developed for solving generalized thermoelasticity problems of the isotropic homogeneous two dimensional media. The coupled thermoelastic equations are derived based on Lord-Shulman (LS) and Green-Lindsay (GL) theories. The temperature and displacement fields are approximated in space domain by linear combinations of Chebyshev polynomials. Also, the direct collocation method is applied to governing differential equations to generate the system of differential equations with respect to time. The resulted set of differential equations are solved in time domain by Wilson method. Both temperature and traction loadings are considered to be applied at the left side of the media. The obtained results from the present paper for the classical coupled thermoelasticity of two dimensional finite domains are compared with the same results extracted analytically in the literature and a very close agreement is observed.

Two-dimensional elastodynamic scattering by a soft/hard strip

A numerical approach for solving of the 2D elastodynamic waves scattering problem by a finite width planar soft/hard strip is considered. The lower face of the strip is perfectly bonded to the matrix and the upper face is modeled as an interface crack. Using Betti's theorem the mixed boundary value problem is reduced to the hypersingular and weakly singular systems of integral equations. An efficient numerical collocation method for the solution of this problem is developed and its numerical results are presented.

Application of Chebyshev collocation method to unified generalized thermoelasticity of a finite domain

In this article, the Chebyshev collocation numerical method is developed for solving generalized thermoelasticity problems of the isotropic layer. The coupled thermoelastic equations are derived based on Lord-Shulman (LS), Green-Lindsay (GL) and Green-Naghdi (GN) theories. Two kinds of shock loading are considered. In the first model, a temperature shock is applied at the left side of the layer, and in the second model a heat flux shock is applied at the mentioned side. The displacement and temperature fields are approximated in the layer by Chebyshev polynomials and then the collocation is imposed on some collocation points to derive the system of ordinary differential equations from the coupled partial differential equations. The derived system of differential equations is then solved by the Wilson method to achieve the displacement and temperature and also stress at any location and time. The obtained results from the present article are compared with the same results in the open literature and a very close agreement is observed. Finally, it is concluded that the Chebyshev collocation numerical method can be utilized as a very strong method for solving the transient wave propagation problems.

The long wave fluid flows on inclined porous media with nonlinear Forchheimer’s law

The surface fluid flows coupled with porous media flows in substrates occur in many circumstances in industry and natural settings. In this paper, we investigate the long wave solutions for the surface flows on inclined porous media. The important feature is that such flows are derived by the Navier-Stokes equations governing the clear flows in the surface fluids and the nonlinear Forchheimer’s equations for the porous media flows in substrates. The problem is reduced to a corresponding Orr-Sommerfeld problem by linearizing the infinitesimal perturbations in the system of coupled equations for analyzing long wave solutions of surface flows. Numerical analysis is taken by using Chebyshev collocation numerical method to the eigenvalue problems of the Orr-Sommerfeld systems for analyzing critical condition and stable region of long wave solutions. We compare the result with that for very small drag constant by Darcy’s law and study numerically the effects of parameters including various drag constants on the long wave solutions with Forchheimer’s law.

Flexural wave scattering by varying-thickness annular inclusions on infinite thin plates

This paper proposes a semi-analytical method for flexural wave scattering by cylindrical annular inclusions of varying thickness on an infinite thin plate. The thickness of inclusions is assumed to vary in arbitrary power form along the radial direction. In the numerical method, analytically linear relationships between expansion coefficients of exciting and scattered fields around each inclusion are firstly derived by means of closed form solutions of a hypergeometric governing equation of varying-thickness thin plates, and boundary conditions at the interface between different media. Then, linear algebraic equations containing only expansion coefficients of exciting fields are set up with the aid of foregoing linear relations. Expansion coefficients of exciting fields are finally solved by a numerical collocation method. Numerical examples of single and multiple scattering are designed to demonstrate the applicability of the method. The results of single scattering show the change of thickness variations allows tuning local resonances of inclusions to specified frequencies. Furthermore, displacement amplitude spectra of flexural waves for multiple scattering have captured the details of stop band formation of square and hexagonal arrangements of varying-thickness inclusions. It is also demonstrated that stop bands of flexural wave propagation can be controlled by changing stiffness gradients of varying-thickness inclusions.

Parabolized stability analysis of chemically reacting boundary-layer flows in equilibrium conditions

Stability of self-similar hypersonic flows over a flat plate in local thermodynamic equilibrium is studied by means of linear parabolized stability equations. A spectral collocation numerical method together with a finite-difference-based marching scheme are used to solve the equations, while the gas mixture and transport properties are computed with a table look-up procedure. High-temperature effects are found to destabilize the second-Mack-mode regardless of the adiabatic/isothermal wall condition considered. On the other hand, nonlocal effects show a Mach-dependence influence: stabilizing at lower Mach, while destabilizing at higher ones. Strong oscillations in the growth rate of perturbations in local thermodynamic equilibrium are found to be caused by damped supersonic modes.

Axisymmetric creeping motion caused by a spherical particle in a micropolar fluid within a nonconcentric spherical cavity

Abstract The problems of the quasisteady translation and steady rotation of a solid spherical particle located at a non-concentric position of a spherical cavity filled with an incompressible micropolar fluid are investigated semi-analytically in the limit of low Reynolds numbers. General solutions are constructed from the superposition of the basic solutions in the two spherical coordinate systems based at the centers of the particle and cavity. The boundary conditions on the particle surface and cavity wall are satisfied by a collocation numerical method. The hydrodynamic drag force and torque exerted by the fluid on the particle which are proportional to the translational and angular velocities respectively are obtained numerically with good convergence for a range of values of the ratio of particle-to-cavity radii, the relative distance between the centers of the particle and cavity and micropolarity parameter. In the limit of the motion of a spherical particle in a concentric position in the cavity and in the lubrication limit, the hydrodynamic drag force and torque are in good agreement with the available results in the literature. As expected, the boundary-corrected drag force and torque exerted on the particle is a monotonic increasing function of the micropolarity parameter.

B-SPLINE COLLOCATION SIMULATION OF NON-LINEAR TRANSIENT MAGNETIC NANOBIO-TRIBOLOGICAL SQUEEZE-FILM FLOW

A mathematical model is presented for magnetized nanofluid bio-tribological squeeze-film flow between two approaching disks. The nanofluid comprises a suspension of metal oxide nanoparticles with an electrically-conducting base fluid, making the nanosuspension responsive to applied magnetic field. The governing viscous momentum, heat and species (nanoparticle) conservation equations are normalized with appropriate transformations which renders the original coupled, non-linear partial differential equation system into a more amenable ordinary differential boundary value problem. The emerging model is shown to be controlled by a number of parameters, viz nanoparticle volume fraction, squeeze number, Hartmann magnetic body force number, disk surface transpiration parameter, Brownian motion parameter, thermophoretic parameter, Prandtl number and Lewis number. Computations are conducted with a B-spline collocation numerical method. Validation with previous homotopy solutions is included. The numerical spline algorithm is shown to achieve excellent convergence and stability in non-linear bio-tribological boundary value problems. The interaction of heat and mass transfer with nanofluid velocity characteristics is explored. In particular, smaller nanoparticle (high Brownian motion parameter) suspensions are studied. The study is relevant to enhanced lubrication performance in novel bio-sensors and intelligent knee joint (orthopaedic) systems.

Multiple scattering and stop band characteristics of flexural waves on a thin plate with circular holes

A numerical procedure is proposed for the multiple scattering analysis of flexural waves on a thin plate with circular holes based on the Kirchhoff plate theory. The numerical procedure utilizes the wave function expansion of the exciting as well as scattered fields, and the boundary conditions at the periphery of holes are incorporated as the relations between the expansion coefficients of exciting and scattered fields. A set of linear algebraic equations with respect to the wave expansion coefficients of the exciting field alone is established by the numerical collocation method. To demonstrate the applicability of the procedure, the stop band characteristics of flexural waves are analyzed for different arrangements and concentrations of circular holes on a steel plate. The energy transmission spectra of flexural waves are shown to capture the detailed features of the stop band formation of regular and random arrangements of holes. The increase of the concentration of holes is found to shift the dips of the energy transmission spectra toward higher frequencies as well as deepen them. The hexagonal hole arrangement can form a much broader stop band than the square hole arrangement for flexural wave transmission. It is also demonstrated that random arrangements of holes make the transmission spectrum more complicated.

Two‐dimensional shifted Legendre polynomial collocation method for electromagnetic waves in dielectric media via almost operational matrices

In this paper, a numerical solution of fractional partial differential equations (FPDEs) for electromagnetic waves in dielectric media will be discussed. For the solution of FPDEs, we developed a numerical collocation method using an algorithm based on two‐dimensional shifted Legendre polynomials approximation, which is proposed for electromagnetic waves in dielectric media. By implementing the partial Riemann–Liouville fractional derivative operators, two‐dimensional shifted Legendre polynomials approximation and its operational matrix along with collocation method are used to convert FPDEs first into weakly singular fractional partial integro‐differential equations and then converted weakly singular fractional partial integro‐differential equations into system of algebraic equation. Some results concerning the convergence analysis and error analysis are obtained. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2017 John Wiley & Sons, Ltd.

Simulation of shear bands with Soft PARticle Code (SPARC) and FE

The aim of this paper is to numerically investigate the development, thickness and orientation of shear bands, in biaxial test with two approaches towards solving problems of continuum mechanics, namely the meshless “Soft PARticle” method and the mesh based Finite Element method. Soft PArticle Code (SPARC) is a straightforward collocation numerical method based on strong formulation, in which a first order polynomial basis is adopted for the evaluation of spatial derivatives in partial differential equations. A novel nonlinear constitutive model— barodesy for clay, is adopted in this study. The biaxial test, which involves homogeneous, and later inhomogeneous localized deformation is simulated using the Soft PArticle Code and the Finite Element method. The inclination and thickness of the shear bands are evaluated and analysed with the earlier experimental, theoretical and numerical investigations. Furthermore, simulation results are compared and presented to demonstrate the advantages and limitations of SPARC in comparison to FE method.

A collocation method for solving nonlinear Volterra integro-differential equations of neutral type by sigmoidal functions

A numerical collocation method is developed for solving nonlinear Volterra integro-differential equations (VIDEs) of the neutral type, as well as other non-standard and classical VIDEs. A sigmoidal functions approximation is used to suitably represent the solutions. Special computational advantages are obtained using unit step functions, and important applications can also be obtained by using other sigmoidal functions, such as logistic and Gompertz functions. The method allows one to obtain a simultaneous approximation of the solution to a given VIDE and its first derivative, by means of an explicit formula. A priori as well as a posteriori estimates are derived for the numerical errors, and numerical examples are given for the purpose of illustration. A comparison is made with the classical piecewise polynomial collocation method as for accuracy and CPU time.

Solving Volterra integral equations of the second kind by sigmoidal functions approximation

In this paper, a numerical collocation method is developed for solving linear and nonlinear Volterra integral equations of the second kind. The method is based on the approximation of the (exact) solution by a superposition of sigmoidal functions and allows one to solve a large class of integral equations having either continuous or $L^p$ solutions. Special computational advantages are obtained using unit step functions, and analytical approximations of the solution are also at hand. The numerical errors are discussed, and a priori as well as a posteriori estimates are derived for them. Numerical examples are given for the purpose of illustration.