Chebyshev Collocation Method Research Articles

On Mixed Steps-Collocation Schemes for Nonlinear Fractional Delay Differential Equations

This research deals with the numerical solution of fractional differential equations with delay using the method of steps and shifted Legendre (Chebyshev) collocation method. This article presents a new formula for the fractional derivatives (in the Caputo sense) of shifted Legendre polynomials. With the help of this tool and previous work of the authors, efficient numerical schemes for solving nonlinear continuous fractional delay differential equations are proposed. The proposed schemes transform the nonlinear fractional delay differential equations to a non-delay one by employing the method of steps. Then, the approximate solution is expanded in terms of Legendre (Chebyshev) basis functions. Furthermore, the convergence analysis of the proposed schemes is provided. Several practical model examples are considered to illustrate the efficiency and accuracy of the proposed schemes.

An efficient numerical approach for solving three-point Lane–Emden–Fowler boundary value problem

For three-point Lane–Emden–Fowler boundary value problems (LEFBVPs), we propose two robust algorithms consisting of Bernstein and shifted Chebyshev polynomials coupled with the collocation technique. The first algorithm uses the Bernstein collocation method with uniform collocation points, while the second is based on the shifted Chebyshev collocation method with roots as its collocation points. In both algorithms, the equivalent integral equations of three-point LEFBVPs are constructed. Then the approximation and collocation approach are used to generate a system of nonlinear equations. Then we implement Newton’s method to solve this system. The current method is different from the traditional method as we do not require to approximate u′ and u′′ appearing in the problem. It reduces not only the computational time but also the truncation error. The existence of a unique solution is discussed. Error analysis of both algorithms is demonstrated. A well known property of the proposed techniques is that it provides efficient solution for a few collocation points. The numerical results verify the same.

A novel numerical approach and stability analysis for a class of pantograph delay differential equation

The goal of this research is to introduce a numerical technique based on shifted Chebyshev polynomial and collocation method for a second order non-linear Lane–Emden pantograph delay differential equation (PDDE) subject to initial conditions. The proposed formulation is based on converting the initial value problem (IVP) into an equivalent fundamental algebraic equation which reduces computational expense. The convergence analysis of the proposed numerical technique demonstrates the efficiency of the proposed scheme. The uniqueness and existence, regularity and stability analysis of the Lane–Emden PDDE is discussed and investigated thoroughly by providing sufficient theorems. Due to the non-availability of abundant literature, the new approach is implemented and tested against existing approaches on standard and newly constructed nonlinear examples. The comparison demonstrates that the proposed method is more accurate and consumes lesser CPU time to compute the results than the existing methods.

Instability mechanism of shear-layered fluid in the presence of a floating elastic plate

In this study, linear stability analysis in the two-dimensional Cartesian coordinate system is used to analyze the flow dynamics underneath a large floating elastic plate over a slippery surface in the presence of external shear. For both viscous and inviscid flows, the Orr–Sommerfeld equation and the Rayleigh equation, respectively, are obtained using normal mode analysis. The Chebyshev collocation method is used to solve both equations numerically. Analysis of the growth rate and energy distributions is performed to understand the flow instability at various flow and structural parameters. The study reveals that the flow below the floating elastic plate dampens for larger uniform mass and structural rigidity in the viscous fluid. On the other hand, there is no effect of structural rigidity on the flow stability in the case of inviscid flow. However, the plate of larger uniform mass stabilizes the growing disturbance generated due to the externally imposed shear at the surface of the plate. The present study is analogous to the simple geographical model of external shearing on the surface of a large ice cover zone caused by atmospheric air. This study can be extended to understand the flow stability below other large floating structures like a floating island and a floating airport.

Numerical evaluation of heat irreversiblity in porous medium combustion of third-grade fluid subjected to Newtonian cooling

Understanding combustion efficiency and reduction of environmental pollution are one of the main challenges encountered in porous medium combustion in many industrial and chemical engineering applications in recent times. In this regard, we present a numerical investigation of the thermal stability of the process and inherent irreversibility analysis for combustible third-grade fluid through a non-Darcian porous medium. The heated fluid flow is assumed to flow steadily through parallel Riga plates with asymmetrical convective cooling. The equations governing the flow are formulated and converted to a set of nonlinear Boundary-value problems by dimensional analysis. The dimensionless equations are solved numerically by spectral Chebyshev collocation method and validated with Shooting-Runge-Kutta method. Graphical and tabular illustrations are provided with sufficient explanations for the velocity, temperature, entropy generation, and irreversibility ratio profiles.

Bernoulli collocation method for the third-order Lane-Emden-Fowler boundary value problem

We propose two efficient numerical algorithms, Bernoulli uniform collocation method and Bernoulli Chebyshev collocation method for solving 3rd-order Lane-Emden-Fowler boundary value problems. To avoid singularity at x=0, we transform the concerned problem into its integral form. By using the Bernoulli collocation method, the resulting integral equation is converted into a system of nonlinear equations to be solved numerically by any suitable iteration method. The error bound of the algorithm and the existence of a unique solution of the problem are also discussed. The high accuracy and efficiency of the proposed method are shown by comparing the results of L∞ and L2 errors of several examples. Moreover, the numerical results of our method are compared with the results obtained by the other known techniques.

(Spectral) Chebyshev collocation methods for solving differential equations

Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along the Legendre orthonormal basis. Interestingly, this approach can be extended to cope with other orthonormal bases and, in particular, we here consider the case of the Chebyshev polynomial basis. The corresponding Runge-Kutta methods were previously obtained by Costabile and Napoli [33]. In this paper, the use of a different framework allows us to carry out a novel analysis of the methods also when they are used as spectral formulae in time, along with some generalizations of the methods.

Dynamics of the coherent structure for incompressible fluid flow in turbulent boundary layers

We consider the nonlinear interaction system of waves to identify discrete clusters of resonant triads, which are classified on the basis of the resonance condition. This study is conducted to investigate the coherent structure of incompressible fluid flow in the turbulent boundary layer. The discrete wave turbulence is characterised by weakly nonlinear interaction modes for amplitude Tollmien Schlichting in a single-mode approximation. Within the framework of multiple-scale analysis, the coherent part of the amplitude equation is defined in the case of multiple three-wave resonance. The resonance condition is defined from the dispersion relation of these amplitudes, which are determined from solving the spectral problem of Orr–Sommerfeld equation by the Chebyshev collocation method. The spectral characteristics of these amplitudes are investigated to define the condition of multiple three-wave resonance. This condition is also defined for a resonant cluster made out of triads sharing a common mode, where all triads satisfy the resonance condition. The coherent amplitudes are represented dynamically by an autonomous system of ordinary differential equations. Non-integrable system of a single triad and a cluster of triads is noted, where the interaction coefficients in the given system do not have the same complex phase. The obtained dynamical system admits a number of invariants, similar to the classical Manley–Rowe invariants but of a different nature. One of these invariants is called the energy invariant manifold that represents the sum of modules square amplitudes of the dynamical system, this invariant is normalised to be defined on the unit sphere. Therefore, Birkhoff–Khinchin theory is applied to calculate the time average of square harmonic and sub harmonic amplitudes. Moreover, this paper is also focused on studying the numerical solutions of both simple and complex structure of the dynamical system by using Runge–Kutta method with random initial conditions. The solution of the dynamical system is examined at different signs of the weight factors, where the bounded solutions of this system are found at both positive and negative signs. However, in another study of a dynamical system, an explosive instability is noted at a negative sign for only one of the weight factors, where all study cases are related to the choice of wave vectors. The random initial conditions are applied to both simple and complex dynamical system to study the behaviour of system solutions. The coupling different triads within the dynamical system lead to chaotic turbulence regime.

Thermal radiation effect on Non-Newtonian Casson fluid through a porous material over a magnetic field with buoyancy

The magnetohydrodynamic (MHD) chemically reactive of Casson non-Newtonian nanofluid flow on a two-dimensional incompressible steady from stretched sheet in a porous quiescent medium with buoyancy effect is investigated numerically. Additional effects included in the originality of the model are the applied magnetic field and solar radiation effect. The Chebyshev collocation method (CCM) was used to solve the ordinary differential equations (ODEs) with MATHEMATICA 11.3 software. The present tables and graphs show the performance of fluid physical quantities, momentum flow, energy distribution, nanoparticle concentration, and velocity for various values of applicable dimensionless numbers. The numerical outcomes demonstrate the effect of different physical parameters of the fluid, and it was observed that the velocity profile increased as the thermal and mass Grashof number increased due to an increase in buoyant force caused by heat transferred from the vertical plate to the fluid but decreased as the Casson parameter increased due to a decrease in its yield stress, porosity, and magnetic parameter. Also Analyses reveal that the thermal profile reduce with an increase in variable thermal conductivity parameter. This study will be of considerable economic value to marine engineers, mechanical engineers, physicists, chemical engineers, and others since its application will help them improve their operations.

Analysis of unsteady boundary layer flow of nanofluids with heat transfer over a permeable stretching/shrinking sheet via a shifted Chebyshev collocation method

In this article, the detailed study of unsteady boundary layer flow over a permeable stretching/shrinking sheet problem of a nanofluid is carried out. The appropriate similarity transformations are utilized to transform the governing equations into nonlinear ordinary differential equations with variable coefficients over an infinite domain. The numerical solution of these equations is achieved by using semi-numerical technique viz. Shifted Chebyshev Collocation Method (SCCM). The variations in velocity, temperature, and concentration profiles under the influence of various physical parameters are analysed and outcomes are expressed in terms of tables and graphs. The obtained results are compared with previous finding for validation along with error analysis. For particular range of stretching/shrinking parameters the dual solutions are exhibited. Also, the range increases with increment in mass suction and unsteadiness parameter, which results into enhancement of skin friction coefficient, local Nusselt number and local Sherwood number. From the graphs it is anticipated that, both temperature and nanoparticle fraction profiles decrease with raise in the values of parameters viz. unsteadiness, suction/injection, Lewis number. Reverse nature of these profiles is observed with the enhancement in thermophoresis parameter. Whereas, temperature increases with increase of Brownian motion and reverse behavior is observed for nanoparticle fraction.

A new linear stability analysis approach for microchannel flow based on the Boltzmann Bhatnagar–Gross–Krook equation

Microchannels are important components of microelectromechanical systems (MEMSs) that encounter rarefaction effects due to their small-scale characteristics. The influence of rarefaction effects on the flow stability of microchannels should be investigated to improve MEMS performance. Based on kinetic theory, a linear stability analysis approach for low-speed rarefied flows was developed by using the Bhatnagar–Gross–Krook (BGK) model of the Boltzmann equation with an external force term. This approach was applied to study the linear temporal stability of microchannel flows. A slip flow model was introduced for comparison. The corresponding eigenvalue problem was solved with a Chebyshev collocation method. This novel approach yielded a critical Reynolds number of 5778. Analysis of the validity and accuracy of the slip flow model shows that although this model cannot capture the Knudsen layer structure, this approach effectively improves the prediction accuracy of the growth rate of the least stable mode. However, the prediction accuracy gradually decreases with increasing Knudsen number. Compared with the stability results obtained from the BGK equation, the Navier–Stokes equations-based stability analysis method always underestimates the disturbance growth rate, regardless of whether a slip flow model is used. The stability analysis results show that rarefaction effects stabilize the flow. The degree of rarefaction does not affect the trends of growth rate and phase velocity with wavenumber, nor does it affect the shape of the velocity eigenfunctions. For a rarefied case, increasing the Mach number has a destabilizing effect on low-speed microchannel flows.

Localized Chebyshev and MLS collocation methods for solving 2D steady state nonlocal diffusion and peridynamic equations

The localized Chebyshev collocation method and MLS collocation method are presented to obtain the solution of two-dimensional nonlocal diffusion and peridynamic equations. The Chebyshev polynomial and MLS interpolation techniques are used to construct shape functions in the frame of the collocation method. Low computational cost and high accuracy are the main advantages of these two methods for solving nonlocal diffusion and peridynamic equations. Several numerical examples are provided to show the validity and applicability of the proposed method with the regular and irregular domains. Numerical experiments indicate that the localized Chebyshev collocation method has high accuracy for nonlocal problems with continuous solutions. The MLS collocation method is more efficient and can maintain good behavior for problems with discontinuous solutions.

Nonlinear Mixed Convection in a Reactive Third-Grade Fluid Flow with Convective Wall Cooling and Variable Properties

Energy management and heat control whenever a reactive viscous fluid is the working medium has been one of the greatest challenges encountered by many in the field of chemical and industrial engineering. A mathematical approach to thedetermination of critical points beyond which the working environment becomes hazardous is presented in the present investigation together with the entropy generation analysis that guarantees the efficient management of expensive energy resources. In this regard, the nonlinear mixed convective flow behavior of a combustible third-grade fluid through a vertical channel with wall cooling by convection is investigated. The mathematical formulation captures the nonlinearities arising from second-order Boussinesq approximation and exponential dependence of internal heat generation, viscosity, and thermal conductivity on temperature. The resulting nonlinear boundary value problems were solved based on the spectral Chebyshev collocation method (SCCM) and validated with the shooting-Runge–Kutta method (RK4). The nonlinear effects on the flow velocity, temperature distribution, entropy generation, and Bejan heat irreversibility ratio are significant. Further analyses include the thermal stability of the fluid. Findings from the study revealed that flow, temperature, and entropy generation are enhanced byincreasing values of the Grashof number, the quadratic component of buoyancy, and the Frank-Kameneskii parameter, but are reducedbyincreasing the third-grade material parameter. Moreover, it was shown that increasing values of the third-grade parameter encourages the thermal stability of the flow, while increasing values of the linear and nonlinear buoyancy parameter destabilizes the flow. The present result is applicable to thick combustible polymers with increased molecular weight.

Scrutinisation of Chebyshev collocation method for mass transfer on a continuous flat plate moving in parallel to a free stream in the presence of a chemical reaction

The paper presents the boundary layer flow of mass transfer on a continuous flat plate moving in parallel or reversely to a free stream with a chemical reaction. By using suitable similarity transformations, the boundary-layer equations are transformed into coupled nonlinear ordinary differential equations (NODEs) over semi-infinite interval. These equations have been analysed using a novel semi-numerical method, viz. spectral method. The dual solutions for velocity and concentration distributions are determined using Chebyshev collocation method (CCM) and the results are presented in the form of tables and graphs. The obtained spectral solutions are compared with previously published results and are comparable. Many interesting physical properties of the problem are observed and verified through both theoretical as well as semi-numerical approach. The derived quantities show that the mass transfer rate is established to be increased as the Schmidt number increases for the solution of the upper branch and reduces for the solution of lower branch.

A Chebyshev collocation method for solving the non-linear variable-order fractional Bagley–Torvik differential equation

Abstract A numerical approach based on the shifted Chebyshev–Gauss collocation method is proposed for solving the non-linear variable-order fractional Bagley–Torvik differential equation (VO-FBTE), subject to initial and boundary conditions. The shifted fractional Chebyshev–Gauss collocation points are used as interpolation nodes, and the solution of the VO-FBTE is approximated by a truncated series of the shifted Chebyshev polynomials. The residuals are calculated at the shifted fractional Chebyshev–Gauss quadrature points. The original VO-FBTE is converted into a system of algebraic equations. The accuracy of the proposed scheme is confirmed with a set of numerical examples, and the results are compared with those obtained by other methods.

Oblique stagnation point flow of micropolar nanofluid impinge along a vertical surface via modified Chebyshev collocation method

This article contains a study of mixed convection in micropolar nanofluid near an oblique stagnation point in the presence of a magnetic field. Similarity transformations are used to convert governing partial differential equations to non-linear ordinary differential equations. Modified Chebyshev collocation method in computational software Maple is used for the solution of governing nonlinear differential equations. A comparison of numerical results obtained by modified Chebyshev collocation method and finite difference method is made to show the accuracy of the method. Graphical results for velocity components, microrotation, temperature, and flow patterns are part of this study. Numerical values for free parameter (A), skin friction, and Nusselt numbers for different parameters are also calculated. It is found that microrotation profiles are enhanced by increasing the effect of stretching while decline with enhancing angle of strike γ. Also, the temperature of micropolar nanofluid is increased by increasing the value of the magnetic parameter and micropolar coefficient. The temperature gradient of nanofluid shows a decline when values of stretching parameter and the angle of the strike are increased.

Series solution and Chebyshev collocation method for the initial value problem of Emden-Fowler equation

ABSTRACT The Emden-Fowler equation has important applications in some mathematical and physical problems. In this paper, a smoothing transformation is given for the initial value problem of the Emden-Fowler equation, and then the equation is further transformed into an equivalent Volterra integral equation of the second kind. The series expansion of the solution to the integral equation about the origin is obtained by Picard iteration, which shows that the solution of the transformed equation is sufficiently smooth at the origin. The series solution and its Padé approximation are usually only accurate near the origin. Hence, a high-precision Chebyshev collocation method is designed to obtain the numerical solution on a finite interval. The convergence of the scheme is analysed, and the error with the maximum norm is estimated. Numerical examples illustrate the high accuracy of the proposed method for solving the initial value problem of the Emden-Fowler equation.

Instabilities of the natural convection around a cone in thermally stratified medium.

In this paper, we provide a mathematical description of the onset of instability for buoyancy-driven flow around a cone. The self-similar solutions of the basic flow are derived, where the ambient fluid and the cone have independent temperature gradients. The linear instability properties are investigated by utilizing the Chebyshev collocation method. It is demonstrated that the neutral curves have a two-lobed structure and that changing the Grashof number has only quantitative influence on stability. The critical streamwise location increases when the half-apex angle is increased, and the unstable range of the wave number diminishes. According to the examination of eigenfunction profiles and the progression of the two spatial branches in the (α_{i},α_{r}) planes, the primary instabilities on the surface of cone are identified as type-I mode and type-II mode. The energy analysis is investigated for a typical situation to gain a physical insight, where it is demonstrated that besides the viscous dissipation, the streamline curvature and buoyancy-driven effects are dominant for the type-I mode while inviscid effect plays an essential role in type II. These encouraging results are expected to be conducive to understanding buoyancy-driven systems.

Monotonic energy stability for inclined laminar flows

The stability of planar shear flows of an incompressible fluid is fundamental and still a debated topic. In this article we define a family of basic flows which include Couette flow and Poiseuille flow, and use analytical methods and a Chebyshev collocation method with many different approaches to investigate the monotonic behaviour of the energy along perturbations. We obtain in different ways an extension of known results for general shear flows, and we confirm the validity of some approaches that can be found in the literature.

Size dependent generalized thermoelasticity: Green-Lindsay theory with modified strain gradient theory

The present work is a development of the Green-Lindsay (GL) thermoelasticity theory for micro continuum bodies. The micro continuum modified strain gradient theory is used here based on the Green-Lindsay theory of the micro continuum media. In fact, the present work is a combination of Green-Lindsay thermoelasticity theory with modified strain gradient continuum theory. All assumptions of GL and modified strain gradient theory are satisfied and the final general forms of constitutive relations and also heat conduction equation are extracted. As an application of the derived formulation, a one-dimensional microlayer is considered and the newly derived equations are applied to it. Finally, the governing equations of coupled generalized thermoelasticity of the microlayer are solved using the Chebyshev collocation method. Effects of microscale parameters appearing in modified strain gradient theory on the dynamic thermo-elastic response of microlayer are investigated.