Chebyshev Pseudospectral Research Articles

Chebyshev Pseudospectral Method for Fractional Differential Equations in Non-Overlapping Partitioned Domains

Chebyshev Pseudospectral Method for Fractional Differential Equations in Non-Overlapping Partitioned Domains

High-Order Chebyshev Pseudospectral Tempered Fractional Operational Matrices and Tempered Fractional Differential Problems

This paper focuses on presenting an accurate, stable, efficient, and fast pseudospectral method to solve tempered fractional differential equations (TFDEs) in both spatial and temporal dimensions. We employ the Chebyshev interpolating polynomial for g at Gauss–Lobatto (GL) points in the range [−1,1] and any identically shifted range. The proposed method carries with it a recast of the TFDE into integration formulas to take advantage of the adaptation of the integral operators, hence avoiding the ill-conditioning and reduction in the convergence rate of integer differential operators. Via various tempered fractional differential applications, the present technique shows many advantages; for instance, spectral accuracy, a much higher rate of running, fewer computational hurdles and programming, calculating the tempered-derivative/integral of fractional order, and its spectral accuracy in comparison with other competitive numerical schemes. The study includes stability and convergence analyses and the elapsed times taken to construct the collocation matrices and obtain the numerical solutions, as well as a numerical examination of the produced condition number κ(A) of the resulting linear systems. The accuracy and efficiency of the proposed method are studied from the standpoint of the L2 and L∞-norms error and the fast rate of spectral convergence.

How do the Intravenously Injection of Nanoparticle and Oxytatic Bacteria Affect Through Circular Cylinder Cell for Non-Newtonian Micropolar Fluid: Mathematical Approach

In this paper, we have researched the conduct of non-Newtonian micropolar nanofluid flow through horizontal circular under the impacts of nanoparticles, oxytatic bacterial and Hall current effects. The utilizations of the current investigation are exceptionally powerful in biomedical therapies, for example, obliteration of malignant growth over biological cells by utilizing drug conveyance of nanoparticles and oxytatic microscopic organisms. If there should arise an occurrence of biological nanofluid it’s accepted to concern variable physical parameters which rely on the nanofluid temperature. Implicit Chebyshev pseudospectral (ICPS) technique by helping MATHEMATICA software has been applied to governing nonlinear system of dimensionless partial differential equations (PDEs). The nanofluid velocity, microrotation angular velocity, temperature, motile bacterial density distributions, oxygen concentration, local skin friction coefficient, Nusselt number, and wall motile density gradient distributions are delineated graphically for various variable physical parameters, likewise comparison between certain results in literature and our current output is introduced and great arrangement is found.

Modification of Chebyshev Pseudospectral Method to Minimize the Gibbs Oscillatory Behaviour in Resynthesizing Process

Modification of Chebyshev Pseudospectral Method to Minimize the Gibbs Oscillatory Behaviour in Resynthesizing Process

Dynamical Analysis of Fractional Integro-Differential Equations

In this article, we solve fractional Integro differential equations (FIDEs) through a well-known technique known as the Chebyshev Pseudospectral method. In the Caputo manner, the fractional derivative is taken. The main advantage of the proposed technique is that it reduces such types of equations to linear or nonlinear algebraic equations. The acquired results demonstrate the accuracy and reliability of the current approach. The results are compared to those obtained by other approaches and the exact solution. Three test problems were used to demonstrate the effectiveness of the proposed technique. For different fractional orders, the results of the proposed technique are plotted. Plotting absolute error figures and comparing results to some existing solutions reveals the accuracy of the proposed technique. The comparison with the exact solution, hybrid Legendre polynomials, and block-pulse functions approach, Reproducing Kernel Hilbert Space method, Haar wavelet method, and Pseudo-operational matrix method confirm that Chebyshev Pseudospectral method is more accurate and straightforward as compared to other methods.

Usabilitiy of Chebyshev Pseudospectral method with finite element method on circular discs

Usabilitiy of Chebyshev Pseudospectral method with finite element method on circular discs

Analysis of the Fractional-Order Delay Differential Equations by the Numerical Method

In this study, we implemented a new numerical method known as the Chebyshev Pseudospectral method for solving nonlinear delay differential equations having fractional order. The fractional derivative is defined in Caputo manner. The proposed method is simple, effective, and straightforward as compared to other numerical techniques. To check the validity and accuracy of the proposed method, some illustrative examples are solved by using the present scenario. The obtained results have confirmed the greater accuracy than the modified Laguerre wavelet method, the Chebyshev wavelet method, and the modified wavelet-based algorithm. Moreover, based on the novelty and scientific importance, the present method can be extended to solve other nonlinear fractional-order delay differential equations.

Bending Analysis of Functionally Graded Nanobeam Using Chebyshev Pseudospectral Method

Static performance of functionally graded cantilever nanobeams exposed to lateral and axial loads from the end was examined by applying the Pseudospectral Chebyshev Method. A solution is given for bending analysis using Euler-Bernoulli beam theory. The nonlocal elasticity theory was first introduced by Eringen and is used to represent effect on a small scale. Using the aforementioned theory, the governing differential equations the phenomenon for functionally graded nanobeams are reproduced. It is supposed that the modulus of elasticity of the beam changes exponentially in the x-axis direction, except for the values taken as constant. The exponential change of material properties may not allow analytical problems to be solved with known methods. Therefore, numerical approach is inevitable for the solution of the problem.

The influence of temperature on physical properties of a hybrid nanofluid flow in a non-Darcy porous medium

The purpose of this paper is to debate the influences of variable thermal conductivity as well as viscosity on a hybrid nanofluid flow through a non-Darcy porous medium along a wavy surface. Two kinds of base fluids are considered; the first is water (H2O) with [Formula: see text] and the second is ethylene glycol–water mixture (C2H6O[Formula: see text]H2O) with [Formula: see text]. Furthermore, the physical properties of the base fluid have been modified via the addition of two kinds of suspended nanoparticles. The suspended nanoparticles of alumina Al2O3were firstly added to the base fluid. Then, titania TiO2suspended nanoparticles are immersed into this nanofluid to form two hybrid nanofluids (TiO[Formula: see text]Al2O3/H2O) and (TiO2-Al2O3/C2H6O2–H2O). Dupuit Forchheimer model as a non-Darcy porous medium is utilized. The mathematical model for the present physical problem with convective boundary condition is studied numerically via the Chebyshev pseudospectral (CPS) technique. Nusselt number, velocity and temperature profiles are investigated under different parameters. The present outcomes clarify that adding the alumina nanoparticle supports and improves the efficiency of the titania nanoparticle. Also, taking both of the hybrid nanofluid viscosity/thermal conductivity as variable functions of temperature has an important role to support the studies of heat transfer.

Group formalism of Lie transformations, conservation laws, exact and numerical solutions of non-linear time-fractional Black–Scholes equation

The topic of the present paper concentrates on a systematic investigation of Lie group analysis of non-linear time-fractional Black–Scholes equation including numerical approximations. Lie point symmetries of the equation are found via Riemann–Liouville derivative with four different volatility models. Also the invariant solutions as a kind of exact solutions are constructed by a useful explained method. Conservation laws for the intended equation are derived by using a modified version of Noether’s theorem. A numerical simulation by Chebyshev pseudo-spectral (CPS) method is computed for the equation. Approximate solutions together with their absolute errors are found and the graphs of solutions are plotted by the action of order of derivative α.

Accelerate the ePC-SAFT-DFT Calculation with the Chebyshev Pseudospectral Collocation Method

ePC-SAFT-DFT is a powerful tool for studying the properties of confined ionic liquids for CO2 separation, in which efficient algorithms are required to obtain the calculation efficiently. In this work, the feasibility of accelerating the ePC-SAFT-DFT calculation with the Chebyshev pseudo-spectral collocation method was discussed for the confined ionic liquid (IL)–CO2 systems. In addition, a general scheme was proposed to search the electrical boundary potential. The new algorithm was further combined with the general scheme to model the confined IL–CO2 systems. It was found that the Chebyshev pseudo-spectral collocation method can improve the efficiency of the ePC-SAFT-DFT calculation significantly. Moreover, the new algorithm can be further combined with the general scheme to efficiently describe the density profile of the IL–CO2 system inside the electroneutral nanopores.

Impacts of Newtonian heating, variable fluid properties and Cattaneo–Christov model on MHD stagnation point flow of Walters’ B fluid induced by stretching surface

MHD viscoelastic (Walters’-B) fluid flow close to the stagnation point region along an extending plate with the changeable fluid properties’ influences has been debated. Heat transfer’s features are scrutinized via Cattaneo–Christov (CC) theory. The mathematical model for the physical problem is tackled numerically via Chebyshev pseudospectral (CPS) technique. The existing outcomes are supported by recent research and have acquired a suitable agreement. The numerical outcomes reveal that temperature fields are more pronounced for Fourier’s law case. Further, the opposite behavior is noticed with the heat transfer rate. Higher values of the conjugate parameter result in an increment of the heat transfer rate and temperature field. Fluid flow’s features, as well as physical quantities, are substantially varied via variable fluid properties.

A Chebyshev pseudo-spectral approach for simulating magnetotelluric TM-mode responses on 2D structures

We introduce a Chebyshev pseudo-spectral (CPS) approach for 2D TM-mode magnetotelluric modelling, which was the first applied on the magnetotellric data in 2D model. In this method, 2D partial differential equations are solved using the Chebyshev differentiation matrix over the Gauss-Chebysev-Lobatto points. In order to test computational efficiency and its accuracy, we compare the simulatation of this method to the analytical solution and finite element (FE) and finite difference (FD) methods. The comparison of CPS numerical solution and corresponding analytical solution on a homogeneous half-space model and a layered model show that the CPS approach can provide an accurate modelling result. The discrete nodes for modelling were also analyzed on two models, which suggest that a few nodes can produce modelling results with a high accuracy. Especially for 2D and 3D modelling problems, the reduction in the number of nodes can improve the computational efficiency significantly. However, the CPS approach will lead to an unsymmetrical and non-sparse matrix, which is not favored by the iterative solver. We further compare the numerical result of the CPS method to the FD method and the averaged numerical solutions of the COMMEMI on the COMMEMI 2D-1 model. We find the numerical result of CPS method is closer to the averaged numerical solutions of the COMMEMI than that of observed by the FD. Finally, a realistic smooth model is considered to compare the modelling results using the CPS method to the published data. All these comparison results suggest that the CPS method can not only give a high accuracy for modelling results and also provide more detail information. Especially, a few nodes can be required in this method relative to the FD and FE method, which can decrease the relative errors. We then deduce that CPS method might be an alternative approach to simulate the magnetotelluric responses in 2D structures.

Optimal Motion Planning for Differential Drive Mobile Robots based on Multiple-Interval Chebyshev Pseudospectral Methods

SUMMARYThis paper presents a Chebyshev Pseudospectral (PS) method for solving the motion planning problem of nonholonomic mobile robots with kinematic and dynamic constraints. The state and control variables are expanded in the Chebyshev polynomial of orderN, and Chebyshev–Gauss–Lobatto (CGL) nodes are provided for approximating the system dynamics, boundary conditions, and performance index. For the lack of enough nodes nearby the obstacles, the interpolation of trajectory may violate the obstacles and the multiple-interval strategy is proposed to deal with the violation. Numerical examples demonstrate that multiple-interval strategy yields more accurate results than the single-interval Chebyshev PS method.

Multiple solutions for non-Newtonian nanofluid flow over a stretching sheet with nonlinear thermal radiation: Application in transdermal drug delivery

We have explored multiple solutions for non-Newtonian Casson nanofluid flow past a moving extending sheet under the influence of variable thermal conductivity and nonlinear radiation through a permeable medium with convective boundary conditions. The governing equations are transformed to ODEs by similarity transformations and then solved numerically by the Chebyshev pseudospectral (CPS) method. Dual solutions are obtained for velocity, temperature and nanoparticle concentration distributions with different values of physical parameters. In the present analysis, it was found that, the nonlinearity formula for thermal radiation gives a realistic description of nanofluid mathematical model depending on the existence of nanoscale particles. Furthermore, the concentration of nanoparticles is highly influenced by nonlinear thermal radiation due to the sizes of nanofluid, where linear radiation has a weak effect on the concentration distributions of nanoparticles. These results are very important in medicine, and more specifically for reinforcing the delivery of drugs through the skin, as the nanoparticle entrapment of drugs enhances delivery to, or absorption by, target cells. The transdermal drug delivery system offers huge clinical advantages over other dosage forms. As transdermal drug delivery offers controlled as well as predetermined rate of release of the drug into the patient, it can keep up steady-state nanofluid concentration.

A new hybrid approach for transient heat transfer analysis of convective-radiative fin of functionally graded material under Lorentz force

In the present paper, a new hybrid method via the Laplace transform and Chebyshev Pseudospectral collocation method is proposed. The proposed method is applied to analyse the transient thermal performance of a convective-radiative fin of functionally graded material under the effect of Lorentz force. The effects of Lorentz force, convective and radiative heat transfer on the thermal performance of the fin are analysed. From the parametric study, the thermal behaviour of the fin of functionally-graded material is established. The study shows that increase in the radiative and magnetic field parameters, as well as the non-homogeneity index, enhances the fin performance. Moreover, the transient thermal response of the fin is low at linear-law function whilst the thermal response is high at exponential-law function. The accuracy of the developed algorithm for the proposed method is validated using an established exact analytical solution. Excellent agreement is established from the result of the present study and the exact analytical method. The present study provides reliable physical insights into the heat transfer enhancement of microelectronic systems components operating in a convective-radiative environment.

Flow of non-Newtonian magneto-fluid with gold and alumina nanoparticles through a non-Darcian porous medium

This paper is centered on a numerical solution of non-Newtonian Casson magneto-nanofluid flow underlying an axisymmetric surface through a non-Darcian porous medium with heat generation/absorption. Using similarity transformations, the system of PDEs with the corresponding boundary conditions are reduced to system of nonlinear ODEs. The Chebyshev pseudospectral (CPS) method is used to get a numerical solution for the formulated differential system. Comparisons of the present numerical results with previously published results are made, and fine agreements for some the considered values of parameters were noted. Two cases of nanofluid are considered. The first case is Newtonian nanofluid, water with suspended gold (Au) or alumina (Al2O3) nanoparticles, and representative results are obtained for β → ∞ and Pr = 6.785 (the Prandtl number of water). The second case is non-Newtonian bio-nanofluid, blood with suspended gold (Au) or alumina (Al2O3) nanoparticles, and representative results are obtained for β = 0.1 and Pr = 25 (the Prandtl number of blood). The variation of different physical parameters on non-dimensional velocity and temperature fields as well as the skin friction coefficient and the Nusselt number are discussed. It is demonstrated that the implication of a nanoparticle into bio-fluid can modify the stream design. Also, the nanoparticles with high thermal conductivity (gold) have better enhancement on heat transfer compared to alumina, i.e., the effectiveness of adding gold to the water and blood is higher than adding alumina. One of the most important applications of nanotechnology in the field of medicine is the use of nanoparticles (gold molecules) in chemotherapy to get rid of cancer cells.

A pseudospectral method based robust-optimal attitude control strategy for spacecraft

This paper proposes an optimal integral sliding mode control (ISMC) scheme for attitude regulation of the rigid spacecraft. This control technique is capable of handling inertial matrix uncertainties as well as external disturbances. To incorporate optimality into the robust control law, the ISMC is integrated with Legendre pseudospectral method (LPSM). The minimization of the cost function and constraint handling of the spacecraft is obtained by LPSM, while the ISMC provides disturbance rejection. LPSM is chosen for its relatively high rate of convergence and its capability of solving a wide range of challenging optimal control problems. Theoretical stability analysis of closed loop system using Lyapunov theorem guarantees the convergence of attitude states. A comparative analysis between the proposed LPSM-ISMC and Chebyshev Pseudospectral Method (CPSM) based ISMC, is also presented in this paper. The effectiveness of the proposed robust-optimal control strategy is established through simulation results.

Fast Calculation Method for Transient Response of Transmission Line Based on Chebyshev Pseudospectral–Two-Step Three-Order Boundary Value Coupled Method

A Chebyshev pseudospectral–two-step three-order boundary value coupled method is proposed and presented for handling the issue associated with complicated calculation, low precision, and poor stability in the process of transient response of transmission line. The first order differential equation in time domain is obtained via dispersing the telegraph equation in space domain by utilizing the pseudospectral method (PSM) based on Chebyshev polynomial. Then the two-step three-order boundary value method (BVM3) is presented and employed to resolve the obtained differential equation, so the numerical solution of the space discrete points can be obtained. Furthermore, the Chebyshev pseudospectral–two-step three-order boundary value coupled method (PSM-BVM3) is presented and compared with the Chebyshev pseudospectral–two-step two order boundary value coupled method (PSM-BVM2), the pseudospectral–differential quadrature method (PSM-DQM), and the pseudospectral method–trapezoid rule (PSM-TR) to validate the feasibility of the new proposed method. Theoretical analysis and numerical simulation reveal that the proposed Chebyshev PSM-BVM3 has a higher performance than the conventional method. For the proposed Chebyshev PSM-BVM3, the higher precision, efficiency, and numerical stability can be obtained and achieved only with fewer discrete points in the space domain, which is suitable for solving the transient response of transmission line. The proposed PSM-BVM3 can improve the drawback of numerical instability of the PSM and can also improve the disadvantage of the BVM as it is not easy to change the latter’s timestep size.

Penetrative Internally Heated Convection in Magnetic Fluids

Convection in an internally heated magnetic fluid is analyzed for rigid – rigid and rigid – free boundary surfaces with zero temperature at top and no heat flux at the lower surface. Linear stability analysis is performed for a variety of internal heat source models and the corresponding Eigen value problem is solved by Chebyshev Pseudospectral method.