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The multi-term time-fractional order diffusion-wave equation (MT-TFDWE) is an important mathematical model for processes exhibiting anomalous diffusion and wave propagation with memory effects. This article develops a robust numerical technique based on the Chebyshev collocation method (CCM) coupled with the Laplace transform (LT) to solve the time-fractional diffusion-wave equation. The CCM is utilized to discretize the spatial domain, which ensures remarkable accuracy and excellent efficiency in capturing the variations of spatial solutions. The LT is used to handle the time-fractional derivative, which converts the problem into an algebraic equation in a simple form. However, while using the LT, the main difficulty arises in calculating its inverse. In many situations, the analytical inversion of LT becomes a cumbersome job. Therefore, the numerical techniques are then used to obtain the time domain solution from the frequency domain solution. Various numerical inverse Laplace transform methods (NILTMs) have been developed by the researchers. In this work, we use the contour integration method (CIM), which is capable of handling complex inversion tasks efficiently. This hybrid technique provides a powerful tool for the numerical solution of the time-fractional diffusion-wave equation. The accuracy and efficiency of the proposed technique are validated through four test problems. [ABSTRACT FROM AUTHOR]

The multi-term time-fractional order diffusion-wave equation (MT-TFDWE) is an important mathematical model for processes exhibiting anomalous diffusion and wave propagation with memory effects. This article develops a robust numerical technique based on the Chebyshev collocation method (CCM) coupled with the Laplace transform (LT) to solve the time-fractional diffusion-wave equation. The CCM is utilized to discretize the spatial domain, which ensures remarkable accuracy and excellent efficiency in capturing the variations of spatial solutions. The LT is used to handle the time-fractional derivative, which converts the problem into an algebraic equation in a simple form. However, while using the LT, the main difficulty arises in calculating its inverse. In many situations, the analytical inversion of LT becomes a cumbersome job. Therefore, the numerical techniques are then used to obtain the time domain solution from the frequency domain solution. Various numerical inverse Laplace transform methods (NILTMs) have been developed by the researchers. In this work, we use the contour integration method (CIM), which is capable of handling complex inversion tasks efficiently. This hybrid technique provides a powerful tool for the numerical solution of the time-fractional diffusion-wave equation. The accuracy and efficiency of the proposed technique are validated through four test problems.

This research focuses on the stability analysis of an odd viscosity-induced shear-imposed Newtonian fluid flowing down an inclined slippery bed having an insoluble surfactant at the top of the liquid surface. The Orr-Sommerfeld boundary value problem is developed by applying the normal mode approach to the infinitesimal perturbed fluid flow and solved using the numerical method Chebyshev spectral collocation. The numerical results confirm the existence of Yih mode and Marangoni mode in the longwave zone. For the clean/contaminated surface of the film flow, the presence of an odd or Hall viscosity coefficient reduces the surface wave energy and delays the transition from laminar to perturbed flow. Also, it has stabilizing nature on the unstable Marangoni mode as well. The growth rate of both clean and contaminated liquid surfaces becomes more/less when the stronger external shear acts along the downstream/upstream direction of fluid flow. Further, the slip parameter leads to a lower critical Reynolds number and makes the liquid surface more unstable. An increase in the critical Reynolds number due to the stronger Marangoni force ensures that the insoluble surfactant has the potential to dampen the Yih mode instability. Moreover, the unstable shear mode occurs in the finite wavenumber regime with very high inertial force and a small angle of inclination. The two-fold variation of the shear mode instability is possible with respect to the imposed shear. However, the inclusion of the odd viscosity coefficient in the viscous falling film may advance the shear mode instability. [ABSTRACT FROM AUTHOR]

The current problem is concerned with the investigation of Casson ternary hybrid nanofluid flow through microchannel. Nanofluids containing three distinct kinds of nanoparticles have a lot of industrial and engineering applications as a result of their amazing thermal properties. The nanoparticles (alumina $A{l_2}{O_3}$Al2O3, titanium oxide $Ti{O_2}$TiO2, and copper oxide $CuO$CuO) are immersed in base fluid (ethylene glycol ${C_2}{H_6}{O_2}$C2H6O2) resulting in ternary hybrid nanofluid ($A{l_2}{O_3} + Ti{O_2} + CuO/{C_2}{H_6}{O_2}$Al2O3+TiO2+CuO/C2H6O2). For the problem under consideration, the momentum distribution, temperature distribution, entropy generation, and Bejan number have been examined. With the aid of tables and graphs, comparisons of ternary hybrid, binary hybrid, and mono nanofluids have also been investigated. With nondimensional variables, the governing equations are transformed into a dimensionless form and solved using the Chebyshev Collocation Method (CCM). Analysis shows that the increment in Reynolds number, Brinkmann number, and exponential heat source parameter results in an enhancement in the dimensionless temperature and the variational improvement of the thermal radiation parameter reduces the thermal distribution. Hence, the findings show that ternary hybrid nanofluids perform better in terms of thermal conductivity performance than either binary hybrid or mono nanofluids. [ABSTRACT FROM AUTHOR]

In this study, we propose a highly accurate technique for solving Volterra and Fredholm integral equations based on the blending of the Chebyshev pseudo methods. The application of the method leads Volterra and Fredholm integral equation to a system of linear algebraic equations that are easy to solve when compared to a integral equations. Some examples are solved and presented through graphs and tables and the obtained results are compared with those methods in the literature to illustrate the ability of the method. The results demonstrate that the new method is more efficient, converges and accurate to the exact solution. [ABSTRACT FROM AUTHOR]

Vibration behaviors of elastically supported functionally graded (FG) sandwich beams resting on elastic foundations under moving loads are investigated. The transformed-section method is first applied to establish the bending vibration equations of FG sandwich beams, then the Chebyshev collocation method is used to study free and forced vibrations. Two types of sandwich beams with FG faces-isotropic core and isotropic faces-FG core are considered. The material properties of FG materials are assumed to vary across the beam thickness according to a simple power function. Regarding the free vibration analysis, bending vibration frequencies are calculated numerically by forming a matrix eigenvalue equation. As for the forced vibration analysis, the backward differentiation formula method is employed to solve the time-dependent ordinary differential equations to obtain the dynamic deformations of the beam. To ensure the accuracy of the proposed model, some calculated results are compared with those in the published literature. Parametric studies are then performed to demonstrate the effects of material gradient indexes, stack types, layer thickness ratios, slenderness ratios, excitation frequency and speed of moving loads, and foundation and support stiffness parameters on the dynamic characteristics of FG sandwich beams. [ABSTRACT FROM AUTHOR]

This paper examines a linear convective instability of an incompressible, Newtonian, viscous fluid rotating over a stretchy spinning disk in an Ekman boundary layer flow. First time in the literature, a numerical analysis of impacts of the stretching mechanism on the stability of the Ekman flow in the presence of the Coriolis force has been carried out, where the bottom disk is allowed to uniformly extend in radial direction. The von Kármán similarity transformations have been used to convert the governing equations of motion into a system of non-linear coupled ordinary differential equations (ODEs) in non-dimensional variables. Then velocity profiles of the flow have been obtained by numerically solving these dimensionless ODEs. The neutral stability curves were then obtained by performing a linear stability analysis for the convective mechanism by using the Chebyshev collocation method. Surface stretching has a globally stabilizing influence on both Type-I and Type-II instability modes of the Ekman flow, according to the stability curves. Then, the result established from the stability curves has been verified by calculating total kinetic energy change due to induced perturbations in the flow system. y. [ABSTRACT FROM AUTHOR]

Chebyshev collocation scheme and Finite difference method plays central roles for solving fractional differential equations (FDE). Therefore purpose of this paper is to solve fractional mathematical problem of diffusion by Chebyshev collocation method which turns the original problems into the system of fractional ordinary differential and algebraic equations by imposing orthogonality property. This system is solved by implementing Finite difference method. The numerical illustrations confirm that the combination of these two methods allow us to establish one of the best truncated solution in the series form.

In this paper, we introduce two conservative Crank–Nicolson type finite difference schemes and a Chebyshev collocation scheme for the nonlinear Schrödinger equation with a Dirac delta potential in 1D. The key to the proposed methods is to transform the original problem into an interface problem. Different treatments on the interface conditions lead to different discrete schemes and it turns out that a simple discrete approximation of the Dirac potential coincides with one of the conservative finite difference schemes. The optimal H 1 error estimates and the conservative properties of the finite difference schemes are investigated. Both Crank-Nicolson finite difference methods enjoy the second-order convergence rate in time, and the first-order/second-order convergence rates in space, depending on the approximation of the interface condition. Furthermore, the Chebyshev collocation method has been established by the domain-decomposition techniques, and it is numerically verified to be second-order convergent in time and spectrally accurate in space. Numerical examples are provided to support our analysis and study the orbital stability and the motion of the solitary solutions. [ABSTRACT FROM AUTHOR]

In the modern world, research in the field of thermal enhancement is going to increasing due to their diverse applications in the field of chemical industries and engineering domains. In the current study, the hydrodynamic or magnetohydrodynamics (MHD) instability of Hartmann flow in the porous medium is considered. Here, the special nature of magnetic field known as the transverse magnetic field is used in the current analysis. The investigation of hydrodynamic stability of electrically conductive for Hartmann flow in channel along with applied magnetize field is analyzed. The fluid layers are penetrated by a constant magnetize field and flow is considered through porous medium. Reynolds number (Re) is utilized to main system of hydrodynamic stability equations. A Chebyshev collocation technique is applied through numerical method to analyze magnetohydrodynamic instability system. The obtained flow equations represent system of ODEs. The current analysis makes use of a unique type of magnetic field known as the transverse magnetic field. The instabilities of nanofluids that contains nanoparticles with water as based fluid for with physical parameters are compared and discussed for growth rate and neutral graphs. These flow equations are solved numerically by using “Cheybeshev Collocation Method”. The mathematical technique “QZ (Qualitat and Zuverlassigkeit)” is applied to find out eigenvalues from comprehensive Orr-sommerfeld technique by using MATLAB software. Different embedded physical parameters Reynolds number (Re), Hartmann number (Ha) wave number (k) are compared and discussed for growth rate and neutral graphs. The instabilities of Hartmann flow in porous medium different embedded physical parameters are compared and discussed using growth rate and neutral graphs. It predicted that the flow become stable due to the magnetic field, Reynolds number and wave number the fluids transportation. The outcomes of current study are utilized in drug-delivery systems, photodynamic therapy and delivery of antitumor.

An effective scheme based on the finite difference and Chebyshev collocation method is introduced to acquire approximate solutions of a mass transfer problem model for transdermal medication delivery. A priori error estimation is derived. The reliability of the proposed scheme is shown by comparing the drug concentration profiles with their counterparts acquired by other analytical and numerical methods.

In this article, the investigation is made to shed a light on the influence of convective boundary conditions on the boundary layer flow over a linearly stretching flat surface. The equations (partial differential equations) describing the model are transformed into system of nonlinear ordinary differential equations using similarity transformations. Equations contain various non-dimensional flow characterizing numbers viz. Prandtl number Pr, Lewis number Le, Biot number Bi, Brownian motion parameter Nb and thermophoresis parameter Nt. The influence of these parameters on thermal boundary layer, concentration distribution and temperature are analyzed in detail, by solving the equations using novel Shifted Chebyshev collocation method. The computed results, reduced Nusselt number, reduced Sherwood number, surface temperature and concentration profiles as functions of dimensionless numbers are validated by comparing the predicted results with available earlier findings (using other methods). To assert the convergence and stability of the scheme used (for much larger, but moderate, parameters values), predicted results are presented in various tabular forms. For presenting finer details of the computed values some results are also given graphically. The innovative semi-numerical scheme is robust and efficient compared with other conventional methods, used in previous studies and enables the analysis of the complex problem adequately.

Analysis and numerical results are presented for the free asymmetric vibrations of functionally graded (FG) annular nanoplates subjected to nonlinearly varying temperature. The mechanical properties of the plate material were assumed to be temperature-dependent and to vary according to the power-law model in the thickness direction. Because the material is asymmetric in the thickness direction of the nanoplate, the physical neutral plane was obtained and incorporated in the analysis. The governing equations for the presented model were derived using Hamilton's principle based on first-order shear deformation theory together with Eringen's nonlocal elasticity theory. Modified size-dependent boundary conditions were derived to handle the paradoxical behavior of the free vibration of nanoplates with a free boundary due to nonlocal parameter and thermal environment. Two different approaches in the quadrature method along with the Chebyshev collocation method were adopted, and it was found that Chebyshev collocation method had a faster rate of convergence than the other two methods. Hence, the Chebyshev collocation method was employed to obtain the numerical values of frequencies. The effect of various parameters together with nonlocal boundary conditions on the nondimensional frequencies was studied. The results were compared with those available in the literature to validate the accuracy of the results and the efficiency of the authors' technique. [ABSTRACT FROM AUTHOR]

Hydrodynamics instability is studied for an electrical conducting fluid against small disturbance between the channels by using normal magnetized force. Chebyshev collocation technique is used to determine the stability of three-dimensional Hartmann flow problem. The distinctive case of the perturbations is calculated. It is also considered that perturbations intensity be determined by just on quantified by different parameters. We have considered one of the flow stabilities conditions to analyze our problem. QZ (Qualitat and Zuverlassigkeit) technique is applied to investigate the problem to draw stability curves. αc,βc are critical wave numbers in streamwise and span-wise respectively and for a big range of Hartmann value (Ha), we obtained critical Reynolds number (Rec). It is found that Couette flow is destabilized for a specific value of Magnetize force while with greater or lesser magnitude than the particular one will stabilize the flow. Disturbances with particular oblique angle θ will grow while the others will decay for the three-dimensional disturbance. We observed from over results that for Ha(>2.0) Rec, increases steadily and when Ha more than 3.886, Rec decreases fast to the minimum. It is also established that for drop down from Ha = 3.886 and Rec becomes very large. For distinct Hartmann values, there exist two Rec numbers due to closed contour. The outcomes of current study are utilized in drug-delivery systems and photodynamic therapy.

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. [ABSTRACT FROM AUTHOR]

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms. [ABSTRACT FROM AUTHOR]

The research focused on nanomaterial solutions and their flow characteristics in relation to their usage. The application of such composites in biological rheological models, in particular, has received a lot of interest. The use of nanofluids in cooling tiny electronic devices like microchips and associated devices cannot be emphasized. Our goal is to explore the influence of a binary chemical reaction and Arrhenius activation energy on a hybrid nanofluid over a melting wedge in a spongy media. It is anticipated that the water‐based nanoparticle contains gyrotactic microbes. By using appropriate similarity variables, the resultant dimensional nonlinear boundary‐layer model is reduced and turned into a dimensionless form. A Chebyshev spectral collocation approach is useful in solving the highly nonlinear model. In terms of physical importance, the effects of important factors on developing profiles are displayed graphically and explained. Computational outcomes are obtained via MATHEMATICA. The plot of residual error is also shown to demonstrate the method's rapid convergence. According to the study's findings, by increasing the melting parameter, the rate of heat transportation at the wall decreases greatly on the average of 12.81%, but the Sherwood number becomes effective for the chemical reaction rate with a rate of about 24.81%. [ABSTRACT FROM AUTHOR]

Abstract The chemical entropy generation analysis is an approach to optimize the performance of different thermal systems by investigating the related irreversibility of the system. The influences of second-order slip with the chemical reaction on the boundary layer flow and heat transfer of a non-Newtonian nanofluid in a non-Darcian porous medium have been investigated numerically. Simultaneous solutions are presented for first and second-order velocity slips. The second-order boundary conditions serve as a closure of a system of the continuity, transport, and energy differential equations. The current work differs from the previous studies in the application of a new second-order slip velocity model. The Casson fluid model is applied to characterize the non-Newtonian fluid behavior. The effect of the second slip parameter on the present physical parameters was discussed through graphs and it was found that this type of slip is a very important one to predict the investigated physical model. The present study provides two fast convergent methods on the semi-infinite interval, namely Chebyshev collocation method and optimal homotopy analysis method are used to analyze the fluid flow, heat, and mass transport. Compared with available analytical and numerical solutions, current methods are effective, quickly converging, and with great accuracy. It was shown that the account for the second-order terms in the boundary conditions noticeably affects the fluid flow characteristics and does not influence on the heat transfer characteristics.

Linear convective instability analysis of non-Newtonian fluids has immense practical applications in the field of aerodynamics and engineering mechanics. The paper deals with linear convective instability analysis of laminar Kármán swirling flow of a non-Newtonian Carreau fluid over a radially stretchable rotating disk of infinite radius when the Coriolis force is significant in the boundary layer. In this paper, the velocity profiles for both shear-thinning and shear-thickening fluids the above-mentioned flow of Carreau model have been determined under stretch boundary condition. By using the Chebyshev collocation method, a study of convective instability has been carried out in order to perform a stability analysis of the flow and determine the neutral stability curves. The stability curves reveal that the bottom disk's surface stretching has a globally stabilizing effect on the fluid exhibiting shear-thinning flow behaviour and a globally destabilizing effect on shear-thickening flow behaviour. To verify the above physical facts, the flow has been subjected to an energy analysis at the same time. [ABSTRACT FROM AUTHOR]

The aim of this work is to provide an efficient technique for numerical solutions of the reaction–dispersion equations including convection and diffusion. The Chebyshev spectral collocation method (ChSCM) is used in both space and time to exhibit anti-kink-type waves that appear as the result of the proposed model equations. It is shown that the revealed solutions are promising to efficiently catch such challenging physical behavior. Quantitative convergence results in both time and space are illustrated for the reaction–dispersion problems. [ABSTRACT FROM AUTHOR]

The main aim of the present study is to develop a mathematical formulation that can describe the grain-size distribution (GSD) of non-uniform sediments in suspension over erodible sediment beds in a wide open channel under non-equilibrium conditions. The two important characteristics of sediment particles, such as non-local mixing and the hiding-hindering effect during settling, has been considered in the current study. The traditional advection–diffusion equation (ADE) is taken in the modified form as fractional ADE which is capable of capturing non-local movement of particles unlike the traditional diffusion theory where particles jump within a restricted distance along the vertical direction over a small time interval. The equation is further modified for any k th class of non-uniform sediment and the effect of non-local movement is included in the expression of depth averaged sediment diffusivity also. The settling velocity of a particle is taken in a form that contains the effect of hiding and hindering due to the presence of non-uniform particles in the flow. The space-fractional derivative is taken in the Caputo sense where the order α of the fractional derivative varies from 1 < α ≤ 2. The non-local effect is considered in the bottom and top boundary conditions also and the governing equation with boundary and initial condition have been solved by Chebyshev collocation method along with Euler's backward method. The temporal variation of concentration profiles for different size particles is studied by varying α and it is observed that the magnitude of concentration decreases for each size as α increases. Non-local effect on bottom concentration for different size particle is also studied and it is seen that overshooting of bottom concentration gradually decreases as α increases for a particular size. Sensitivity analysis of α on GSD at different heights also has been done. The model has been validated for GSD under equilibrium conditions with available experimental data and it is found that the model aligns well when the non-local mixing effect is taken into account. At the end, an error analysis has been conducted to confirm the accuracy of the presented model. • Suspended grain-size distribution using space fractional advection-diffusion equation • Model incorporates non-locality due to turbulent bursting phenomenon. • First kind shifted Chebyshev polynomials & Euler backward method used to solve model. [ABSTRACT FROM AUTHOR]

The chemical entropy generation analysis is an approach to optimize the performance of different thermal systems by investigating the related irreversibility of the system. The influences of second-order slip with the chemical reaction on the boundary layer flow and heat transfer of a non-Newtonian nanofluid in a non-Darcian porous medium have been investigated numerically. Simultaneous solutions are presented for first and second-order velocity slips. The second-order boundary conditions serve as a closure of a system of the continuity, transport, and energy differential equations. The current work differs from the previous studies in the application of a new second-order slip velocity model. The Casson fluid model is applied to characterize the non-Newtonian fluid behavior. The effect of the second slip parameter on the present physical parameters was discussed through graphs and it was found that this type of slip is a very important one to predict the investigated physical model. The present study provides two fast convergent methods on the semi-infinite interval, namely Chebyshev collocation method and optimal homotopy analysis method are used to analyze the fluid flow, heat, and mass transport. Compared with available analytical and numerical solutions, current methods are effective, quickly converging, and with great accuracy. It was shown that the account for the second-order terms in the boundary conditions noticeably affects the fluid flow characteristics and does not influence on the heat transfer characteristics. [ABSTRACT FROM AUTHOR]

In this paper the effect of stratification on vertical variation of sediment concentration in the suspended-load layer in turbulent flows through open channels under unsteady and nonequilibrium condition is investigated. Unlike previous studies, the model includes the effect of stratification in terms of reduction of the eddy diffusivity. A new approach for solving the governing advection-diffusion equation using orthogonal shifted Chebyshev polynomials of fourth kind is presented. In this approach, the solution of the problem is obtained using a finite degree Chebyshev polynomial. The distribution of the sediment diffusivity is considered including the effect of stratification. Three different types (constant, linear and parabolic) of sediment diffusivity distribution is considered both in neutral and stratified flow condition. The final transport equation is solved under both neutral and stratified condition and appropriate bottom boundary condition using the conventional backward Euler scheme. The method is unconditionally convergent and converges more rapidly than previous methods as Laplace transformation and generalized integral transformation technique. The solutions are compared with previous methods and satisfactory results are obtained. The proposed solutions are also validated with experimental data under steady flow conditions. Results show that under stratified flow condition, sediment concentration also decreases when flow is still unsteady. [ABSTRACT FROM AUTHOR]

In this paper, we propose an efficient Chebyshev collocation scheme to solve diffusion problem including time fractional diffusion equation considering the fractional derivative in the Liouville-Caputo sense. By making use of shifted Chebyshev polynomial series and their orthogonality properties, the problem is reduced to the system of fractional ordinary differential equations which can be solved by residual power series method (RPSM) with the help of the given scheme and boundary conditions. The numerical examples shows that the method is reliable and effective to construct the numerical solution of fractional diffusion equation.

Abstract In the present work, a numerical technique for solving a general form of nonlinear fractional order integro-differential equations (GNFIDEs) with linear functional arguments using Chebyshev series is presented. The recommended equation with its linear functional argument produces a general form of delay, proportional delay, and advanced non-linear arbitrary order Fredholm–Volterra integro-differential equations. Spectral collocation method is extended to study this problem as a matrix discretization scheme, where the fractional derivatives are characterized in the Caputo sense. The collocation method transforms the given equation and conditions to an algebraic nonlinear system of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. The introduced operational matrix of derivatives includes arbitrary order derivatives and the operational matrix of ordinary derivative as a special case. To the best of authors’ knowledge, there is no other work discussing this point. Numerical test examples are given, and the achieved results show that the recommended method is very effective and convenient.

The Bödewadt boundary-layer flow is induced by the rotation of a viscous fluid rotating with a constant angular velocity over a stationary disk. In this paper, the Bödewadt boundary-layer flow has been studied in the presence of the Coriolis force to observe the effect of radial stretch of the lower disk on the flow. For the first time in the literature, a numerical investigation of the effects of both stretching mechanism and the Coriolis force on the flow behaviour and on the convective instability characteristics of the above flow has been carried out. In this paper, the Kármán similarity transformations have been considered in order to convert the system of PDEs representing the momentum equations of the flow into a system of highly non-linear coupled ODEs and solved numerically to obtain the velocity profiles of the Bödewadt flow. Then, a convective instability analysis has been performed by using the Chebyshev collocation method in order to obtain the neutral curves. From the neutral curves it is observed that radial stretch has a globally stabilising effect on both the inviscid Type-I and the viscous Type-II instability modes. This underlying physical phenomena has been verified by performing an energy analysis of the flow. The results obtained excellently supports the previous works and will be prominently treated as a benchmark for our future studies. [ABSTRACT FROM AUTHOR]

Theoretical investigation of variable mass diffusivity, thermal conductivity, and viscosity on unsteady squeezed flow of dissipative Casson fluid is presented. Physically, for any effective heat and mass transfer process, a proper account of thermophysical properties in such a system is required to attain the desired production output. The magnetized free convective flow of unsteady Casson fluid encompassing Joule dissipation, radiation, and chemical reactive influence is induced as a result of squeezing property. The governing model assisting the magnetized flow is formulated and transformed via an appropriate similarity transformation. The resulting set of ordinary differential equations is solved numerically using Chebyshev based Collocation Approach (CCA). However, variable viscosity, thermal conductivity, and mass diffusivity effects are seen to diminish the fluid flow velocities, temperature, and concentration respectively along with the lower plate. Heat and mass transfer coefficient, skin friction downsized to an increasing value of variable thermal and mass diffusivity parameters while variable viscosity pronounces the skin friction coefficient. Furthermore, the present analysis is applicable in polymer processing, such as injection molding, extrusion, thermoforming among others. [ABSTRACT FROM AUTHOR]

This study deals with the quartic autocatalysis of homogeneous–heterogeneous chemical reaction that occurs in the bioconvective flow of micropolar nanofluid between two horizontally parallel plates. The quartic autocatalysis is found to be more effective than cubic autocatalysis since the concentration of the homogeneous species is substantially high. The upper plate is assumed to be in motion and the lower plate is kept stationary. Such a flow of micropolar fluid finds application in the pharmaceutical industry, microbial enhanced oil recovery, hydrodynamical machines, chemical processing, and so forth. The governing equations for this flow are in the form of the partial differential equation and their corresponding similarity transformation is obtained through Lie group analysis. The governing equations are further transformed to coupled nonlinear differential equations that are linearized through the Successive linearization method and are solved using the Chebyshev Collocation method. The effects of various parameters, such as micropolar coupling parameter, spin gradient parameter, reaction rates, and so forth, are analyzed. It is observed that the fluid flows with a greater velocity away from the channel walls, whereas near the channel walls the velocity decreases with an increase in the coupling parameter. Furthermore, the spin parameter increases the spin gradient viscosity that reduces the microrotation of particles that further decreases the microrotation profile. [ABSTRACT FROM AUTHOR]

The influence of the uniform longitudinal magnetic field on the stability against small disturbances of an electrically conducting Newtonian fluid flow between two parallel horizontal plates is investigated. The sixth‐ order system of disturbance equations is solved by the Chebyshev collocation method, and the critical Reynolds number Rec, the critical wave number αc, and the critical wave speed cc are computed for a wide range of the magnetic Reynolds number Rm and Alfven number A. Curves of wave number against Reynolds number for neutral stability are presented for different values of the parameters. The onset of instability is also discussed in detail using the growth rate curves for various parameters of the problem. It is observed that the effect of both conductivity of the fluid and the strength of the magnetic field is to decay the onset of instability. A comprehensive study is carried out at the critical state of the fluid using the graph of Rec, αc, and cc with respect to Rm for various values of A. The critical values at the onset of instability are also presented for both the Galerkin method and the Chebyshev collocation method. [ABSTRACT FROM AUTHOR]

The present paper investigates the transverse vibration of a non-uniform axially functionally graded Timoshenko beam with cross-sectional and material properties varying in the beam length direction. The Chebyshev collocation method is used to spatially discretize the governing partial differential equations of motion of the beam into time-dependent ordinary differential equations in terms of Chebyshev differentiation matrices. An algebraic eigenvalue equation in matrix form is then formed to study the free vibration behavior of non-uniform axially functionally graded Timoshenko beams. Several results of natural frequencies of the beams are evaluated and compared with those in the published literature to assure the accuracy of the proposed model. The effects of taper ratio, material graded index, slenderness ratio, material compositions and restraint types on the natural frequencies of tapered axially functionally graded Timoshenko beams are examined. [ABSTRACT FROM AUTHOR]

• Study includes the effect of non-local mixing due to turbulent bursting phenomena. • The non-locality is included in unsteady fractional governing equation as well as in boundary conditions. • The novel Chebyshev spectral method is applied to solve the non-linear problem. • Non-locality enhances the bottom and transient sediment concentration. • Model shows faster convergence than other previous methods. This study aims to develop a mathematical model that can describe the vertical distribution of suspended sediment particles in an open channel turbulent flow under unsteady nonequilibrium condition with non-local mixing effect. A space non-local transport of sediment is considered that arises due to turbulent bursting where the hopping height of a particle is not limited to a small distance as described by classical theory of turbulence, rather it can jump upto any height. To take into account this fact unlike previous researchers who used Rouse equation, Hunt equation or traditional advection–diffusion equation as the governing equation, the present study uses fractional advection–diffusion equation (fADE) where space fractional derivative for the local diffusive flux has been used that counts the effect of non-locality. The fractional derivative is considered in Caputo sense and the order α of the fractional derivative corresponds to α -order L e ´ vy stable distribution with value 1 < α ≤ 2. The non-local effect has been considered in the boundary conditions as well as in the non-linear models of sediment diffusivity. Four types of sediment diffusivity are considered which are generalized non-linear models for broad applicability of the study. The fADE together with the boundary and initial conditions, is solved by Chebyshev collocation method and Euler backward method. This proposed method is unconditionally convergent and converges more rapidly than other previous methods used in similar studies. Effects of non-local mixing have been investigated for transient and bottom concentration distributions. Proposed models have been validated for sediment distribution under steady and unsteady condition with existing experimental data and satisfactory results are obtained for all choices of sediment diffusivity. The variation of transient and bottom concentration with non-local effects are physically justified. Validation results show that the model can be applied to describe vertical concentration distribution in unsteady and steady turbulent flows in practical situation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Solving the generalized density evolution equation (GDEE) is essential to obtain the structural random response by the probability density evolution method (PDEM). To improve the collocation method, this study proposes a modified Chebyshev collocation method (MCCM) scheme for the solution of the GDEE based on the pseudo-spectral method, and derives a refine time integration format for GDEE. The GDEE is numerically solved by combining the fine time integration method and MCCM, and the accuracy, efficiency and numerical stability of the algorithm are investigated. Two numerical examples indicate that the results for the proposed method are in good agreement with analytical solutions and Monte Carlo simulation results. The proposed four numerical collocation schemes could effectively suppress numerical dispersion and numerical dissipation. In addition, the modified Chebyshev collocation method based fourth-order Runge-Kutta scheme presents excellent computational accuracy, efficiency and numerical stability, especially highlighting significant advantages in the computational efficiency. • A modified Chebyshev collocation method has been proposed for solving the generalized density evolution equation. • The modified Chebyshev collocation method based fourth-order Runge-Kutta scheme presents excellent computational accuracy, efficiency and numerical stability, especially highlighting significant advantages in the computational efficiency. • The modified Chebyshev collocation schemes could effectively suppress numerical dispersion and numerical dissipation. [ABSTRACT FROM AUTHOR]

In this article, the solution for general form of fractional order delay-differential equations (GFDDEs) is introduced. The proposed GFDDEs have multi-term of integer and fractional order derivatives for delayed or non delayed terms. An operational matrix is presented for all terms. The spectral collocation method is used to solve the proposed GFDDEs as a matrix discretization method. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.

The nonlinear convection study on the flow of a dissipative Casson nanofluid through a porous medium of an inclined micro-annular channel is presented. The cylindrical surfaces were conditioned to temperature increase and velocity slip effects. A uniform magnetic field strength was applied perpendicular to the cylinder surface. The heat source and Darcy number influence are explored in the examination of the blood rheological model (Casson) through the annular cylinder. Appropriate dimensionless variables are imposed on the dimensional equations encompassing Casson nanofluid rheology through an annular microchannel. The resulting systems of equations were solved and computed numerically via Chebyshev-based collocation approach. Thus, the solutions of flow distributions, volumetric flow rate, and other flow characteristics were obtained. The result shows that both nonlinear convection parameters decrease the nanoparticle volume fraction, whereas they increase the energy and momentum distributions. Moreover, the volumetric flow rate is upsurged significantly by a wider porous medium, annular gap, a higher Casson parameter, and nonlinear convection influence. [ABSTRACT FROM AUTHOR]

The parabolic approximation has been used extensively for underwater acoustic propagation and is attractive because it is computationally efficient. Widely used parabolic equation (PE) model programs such as the range-dependent acoustic model (RAM) are discretized by the finite difference method. Based on the idea of the Pad e ´ series expansion of the depth operator, a new discrete PE model using the Chebyshev collocation method (CCM) is derived, and the code (CCMPE) is developed. Taking the problems of four ideal fluid waveguides as experiments, the correctness of the discrete PE model using the CCM to solve a simple underwater acoustic propagation problem is verified. The test results show that the CCMPE developed in this article achieves higher accuracy in the calculation of underwater acoustic propagation in a simple marine environment and requires fewer discrete grid points than the finite difference discrete PE model. Furthermore, although the running time of the proposed method is longer than that of the finite difference discrete PE program (RAM), it is shorter than that of the Chebyshev–Tau spectral method. [ABSTRACT FROM AUTHOR]

The method of matched asymptotic expansions is applied to the investigation of transitional separation bubbles. The problem-specific Reynolds number is assumed to be large and acts as the primary perturbation parameter. Four subsequent stages can be identified as playing key roles in the characterization of the incipient laminar–turbulent transition process: due to the action of an adverse pressure gradient, a classical laminar boundary layer is forced to separate marginally (I). Taking into account viscous–inviscid interaction then enables the description of localized, predominantly steady, reverse flow regions (II). However, certain conditions (e.g. imposed perturbations) may lead to a finite-time breakdown of the underlying reduced set of equations. The ensuing consideration of even shorter spatio-temporal scales results in the flow being governed by another triple-deck interaction. This model is capable of both resolving the finite-time singularity and reproducing the spike formation (III) that, as known from experimental observations and direct numerical simulations, sets in prior to vortex shedding at the rear of the bubble. Usually, the triple-deck stage again terminates in the form of a finite-time blow-up. The study of this event gives rise to a noninteracting Euler–Prandtl stage (IV) associated with unsteady separation, where the vortex wind-up and shedding process takes place. The focus of the present paper lies on the triple-deck stage III and is twofold: firstly, a comprehensive numerical investigation based on a Chebyshev collocation method is presented. Secondly, a composite asymptotic model for the regularization of the ill-posed Cauchy problem is developed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Targeted drug delivery is one of the promising applications for cancer diagnosis and therapy, as magnetic nanoparticles can be used as therapeutic agents in presence of thermal radiation, and an inclined magnetic field. This article therefore aims to perform the stability analysis of the blood flow considered as a non-Newtonian Casson fluid transporting two types of magnetic nanoparticles, namely hematite and magnetite through an anisotropic porous artery. Starting from the Navier-Stokes equations to which the energy and Maxwell equations are added, a set of two eigenvalue equations governing the stability of the flow are obtained and solved numerically by the spectral collocation method. The impacts of different parameters such as volume fraction of magnetic nanoparticles, strouhal number, Casson, permeability, mechanical anisotropy, wave number, Hartmann number, direction of magnetic field and thermal radiation parameters on the stability of non-hybrid and hybrid suspensions are shown. The results show that the infusion of magnetic nanoparticles in the blood increases its inertia, which dampens the disturbances and delays the transition in the flow of the suspension. The Casson and mechanical anisotropy parameters maintain the instabilities which precipitate the nanoparticles and allow the flow to take place effortlessly in the horizontal direction. The wave number, Strouhal number and the permeability parameter have stabilizing effects on the dynamics of the two suspensions which prevents the approximation between the magnetic nanoparticles, thus avoiding the phenomenon of sedimentation of the nanoparticles in the blood vessels which would arise if the flow becomes turbulent. The direction of the magnetic field controls the flow by increasing the intensity of the Lorentz force. The magnetic field through the Lorentz force absorbs the kinetic energy of the flow which dampens the disturbances and thus prevents the transition in the flow. Thermal radiation dissipates temperature fluctuations which increases the volume of magnetic nanoparticles in the area where tumor tissue is located. The results have an important influence in medicine for the treatment of cancerous disease and arterial disease without the need for surgery, with can minimize the expenditures and post-surgical complications in patients.

In this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient. [ABSTRACT FROM AUTHOR]

The stability of fluid flow in an anisotropic porous medium of Brinkman type is investigated. Anisotropy in the permeability is considered such that its longitudinal principal axis is oriented arbitrarily with the horizontal, while transversely it is isotropic. A fourth‐order eigenvalue problem obtained by performing a linear stability analysis is solved numerically using the Chebyshev collocation and the compound matrix methods. The critical Reynolds number and the critical wave number are computed for different values of porous parameter, orientation angle of the principal axis and the mechanical anisotropy parameter. The porous and the mechanical anisotropy parameters disclose contrasting contributions to the stability of fluid flow. The orientation angle instills either stabilizing or destabilizing effects on the fluid flow depending on the value of anisotropy parameter. Besides, the streamlines of the perturbation modes are presented for various values of orientation angle and anisotropy parameter. For the Darcy case, a simple analytical method is employed and established that the flow is stable for all infinitesimal perturbations. [ABSTRACT FROM AUTHOR]

This paper studies the vibration behaviors of bidirectional functionally graded (BDFG) Timoshenko beams based on the Chebyshev collocation method. The material properties of the beam are assumed to vary simultaneously in the beam length and thickness directions. The Chebyshev differentiation matrices are used to reduce the ordinary differential equations into a set of algebraic equations to form the eigenvalue problem for free vibration analysis. To validate the accuracy of the proposed model, some calculated results are compared with those obtained by other investigators. Good agreement has been achieved. Then the effects of slenderness ratios, material distribution types, gradient indexes, and restraint types on the natural frequency of BDFG beams are examined. Through the parametric study, the influences of the various geometric and material parameters on the vibration characteristics of BDFG beams are evaluated. [ABSTRACT FROM AUTHOR]

This paper studies a new computational method for the approximate solution of the space fractional advection–dispersion equation in sense of Caputo derivatives. In the first method, a time discretization is accomplished via the compact finite difference, while the fourth kind shifted Chebyshev polynomials are used to discretize the spatial derivative. The unconditional stability and convergence order of the method are studied via the energy method. Three examples are given for illustrating the effectiveness and accuracy of the new scheme when compared with existing numerical methods reported in the literature. [ABSTRACT FROM AUTHOR]

The solution of singular two-point boundary value problem is usually not sufficiently smooth at one or two endpoints of the interval, which leads to a great difficulty when the problem is solved numerically. In this paper, an algorithm is designed to recognize the singular behavior of the solution and then solve the equation efficiently. First, the singular problem is transformed to a Fredholm integral equation of the second kind via Green’s function. Second, the truncated fractional series of the solution about the singularity is formulated by using Picard iteration and implementing series expansion for the nonlinear function. Third, a suitable variable transformation is performed by using the known singular information of the solution such that the solution of the transformed equation is sufficiently smooth. Fourth, the Chebyshev collocation method is used to solve the deduced equation to obtain approximate solution with high precision. Fifth, the convergence analysis of the collocation method is conducted in weighted Sobolev spaces for linear singular equations. Sixth, numerical examples confirm the effectiveness of the algorithm. Finally, the Thomas–Fermi equation and the Emden–Fowler equation as some applications are accurately solved by the method. [ABSTRACT FROM AUTHOR]