The study is intended to give a brief overview of recent papers on spectral methods in different types of fluid flow especially nanofluid flow. The survey does not presume to be exhaustive; hence some selection is necessary. The papers reviewed herein relate to the recent use of some spectral based algorithms, notably the spectral relaxation method, the spectral local linearization method, the spectral quasi-linearization method, the successive linearization method, the piecewise successive linearization method and the spectral homotopy analysis method, etc. We review the important papers and problems where these methods have been used. The paper is divided into sections related to the use of each method. In addition to reviewing the journal articles in the main body of the paper, we also give details of important books published on spectral methods in fluid dynamics.

Spectral methods to solve nonlinear problems: A review

In this article, a meshless spectral radial point interpolation method is proposed for the numerical solutions of a class of time-fractional advection–diffusion–reaction equations. The current approach utilizes meshless shape functions, having Kronecker delta function property, for approximation of spatial operators. Forward difference along with quadrature formula is used for tempered fractional derivative approximation in the framework of implicit time marching scheme. Assessment of the proposed method is made by applying to various concrete test problems having constant and variable coefficients. Approximation and function reproduction quality are measured via 
$${E}_{\infty }$$

, 
$${E}_{2}$$

 and 
$${E}_{\text {rms}}$$

 error norms. Comparison of simulated results is also made with available exact solutions as well as earlier reported works. Stability analysis of the proposed method is thoroughly discussed and computationally affirmed.

A computational study of variable coefficients fractional advection–diffusion–reaction equations via implicit meshless spectral algorithm

A numerical scheme based on polynomials and finite difference method is developed for numerical solutions of two-dimensional linear and nonlinear Sobolev equations. In this approach, finite difference method is applied for the discretization of time derivative whereas space derivatives are approximated by two-dimensional Lucas polynomials. Applying the procedure and utilizing finite Fibonacci sequence, differentiation matrices are derived. With the help of this technique, the differential equations have been transformed to system of algebraic equations, the solution of which compute unknown coefficients in Lucas polynomials. Substituting the unknowns constants in Lucas series, required solution of targeted equation has been obtained. Performance of the method is verified by studying some test problems and computing E2, E
 $$_{\infty }$$
 and Erms (root mean square) error norms. The obtained accuracy confirms feasibility of the proposed technique.

Approximate solution of two-dimensional Sobolev equation using a mixed Lucas and Fibonacci polynomials

In this article, we investigate the existence, uniqueness, and stability of coupled implicit fractional integro-differential equations with Riemann–Liouville derivatives. We analyze the existence and uniqueness of the projected model with the help of Banach contraction principle, Schauder’s fixed point theorem, and Krasnoselskii’s fixed point theorem. Moreover, we present different types of stability using the classical technique of functional analysis. To illustrate our theoretical results, at the end we give an example.

Hyers–Ulam stability of coupled implicit fractional integro-differential equations with Riemann–Liouville derivatives

L1/LDG method for the generalized time-fractional Burgers equation

We propose a polynomial-based numerical scheme for solving some important nonlinear partial differential equations (PDEs). In the proposed technique, the temporal part is discretized by finite difference method together with θ-weighted scheme. Then, for the approximation of spatial part of unknown function and its spatial derivatives, we use a mixed approach based on Lucas and Fibonacci polynomials. With the help of these approximations, we transform the nonlinear partial differential equation to a system of algebraic equations, which can be easily handled. We test the performance of the method on the generalized Burgers–Huxley and Burgers–Fisher equations, and one- and two-dimensional coupled Burgers equations. To compare the efficiency and accuracy of the proposed scheme, we computed $L_{\infty }$
 , $L_{2}$
 , and root mean square (RMS) error norms. Computations validate that the proposed method produces better results than other numerical methods. We also discussed and confirmed the stability of the technique.

/pdf/an-efficient-numerical-scheme-based-on-lucas-polynomials-for-2dax718qb2.pdf

An efficient numerical scheme based on Lucas polynomials for the study of multidimensional Burgers-type equations

In this paper, an effective numerical technique based on Lucas and Fibonacci polynomials coupled with finite differences is developed for the solution of nonlinear reaction–diffusion Brusselator system. The system arises in modeling of chemical processes such as enzymatic reactions, plasma and laser physics in multiple coupling between modes and in the formation of ozone by atomic oxygen via a triple collision. The proposed scheme first converts the problem to discrete form and then with a collocation approach to a system of linear equations which is easily solvable. Performance of the method is checked by solving one- and two-dimensional test problems. Validation of the results is examined in terms of L ∞ , L 2 and relative error L R norms. In case of no exact solution, quality of computed solution is examined for small value of diffusion coefficient η . It is observed that when 1 - ξ + β > 0 , the solutions converges to equilibrium points ( β , ξ / β ) .

A computational study of two-dimensional reaction–diffusion Brusselator system with applications in chemical processes

In this article, we compute numerical solutions of time-fractional coupled viscous Burgers’ equations using meshfree spectral method. Radial basis functions (RBFs) and spectral collocation approach are used for approximation of the spatial part. Temporal fractional part is approximated via finite differences and quadrature rule. Approximation quality and efficiency of the method are assessed using discrete $$E_{2}$$, $$E_{\infty }$$ and $$E_{\text {rms}}$$ error norms. Varying the number of nodal points M and time step-size $$\Delta t$$, convergence in space and time is numerically studied. The stability of the current method is also discussed, which is an important part of this paper.

Numerical solutions of time-fractional coupled viscous Burgers’ equations using meshfree spectral method

In this work, a numerical scheme based on combined Lucas and Fibonacci polynomials is proposed for one- and two-dimensional nonlinear advection–diffusion–reaction equations. Initially, the given partial differential equation (PDE) reduces to discrete form using finite difference method and $$\theta -$$
 weighted scheme. Thereafter, the unknown functions have been approximated by Lucas polynomial while their derivatives by Fibonacci polynomials. With the help of these approximations, the nonlinear PDE transforms into a system of algebraic equations which can be solved easily. Convergence of the method has been investigated theoretically as well as numerically. Performance of the proposed method has been verified with the help of some test problems. Efficiency of the technique is examined in terms of root mean square (RMS), $$L_2$$
 and $$L_\infty $$
 error norms. The obtained results are then compared with those available in the literature.

/pdf/numerical-study-of-1d-and-2d-advection-diffusion-reaction-4o0msdj145.pdf

Ihteram Ali

Papers

An efficient numerical scheme based on Lucas polynomials for the study of multidimensional Burgers-type equations

Approximate solution of two-dimensional Sobolev equation using a mixed Lucas and Fibonacci polynomials

A computational study of two-dimensional reaction–diffusion Brusselator system with applications in chemical processes

Numerical solutions of time-fractional coupled viscous Burgers’ equations using meshfree spectral method

Numerical study of 1D and 2D advection-diffusion-reaction equations using Lucas and Fibonacci polynomials