Local RBF-FD-Based Mesh-free Scheme for Singularly Perturbed Convection-Diffusion-ReactionModelswithVariableCoefficients

(is work analyze singularly perturbed convection-diffusion-reaction (CDR) models with two parameters and variable coefficients by developing a mesh-free scheme based on local radial basis function-finite difference (LRBF-FD) approximation. In the evolvement of the scheme, time derivative is discretized by forward finite difference. After that, LRBF-FD approximation is used for spatial discretization, and we obtained a system of linear equations. (en, the obtained linear system is solved by LU decomposition method in MATLAB. For numerical simulation, four singularly perturbed models are pondered to check the efficiency and chastity of the proposed scheme.


Introduction
Singularly perturbed models (SPMs) can be seen in different areas of science, medicine, and engineering. Herein, we consider the following class of parabolic SPMs with two small parameters and variable coefficients: with initial and boundary conditions (BCs) w(x, 0) � w 0 (x), x ∈ (0, 1), where ε and μ are small positive parameters and the variable coefficients a(x, t), b(x, t), c(x, t), and f(x, t) are sufficiently smooth functions on Ω and satisfies the following conditions: We also suppose that the initial and BCs have enough smoothness on zΩ and sufficient compatibility conditions on the edges of the domain Ω. ese conditions on the initial and BCs ensure that there exists a unique classical solution w(x, t) ∈ C 4,2 (Ω) [1].
In the SP case, the solutions of the abovementioned models may possess a multiscale character, presenting rapid variations in some narrow regions adjacent to both the left and right lateral boundary depending on the magnitude of the parameters ε and μ. For μ � 1, ε ⟶ 0, model (1) becomes convection dominated model (CDM) and for μ � 0, ε ⟶ 0, model (1) was converted into the reaction-diffusion model (RDM). ere are various numerical techniques available in literature for the simulation of CDM and RDM, e.g., adaptive B-spline collocation method [2], ε-uniform schemes [3], parameter uniform difference scheme [4], uniformly convergent B-spline collocation method [5], finite difference fitted schemes [6], uniformly convergent difference schemes [7,8], SDFEM [9], parameter-uniform hybrid finite difference scheme [10], finite difference domain decomposition algorithms [11], high order methods [12], Layer-adapted meshes and FEM [13], parameter uniform approximations [14], uniformly convergent scheme [15], and finite difference scheme [16]. Except these, there are many schemes for these types of problems [17][18][19][20][21][22]. e singularly perturbed CDR models occurred in mathematical modelling of physical problems, such as fluid dynamics, elasticity, quantum mechanics, heat and mass transfer in mechanical engineering, theory of plates and shells, oil and gas reservoir simulation, and magneto-hydrodynamic flow. Two parameter problems arise in chemical flow reactor theory [23] as well as in the case of boundary layers controlled by suction (or blowing) of some fluids [17,18,24]. It is important to study these CDR models with two parameters and variable coefficients so that other information can be clarify by the interaction between convection and diffusion.
Recently, singularly perturbed convention diffusion problems are solved by Ahmad and Kalarestaghi [25] using LRBFs methods. But, these problems are steady state and with constant coefficients. Secondly, upwind LRBFs and central LRBFs methods are used for solving these problems.
is motivates the authors to develop LRBF-FD scheme for unsteady-state singularly perturbed CDR models. To achieve this aim, we present a mesh-free scheme based on LRBF-FD approximation for the unsteady-state singularly perturbed CDR models with two parameters and variable coefficients. e novelty of the work is that we solved unsteady-state singularly perturbed problems with two parameters and variable coefficients. Second, we used LRBF-FD approximation in place of upwind LRBFs and central LRBFs methods. In the evolvement of the scheme, time derivative is discretized by forward finite difference. After that, RBF-FD approximation is used for spatial discretization, and we obtained a system of linear equations. Finally, the obtained linear system is solved by the LU decomposition method in MATLAB. For numerical simulation, four problems are pondered to check the efficiency and chastity of the proposed scheme.
Rest of the paper is organized as follows: in Section 2, the details of RBFs and Local RBF-FD approximation has been described. Mesh-free numerical scheme based on LRBF-FD approximation will be introduced in Section 3. In Section 4, MATLAB implementation steps are given in form of algorithm of the developed scheme. Numerical simulation and discussion are carried out in Section 5. Concluding remarks of the work are given in the final Section 6.

RBFs and Local RBF-FD Approximation
Herein, we presented the brief introduction of RBFs and LRBF-FD approximation in R n space as follows.

RBFs.
For scattered points Χ ∈ Ξ ⊂ R n , the approximation of a function χ(X) can be written as a linear combination of RBFs [26,27].
where M is the number of centers or sometimes called knots Χ, Χ � (x 1 , x 2 , . . . , x n ), n is the dimension of the problem, λ ′ s are coefficients to be determined and φ is a RBF. (4) can be written without the additional polynomial ψ(Χ). If Ψ n q denotes the space of n-variate polynomials of order not exceeding q, and letting the polynomials P 1 , P 2 , . . . , P m be the basis of Ψ n q in R n , then the polynomial ψ(Χ), in (4), is usually written in the following form: where m � (q − 1 + n)!/(n!(q − 1)!).

LRBF-FD Method Formulation.
In this work, we used LRBF-FD approximation for the discretization of spatial derivatives. LRBF-FD approximation is more general form of classical finite difference method. To explain the concept of LRBF-FD approximation, assume that there are M scattered points in the bounded domain of the problem i.e. Ξ ⊂ R n . In this approximation, the differential operator ℓ .
{ } is approximated at a particular point x c ∈ Ξ as a weighted linear combination of function values at closest n s neighbor stencil of the point x c as follows: where I(x c ) � j ∈ 1, 2, . . . , M { }: ‖x j − x c ‖ 2 ≤ δ and δ is the radius of support domain. Now, the main task is to calculate the unknown coefficients α j occurred in (7). For this purpose, assume the 2 Journal of Mathematics approximation of the unknown solution w(.) is as follows [35,36].
where φ(.) is MQ-RBF discussed in above subsection. Apply the differential operator ℓ . { } over (8) and take Now, using the (7)-(9), we can write as follows: To compute the unknown weights α j , the system algebraic (10) can be written as Solve the (11) and substitute in the (7) to the approximation of the differential operator ℓ .
{ } at the point x c . To simulate the unsteady SPMs model (1), we need to approximate the 1 st and 2 nd order derivatives. For this aim, we will divide the M scattered points Χ � (x 1 , x 2 , . . . , x M ) into M B boundary points and M I interior points and M � M B + M I . en, the weights associated to differential operators z/zx, z 2 /zx 2 in node x i based on stencil I(x i ) for internal points, will be determined by using (11) as follows: e Sparsity pattern of the matrices α (1) and α (2) is plotted in Figure 1.

Mesh-free Numerical Scheme Based on LRBF-FD Approximation for the Parabolic Singularly Perturbed Models
Herein, we will develop a mesh-free scheme based on forward finite difference (FFD) and LRBF-FD for the simulation of parabolic SPMs (1). e steps of the scheme are as follows 3.1. Semidiscretization. In this scheme, the first step is semidiscretization of model (1) in time space. For this purpose, define t n � nτ, n � 0, 1, . . . , K, where τ � T/K is the time step. Now, approximate time derivative by FFD and apply Crank-Nicolson (CN) on other part of the model, then we have where , and s n is the truncation error (T. E.) in semidiscrete problem.
With initial and BCs e system (13) can written as follows:

Journal of Mathematics 3
Equation (15) is the system of linear ordinary differential equations (ODEs) for each value of n and if we drop out the small T. E. s n , we have

Fully Discretization.
For fully discretization, we used LRBF-FD approximation. First of all discretize the given domain by the uniform node x j , j � 1, 2, . . . , M. en, discretize the equation (16) at x � x i and the spatial derivatives are approximated as follows: After that use the above approximation in (16), we have the following system of linear equations: e above system can be written in matrix form as where the elements of the matrices are For simulation purpose, we used the uniform nodes instead of nonuniform nodes since uniform nodes gives better results than nonuniform nodes for the proposed algorithm.

MATLAB Implementation Steps of the
Developed Scheme e implementation steps of the scheme in MATLAB are as follows: A1. Partition the spatial domain (0, 1) into grid points with uniform step length and the grid points are x j � j − 1/N − 1, j � 1, 2, ..., N and enter the constant shape parameter A2. Compute the local first and second order approximation named as sparse differentiation matrices α (1) and α (2) A3. Enter the linear system (19) to get the solution at grid point A4. Solve the entered system in MATLAB via LU decomposition method and iterates upto T (final time)

Numerical Simulation and Discussion
is section ponders some particular SPMs of (1) for simulation purpose. To check the chastity and competence of the developed numerical scheme, maximum absolute error L ∞ and relative error L 2 are computed by the formulas: where u analytic and u appx. denote analytic and approximated solutions respectively. e simulation work has been done in the MATLAB environment on desktop computer (Intel(R) Core(TM) i3-6100T CPU, 4 GB RAM). e shape parameter used in RBFs is calculated by the following algorithm [37]. (Algorithm 1) Input kmin, kmax, increment k � 1; while (k < kmin or k > kmax). produce interpolation matrix say A; use [U, S,V] � svd(A); K � max(S)/min(S) if K < minK shape � shape − increment; elseif K > maxK shape � shape + increment; end end ALGORITHM 1: Local RBF-FD scheme algorithm. Table 1: Maximum absolute L ∞ and L 2 errors at different times and parameters with M � 150, n s � 3.

Time
Errors    (1). e function f(x, t) is taken from exact solution given by [38].
e results of the model are presented in Tables 1 and 2  and Figure 2. e Table 1 submits the maximum absolute L ∞ and L 2 errors at different time and parameter up to ε � 2 − 35 .
e Table concludes that as we decrease the parameter the errors decrease. For ε � 2 − 30 at time t � 1.0, the present L ∞ error with M � 150, ε � 2 − 30 is 6.660E − 06 while the same error with N � 256, 6.72E − 06 (hybrid method) and 2.80E − 04 (backward Euler method in time and central difference approximation in space) [38]. is concludes that the proposed scheme produces good results with small number of grid points. Table 2 reports the CPU time at different time and perturbation parameter ε. Figure 2 depicts the physical behavior of the model at different times with parameter ε � 2 − 10 . e figure shows that, as time increases, the solution decreases and comes closer and closer near x � 0.5. , for μ � 0). Consider the one parameter SP reaction-diffusion model as w t − εw xx + (1 + xe − t )w � f(x, t), w(x, 0) � 0, w(0, t) � 0, w(1, t) � 0, over the domain (x, t) ∈ (0, t) × (0, 1] which is same as the above example except the known function f(x, t). e function f(x, t) is taken from exact solution given by [38].

Time
Errors    Journal of Mathematics 7 Figure 3 offer the results of the model. Table 3 submits the maximum absolute L ∞ and L 2 errors at different time and parameter up to ε � 2 − 35 .

Tables 3 and 4 and
e table concludes that, as we decrease the parameter the errors decrease. For ε � 2 − 30 , at time t � 1.0, the present L ∞ error with M � 150, ε � 2 − 30 is 3.854E − 05, while the same error with N � 256, 5.67E − 03 (hybrid method) [38]. is concludes that the proposed scheme produces good results with small number of grid points. Table 4 reports the CPU time at different time and perturbation parameter ε. Figure 3 depicts the physical behavior in 2D and 3D of the model at different times with parameter ε � 2 − 20 . e figure shows that, as time increases the solution decreases and comes closer and closer between x � 0.4 to x � 0.6. Second, the solution becomes steeper near the right and left boundaries.

SP Model 3.
Consider the two parameter SP reactiondiffusion model with initial and BCs as [39,40].
and w(x, 0) � 0, w(0, t) � 0, w(1, t) � 0 over the domain (x, t) ∈ (0, t) × (0, T]. e exact solution of the model is not given in literature but the solution behaviour is given in [39,40]     goes upto − 0.6316. Second, the solution comes closer and closer between the right and left boundaries.

SP Model 4.
Consider the two parameter SP reactiondiffusion model with initial and BCs as [40,41].
and w(x, 0) � 0, w(0, t) � 0, w(1, t) � 0 over the domain (x, t) ∈ (0, t) × (0, T]. e exact solution of the model is not given in literature but the solution behaviour is given in [40,41] up to time t � 1.0. Here, we present numerical results in 2D and 3D form of the model in Figure 5 with parameter ε � 10 − 4 , μ � 10 − 10 , M � 150. e figure exhibits that, as time increases up to t � 2.0, the solution decreases and goes upto − 0.274287. Second, the solution comes closer and closer between the right and left boundaries. Since for the above model, we do not know the analytical solution; therefore to explain the performance of the proposed scheme, we depicted the solution behavior of the model.

Concluding Remarks
In this article, the authors developed a mesh-free numerical scheme based on FFD and LRBF-FD for the simulation of unsteady-state 1D parabolic singularly perturbed CDR models with two parameters and variable coefficients. e novelty of the work is that we solved unsteady-state singularly perturbed problems with two parameters and variable coefficients. e proposed numerical scheme is tested by taking several numerical examples and the results found satisfactory. e proposed mesh-free scheme is new for the 1D parabolic singularly perturbed CDR models and provides similar results as given in [38][39][40][41]. e developed scheme easily worked for small perturbed parameter ε � 10 − 35 and variable coefficients and produced comparable results with [38][39][40][41]. e main advantage of the scheme is that it is easy to implement and an attractive alternative to the FDM and FEM which are classical solution schemes for the higher dimensional nonlinear PDEs.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.