Implementing Reproducing Kernel Method to Solve Singularly Perturbed Convection-Diffusion Parabolic Problems

In the present paper, reproducing kernel method (RKM) is introduced, which is employed to solve singularly perturbed convection-diffusion parabolic problems (SPCDPPs). It is noteworthy to mention that regarding very serve singularities, there are regular boundary layers in SPCDPPs. On the other hand, getting a reliable approximate solution could be difficult due to the layer behavior of SPCDPPs. The strategy developed in our method is dividing the problem region into two regions, so that one of them would contain a boundary layer behavior. For more illustrations of the method, certain linear and nonlinear SPCDPP are solved.

In this study, the RKM is utilized without the Gram-Schmidt orthogonalization process, which was primarily introduced by Wang et al., [32,33,34]. The strategy developed in order to solve these problems with layer behavior is explained in three steps. The first step is splitting the region into two regions, so that one of them would contain a boundary layer behavior; the second step is shifting the layer region to another region, and as the final step, a proper set of collocation points is also required for the boundary layer and regular regions. Since the RKM is a powerful numerical method, if these three steps are properly applied to the problem, this technique will be able to provide an appropriate approximation of the solution, even with severe singularities.
The inner product and norm in W (m,n) 2 (D) are given as follows,

Reproducing kernel method
(D 1 ) for the regular region in equation (1.1) where L is a invertible bounded linear differential operator, and N (y(x, t)) is a continuous nonlinear differential operator, and y(x, t) is an unknown function. The functionsK η,ξ (x, t) andk η,ξ (x, t) are reproducing kernels of W on the region D 1 and define, where L * is adjoint operator of L. It is clear that, are complete functional systems for reproducing kernel space W [32,33,34].
are countable dense sequence of points on the D 1 and are the unknown coefficients that must be determined and ≡ i, j.

Implementing RKM for SPCDPP
(2.5) Now homogenize the equation (2.5) and solve it by using the RKM in the space W Suppose that x = µz + 1 and y(x, t) = u(z, t) such that ∂ z u(z, t) = µ∂ x y(x, t) and ∂ 2 z u(z, t) = µ 2 ∂ 2 x y(x, t) therefore the equation (2.6) turns into, Now again homogenize the equation (2.7) and solve it by using the RKM in the space W

Preliminaries and notations for the boundary layer region
Suppose y(x, t) is exact solution for the problem (1.1) with a boundary layer behavior and Y n (x, t) is approximate solution as following form, whereẏṅ(x, t) andÿn(x, t) are approximate solutions that obtained from the present method in the regions D 1 and D 2 , respectively. Number of collocation points throughout the region D is N andṅ,n are number of collocation points on D 1 and D 2 , respectively and N =ṅ +n.
Theorem 3. Suppose the problem (1.1) as linear form which has homogeneous initial-boundary conditions and f (x, t) is sufficiently smooth function such that the following conditions are satisfied where k + 2m ≤ 3, then we have the following bound for solution of the problem (1.1), Proof. See [10,31].
Corollary 3. According to Theorem 3 upper bound for derivative of y(x, t) relative to t is independent of the negative powers of the ε and therefore it is finite.
We know that the solution of problem (1.1) and its derivative relative to x (∂ x y(x, t)) has the boundary layer behavior. Moreover, in the boundary layer region, the derivative of the solution relative to x is of great value, see Figure 7 (Appendix). Therefore, in order to solve problem (1.1) region D is split into two regions, so that one of them would contain a boundary layer behavior and uses a proper variable change in this region. It should be remarked that the present method is employed only to approach the solution of problem (1.1) and is not applicable to the approximate derivative of the solution, since on the boundary layer region, the value of the solution derivative (∂ x y(x, t)) is very high.
Remark 3. In summary, since the derivative of solution (1.1) relative to x has the boundary layer behavior and its value is large, the present method is not appropriate to approach the derivation of the solution according to Corollary 2.
Remark 4. According to Remark 3, error analysis theorems are valid if, in the process of proving theorems, we do not use the derivative of the approximate solution (∂ xÿn,n (x, t)). Accordingly, in the present work the proof process for the error analysis theorems are provided without using the derivative of the approximate solution (∂ xÿn,n (x, t)), but using the ∂ tẏṅ,n (x, t), instead. (Theorems 6, 7).

Remark 5.
Since in the process of proving error analysis theorems, uniform convergence to the exact solution is required, the approximate solution is illustrated, and its derivation relative to t (∂ t y(x, t)) is uniformly convergent in Theorem 2 and Corollary 1.
is the approximate solution of the problem (1.1) that has been obtained using RKM in space W (3,2) 2 and y(x, t) is the exact solution, if p(x, t), q(x, t) ∈ C 2 (D) and D ≡ [0, 1] × [0, 1] then, where (x i , t j ) ∈ D and C x , C t are constants, h x = max 1≤i≤N |x i+1 − x i |, and h t = max 1≤j≤N |t j+1 − t j |. Number of collocation points in throughout region D is N , and N =ṅ +n.
Proof. By using Theorems 4 and 7 and definition of Y N,n (x, t) on the region D, and the proof is completed.
Remark 7. According to Corollary 3 and Remark 3 we compared maximum absolute errors for derivative of the approximate solutions relative to t (E ∂tY N,n (x,t) = M ax |∂ t Y N,n (x, t) − ∂ t y(x, t)|) and x (E ∂xY N,n (x,t) = M ax |∂ x Y N,n (x, t) − ∂ x y(x, t)|) throughout D. We presented results in Table 3.
Remark 8. According to error analysis Theorems errors ratio for approximate solution Y N,n (x, t) throughout D must be about 0.5. We showed errors ratio in Tables 1 and 2

Conclusions
Hereby, a technique is introduced based on RKM to solve SPCDPPs. One of the advantages of this technique is that it could be used to solve SPCDPPs, so that SPCDPPs would not be easily solved employing common numerical methods or numerical commands in mathematical softwares that are available for free. Numerical examples demonstrated that the present method has higher precision compared to other methods. That is in account of the fact that the Gram-Schmidt process is removed.      Table 2.

Appendix: tables and figures
Max absolute error (E Y N,n (x,t) = M ax |Y N,n (x, t) − y(x, t)|, (x, t) ∈ D) with n = 10 and error ratio E Y 2N,n /E Y N,n .