Hermite Interpolant Multiscaling Functions for Numerical Solution of the Convection Diffusion Equations

abstract: A numerical technique based on the Hermite interpolant multiscaling functions is presented for the solution of Convection-diffusion equations. The operational matrices of derivative, integration and product are presented for multiscaling functions and are utilized to reduce the solution of linear Convection-diffusion equation to the solution of algebraic equations. Because of sparsity of these matrices, this method is computationally very attractive and reduces the CPU time and computer memory. Illustrative examples are included to demonstrate the validity and applicability of the new technique.


Consider the convection-diffusion equation with variable coefficients
with initial condition u(x, 0) = f (x), 0 Elmira Ashpazzadeh and Mehrdad Lakestani and boundary conditions u(0, t) = g 0 (t), u(T, t) = g 1 (t), 0 ≤ t ≤ T, where α(x) and β(x)( = 0) are continuous functions and f (x), g 0 (t), g 1 (t) are known functions while u is unknown function.Convection diffusion equation appears in many branches of engineering, such as fluid mechanics, heat transfer and problems in structural mechanics posed over thin domains [4].Since it is impossible to solve convection diffusion equations analytically for most application problems, efficient numerical algorithms are becoming increasingly important to numerical simulations involving convection diffusion equations.For this model, some authors have studied the numerical techniques such as finite difference method [1,4,17], the Bessel collocation method [19], the finite element method [6,7], the wavelet-Galerkin method [2,8], the Crank-Nicolson method [18], the piecewise-analytical method [16], ADI method [11] and Second kind Chebyshev wavelet method [20].Among these methods, the wavelet method is more attractive.
Within the last years, wavelet bases have been successfully used for the numerical solution of PDEs.The main characteristic of this method is that it reduces these problems to those of solving systems of algebraic equations, thus it greatly simplifies the problems.
It is worth pointing out that in this work, we propose an alternative approach.We shall apply collocation method to solve convection-diffusion equations numerically, in general case.In the current investigation, we reduce the problem to a set of algebraic equations by expanding the unknown function as Hermite scaling functions, with unknown coefficients.The operational matrices of derivative, integration and product are given.The idea of using operational matrices was used in the literature by several authors [5,[12][13][14][15].These matrices together with the Hermite scaling functions are then utilized to evaluate the unknown coefficients.Our results have good agreement with exact results and demonstrate the viability of the new technique.In this sense,these methods have the potential to provide a wider applicability.On the other hand, the comparison of the results obtained by these methods and others shows that the new method provide more accurate solutions than those obtained by other authors.This article is organized as follows: In Section 2, we describe the formulation of the Hermite scaling functions on [0, ℓ] and derive the operational matrices of derivative, integration and product required for our subsequent development.In Section 3, the proposed method are used to approximate the solution of the problem.As a result, a set of algebraic equations is formed and a solution of the considered problem is introduced.In Section 4, we report our computational results and demonstrate the accuracy of the proposed numerical scheme by presenting several numerical examples.Section 5, completes this article with a brief conclusion.
We need the biorthogonal Hermite functions intrinsically defined on [0, ℓ], so we put

Some formulas for cardinal Hermite functions
In this subsection, we give some formulas for our future computations.Using Eq. (2.3), we can approximate the function in which i = 1, 2, 3, 4 and l = 0, . . ., ℓ × 2 J .Then, the differentiation of vector Φ J in Eq. (2.4) can be expressed as [5,12,13,14] where the D φ is N × N matrix as defined as where Using a similar method, we can find the operational matrix of integration for multiscaling function.The integration of vector Φ J , defined in (2.4), can be achieved as [5,12] x 0 where I Φ is N × N operational matrix of integration for multiscaling functions and can be obtained by the following process.The function x 0 φ i (2 J s − l)ds using Eq.(2.3) can be approximated as for i = 1, 2, 3, 4 and l = 1, . . ., ℓ × 2 J .Then, It can be shown that , where 0 0 0 The following property of the product of two multiscaling function vectors [12] will also be used.Let where T is an N × 1 vector, and Z is a N × N operational matrix of product given by

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where Zk is 4 × 4 matrix given by Theorem.Suppose that the function f and the interpolation operator P J mapping function f into space V J be as Eq. ( 2.3), then the error bound is given by where Proof.See [3].✷

Description of proposed method
In this section, we solve the convection-diffusion equation in Eq. (1.1) with initial and boundary conditions in Eqs.(1.2) and (1.3).At the first we integrate of both side of (1.1) in the interval [0, t] and using Eq.(1.2) we have u(x, t) − f (x) + α(x) Similarly to Eq. (2.6), we expand f (x) and q(x, t) as q(x, t) ≈ Φ T J (t)QΦ J (x), ( Now using (2.6), (3.1), (3.2) and (4.3) we can write The functions α(x) and β(x) can be approximated as By using operational product matrix, the above relation can be written where α and β are N × N known matrices.Now by rearranging the Eq.(4.9) and collocating it in Using Eq. (2.6) in the boundary conditions (1.3) we get where V 1 and V 2 are N × 1 vectors and we know their entries.The entries of Φ J (t) and Φ J (x) are independent, so Eqs.(4.11) and (4.12) give By choosing N equations of Λ 1 = 0 and Λ 2 = 0, we get 2N equations.These equations together with Eqs.(4.10) give a system of algebraic equations with N 2 unknowns and equations, which can be solved for U i,j , i, j = 1, . . ., N .So the unknown function u(x, t) can be found using Eq.(2.6).

Numerical experiment
In this section, we will use the proposed method to solve the initial boundary value problem of convection-diffusion equation with variable or constant coefficients.The following numerical examples are given to show the effectiveness and practicality of the method and the results have been compared with the exact solution.Example 1.Consider the convection-diffusion equation and boundary conditions where α(x) = −0.1 and β(x) = 0.01.The exact solution of this example is u(x, t) = e −x−0.09t .The results for example 1 are displayed in Table I.For the purpose of comparison in Table I, we compare the absolute error of our method with different values of J with the method in [20].The space-time diagram of the exact and approximate solutions are shown in Fig. 1.Comparison of the approximate solution with the exact solution for t = 1 is presented in Fig. 2.    Example 2. Consider the convection-diffusion equation

Elmira Ashpazzadeh and Mehrdad Lakestani
and boundary conditions where α(x) = − x 6 and β(x) = x 2 12 .The exact solution of the above example is u(x, t) = x 3 e t .
The results for example 2 are displayed in Table II, Fig. 3 and Fig. 4. In Table II, the approximate solution obtained by the present method have been compared with the exact solution.The approximate solutions with exact solutions for different values of t are presented in Fig. 3. Also in Fig. 4, the absolute errors at various times of t have been reported.
where the initial and boundary conditions are defined such that the exact solution is u(x, t) = e t−x .
We applied the new method presented in this paper with J = 2 and solve the above equation.The approximate solution with the exact solution have been displayed in Fig. 5. Fig. 6, shows the plot of error for u(x, t) with J = 2.

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where the initial and boundary conditions are defined such that the exact solutionis u(x, t) = e 1.7712434446770x−0.09t .
The logarithm of the absolute error of the present method at the time level t = 0.1 have been displayed in Fig. 7.As the Fig. 7. shows the results of the new method is more effective than the method in [17] (see Fig. 1. in [17]).For the purpose of comparison, the absolute errors between exact and approximate solutions at various times are presented in Fig. 8, Also the approximate and exact solutions are plotted in Fig. 9.

Conclusion
This paper focused on solving the Convection diffusion equations.Hermite interpolant biorthogonal multiscaling functions on [0, 1] were employed and the operational matrices of derivative, integration and product for them are constructed.The obtained results showed that this approach can be solve the problem effectively.

Figure 1 :
Figure 1: Approximate and exact solutions of example 1 with J = 2.

Figure 2 :
Figure 2: Comparison of the approximate solution with the exact solution (left) and absolute error (right) at t = 1 with J = 2 for x ∈ [0, 1], example 1.

HermiteFigure 3 :
Figure 3: Comparison of the Approximate solution with the exact solution at various times with J = 2, example 2.

Figure 4 :
Figure 4: Absolute errors at various times of t, example 2.

Figure 7 :
Figure 7: Accuracy of method with J = 2 for example 4.

Figure 8 :
Figure 8: Absolute error between exact and approximate solutions at various times t with J = 2, example 4.

Table I .
Comparison between the approximate and exact solutions with t = 0.3 for Example 1

Table II .
Comparison between the approximate and exact solutions with t = 0.