Abstract
In the current manuscript, we implement the collocation method to obtain an approximate solution of one-dimensional time-fractional convection equation. The operational matrices of Pell polynomials are applied to solve the fractional partial differential equations. In the Caputo sense, we describe the time-fractional derivatives. So, this algorithm allows us to transform the fractional differential equation with initial (boundary) conditions into a system of algebraic equations in a good form. The convergence and error analysis of Pell polynomials are carefully investigated, and some examples are explored to show the comparison and efficiency of our method with others.
1. Introduction
The investigation of differential and integral equations of fractional order is a very important subject. These kinds of equations describe many phenomena in different fields such as mathematics, mechanics, engineering, physics, biology, fluid, and chemistry. It is difficult to evaluate the exact solution of these equations. It had to evaluate the approximate solution instead. For example, the variational iteration method is used to solve fractional wave and Burgers equations, time-fractional partial differential equations, and fractional Klein-Gordon and Black Scholes equations (see [1–3]). The linear space-time-fractional convection-diffusion equations with fractional derivatives for both space and time, the fractional kinetic equations of diffusion or dispersion with time, and space-fractional differential equations with variable diffusivity coefficient are studied by the finite difference method (see [4–6]). The finite element is another method for solving time-fractional partial differential equations, space-fractional partial advection-diffusion equation with non-homogeneous initial-boundary conditions, and linear Riesz space-fractional partial differential equations with a second-order time derivative (see [7–9]).
Various types of ordinary and partial fractional differential (integral) equations can be solved by the well-known methods which are called spectral methods. These methods decrease the number of unknowns and supply small errors. The way of solution in these methods is expressed as expansion of polynomials. One of the spectral methods is shifted Jacobi polynomials which are used to approximate the solution of Volterra-Fredholm integral equation, the space-fractional advection-dispersion equation, and linear multi-term fractional differential equations (see [10–12]). Another spectral method is shifted Chebyshev which is used for solving one-dimensional linear hyperbolic first-order partial differential equations (see [13]). Solving multi-term fractional differential equations and a system of high-order linear differential equations with variable coefficients are using Lucas method (see [14, 15]). Evaluating the solution of Volterra-Fredholm integral equation, multi-term fractional differential equations, multi-term initial value problems, and fractional pantograph differential equations are studied using generalized Lucas polynomials (see [16–19]). Another way of solving such problems is connecting between generalized Lucas polynomials together with third and fourth kinds of Chebyshev polynomials (see [20]). The coupled systems of a fractional differential equation is solved by generalized Fibonacci method (see [21]). However, for multidimensional Burgers-type equations, one- and two-dimensional nonlinear heat-type equations, two-dimensional linear and nonlinear Sobolev equations, and nonlinear reaction–diffusion Brusselator system with applications in chemical processes, all of them are using mixed Lucas and Fibonacci polynomials (see [22–25]). Also, Fermat polynomials make use for solving the Bagly-Torvik equation (see [26]). In our paper, we shall use Pell collocation (PLC) method to solve partial fractional differential equations and to the best of our knowledge this is the first to be done, also compare our results with the non-uniform L1-discontinuous Galerkin (DG) method (see [27]), and show that our method is more accurate although it takes a longer time.
We considered the fractional derivative in the Caputo sense because it is mathematically rigorous than other derivatives and this was discussed in details (see [17]).
Now our aim is to solve the one-dimensional time-fractional convection equation using Pell polynomials (see [27]) where is an unknown function, is a bounded interval, and is a non zero given const. and are known functions. is the -th fractional order Caputo derivative which is given by
We organize this paper as follows: In Section 2, we demonstrate the basic properties and preliminaries of Pell polynomials which will be frequent. Section 3 displays the derivatives of integer and fractional orders for these polynomials. The spectral collocation algorithm for solving one-dimensional time-fractional convection equation is analyzed and described in Section 4. The basic results which show the investigation of convergence, error analysis, and the evaluation of the global error for Pell expansion are included in Section 5. Two numerical examples are demonstrated and we compare the results with another method which will be in Section 6. The last section deals with the conclusions.
2. Preliminaries and Properties of Pell Polynomials
It is known that Pell polynomials are special cases of Fibonacci polynomials and the recurrence form of Pell polynomials is
Another form of this relation is with initial conditions
The Binet formula of Pell polynomials is where
We propose to choose and . Then, Pell polynomials will take the form The analytical form of these polynomials can be deduced from equation (3) [21], where is the largest integer less than or equal From Theorem (4) [21], we have
3. Integer and Fractional Derivatives of Pell Polynomials
In this section, we get the integer and fractional derivatives of Pell polynomials. This section will be divided into two subsections; the first explores the integer derivatives of Pell polynomials and the second concerns the fractional derivatives of these polynomials.
Let be expressed in terms of Pell polynomials
The approximate solution of this function is where
3.1. The Integer Derivatives of Pell Polynomial
The first derivative of can be written as (see [21]) where is the matrix of derivatives.
If , then has the form
Also, the integer derivatives of can be easily written in the form (see [21])
3.2. The Fractional Derivatives of Pell Polynomial
If is not an integer, then the fractional derivative of takes the form (see [21]) where is the Pell operational matrix of fractional derivatives of order , which has the form
can be written in the form where
Since ,
Clearly, can be rewritten in the form
So the fractional derivative has the relation
4. The Algorithm of the Method
In this section, we study the method for solving one-dimensional time-fractional convection equation with initial (boundary) conditions by using Pell polynomials.
The function of two variables can be expanded in terms of double Pell polynomials and the approximate solution of this function is where , are the unknown coefficients where is any arbitary positive integer. and are Pell polynomials defined in equation (9). Also, is an matrix introduced by where
Substitute equation (26) into equation (1), we obtain
Let
Consequently, equation (30) can be rewritten as
Collocating this equation at roots of Pell polynomials, we have a system of equations
The matrix form of this equation is where and
Finally, we can determine the constants by with initial (boundary) conditions in the forms
Now we summarize the steps of our scheme in Algorithm 1 below.
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5. The Convergence and Error Analysis
In this section, we explore our main results concerning the convergence and error analysis of Pell polynomials for fractional differential equations. Let us start with the following three Lemmas.
Lemma 1. If is an infinitely differential function, then it can be expressed in the form of the Pell polynomials. where .
Proof. From equation (43) (see [21]). where are the generalized Fibonacci polynomials. Taking , , and equations (4.3,5.3) (see [28]), the result holds.
Lemma 2. The modified Bessel function satisfies the following inequality.
Proof. See [29].
Lemma 3. The following relations are satisfied. where
Proof. By simplifying the relation in equation (8) and , we obtain the required inequalities.
The following results are important in itself and will be helpful in discussing the examples given in the coming sections.
Theorem 4. If and the exact solution of equation (1) is defined on , then the following hold: where where is a positive constant. (2)The series converges absolutely
Proof. From Lemma 1. Using the given assumptions, we have By applying Lemma 2, we obtain the desired results of the first part. For the second part, from Lemma 3, we obtain So Thus, the proof completes.
Theorem 5. If Theorem 4 is satisfied and the truncation error is where is the approximate solution of equation (1), then the following estimation is true:
Proof. From the definition of , we have Using Theorem 4 and Lemma 3 and collecting the summation, we get where and are gamma and incomplete gamma functions, respectively. So Then, the right hand side of the above inequality can be simplified as Thus, the inequality holds.
Theorem 6. If satisfies the hypotheses of Theorem 5, and by assuming and (a positive constant), then we have the following global error estimation:
Proof. From equation (1), we have So, we obtain Using Theorem (2) and Lemma (3), this estimation is transformed to By evaluating the summation, we obtain The proof of the required result is similar to the proof in the given last theorem.
6. Numerical Examples
In this section, we demonstrate two examples using Pell polynomials for solving equation (1). Then, the comparison of the used method and the method in [27] shows the efficiency of our method.
Example 1. Consider the one-dimensional time-fractional convection equation (1) with conditions where The exact solution of this equation is .
In Table 1, we observe that the absolute errors obtained by Pell collocation method at =2, 3, and 4 are better than obtained by the L1-discontinuous Galerkin method [27]. Figures 1–3 display the numerical solutions at =, , and . The absolute errors at the same values are plotted in Figure 4. It is clear that the last figure of the absolute errors decreases drastically with decreasing the number of steps.
Example 2. Consider the one-dimensional time-fractional convection equation (1) with conditions where The exact solution of this equation is
Table 2 lists the numerical solutions of our method at and different values of . These solutions and the absolute errors are plotted in Figures 5–8. From Figure 8, we observe that the convergence is exponential, and the errors are better with decreasing the number of steps. We notice that the obtained errors are the best and the least.
7. Conclusion
Due to this method, we demonstrate two examples which show the high accuracy of our suggested method and also it is easy to apply. The Mathematica software was agreeable surprised with the results of this method. We also show that any system of linear equations can borrow the given method. The comparison of the method and the non-uniform L1-discontinuous Galerkin (DG) method shows that the spectral solution in ours is more efficient. The truncation and global errors are discussed. Finally, we conclude that the proposed method can solve different kinds of fractional differential equations and integral equations. Finally, Pell polynomial is not the best choice to approximate the solutions of differential equations. I think—as far as I know—the orthogonal Chebyshev polynomials are better.
Data Availability
No data is associated with this research.
Conflicts of Interest
The author declares that she has no competing interest.
Authors’ Contributions
The author confirms that this is her own work.
Acknowledgments
The author is very grateful to the anonymous referees for careful reviewing and crucial comments, which enabled me to improve the manuscript.