A hybrid technique based on Lucas polynomials for solving fractional diffusion partial differential equation

This paper presents a new numerical technique to approximate solutions of diffusion partial differential equations with Caputo fractional derivatives. We use a spectral collocation method based on Lucas polynomials for time fractional derivatives and a finite difference scheme in space. Stability and error analyses of the proposed technique are established. To demonstrate the reliability and efficiency of our new technique, we applied the method to a number of examples. The new technique is simply applicable, and the results show high efficiency in calculation and approximation precision.


Introduction
During the last four decades, the topic of fractional calculus has gained massive popularity and importance due to its gradually flourishing applications in numerous diverse fields in system biology, physics, chemistry and biochemistry, medicine, and finance.In most of these new models, fractional orders are more ample than previously used integer orders because fractional derivatives describe memory and properties immanent in several materials and processes that are controlled by aberrant diffusion.Hence, we need to find the solution to these fractional differential equations.In general, the analytic solutions of most fractional differential equations cannot be obtained.Therefore, numerical and approximate methods are valuable in identifying the solution conduct of such fractional equations and exploring their applications.
One of the most important groups of partial equations is the diffusion equations, which are used to describe physical phenomena, biology, and economics.This equation describes the behavior of the mass movement of micro-particles in a material.Due to the wider applications of these equations, many researchers in different fields have been interested in studying them [25,35].
Many numerical and approximate methods have been presented, with different techniques for each method.Authors in [25] have developed a technique to solve fractional diffusion problems of variable order.Recently, the Gradient Discretization Method has been proposed to approximate multidimensional diffusion wave and time fractional diffusion equations [13].The finite difference method has been proposed to solve a three-dimensional time fractional diffusion equation [20], and the fractional diffusion equation is solved by a neural network [36][37][38].For the application of the diffusion equation for noise reduction and fractional terminal value problems see [9,10,31,39].Spectral methods are pivotal in treating different types of differential equations, and the most commonly used trial functions are the various orthogonal polynomials, such as Fourier used trigonometric polynomials, Jacobi, Chebyshev and Legendre.The collocation spectral method [12,14,17,18,26] is applied to solve the fractional differential equations.
The golden ratio and Lucas polynomials with their generalizations are of major interest; both appear in different applications in various disciplines such as physics, biology, computer science, and statistics.Therefore, many researchers have written about them in a variety of papers.The main motivation for using the suggested scheme is that both fractional and higher-order derivatives can be easily calculated using Lucas and Fibonacci polynomial relations.Meanwhile, the promising results for these polynomials, which have been considered in several applications in the area of ordinary differential equations, gave us strong motivation for discussing these polynomials for solving partial fractional differential equations.Moreover, the proposed technique achieves better accuracy for a small number of collocation points, which reduces the computational procedures time and cost.In the area of ordinary differential equations, Elhameed and Youssri [2,4,5] introduce a relation between Lucas polynomials and Chebyshev and give accurate solutions to boundary value problems.[3,4,6] Lucas polynomials have been used for solving coupled fractional differential equations.For integro differential equations [32] presents Lucas sequence, and [15] proposes Lucas polynomial approach to get an approximate solution of higher-order differential equations.This article is organized as follows: In Sect.2, we introduce the basic principles and notations of fractional calculus with Lucas polynomials.In Sect.3, we demonstrate the formulation of the proposed numerical scheme.And Sect. 4 provides stability and error analysis.In Sect.5, numerical examples are introduced to ensure the accuracy of the presented method, and a conclusion is given in Sect.6.

Basic principles and notations
In this section, we will introduce some necessary definitions and notations used to describe the numerical schemes.

Caputo fractional derivative
There are different ways to define fractional derivatives, and the most commonly used fractional derivatives are the Grünwald-Letnikov derivative, the Riemann-Liouville derivative and the Caputo derivative [33] Definition 2.1 Caputo fractional derivatives with order α > 0 of the given function f (t) are defined as: where n is a non-negative integer and n − 1 < α < n.

Lucas polynomials
The well-known Lucas polynomials of ordern which are defined on interval (0,1) have an explicit form: Also, Lucas polynomials may be generated by recurrence relations: 3) The Lucas polynomials have the power form: ) (2.5) For an arbitrary function u(t) can be written in terms of the Lucas polynomials as: Applying the fractional differential operatorD α to Eq. (2.6) and with the aid of relation (2.5), we will get the following relation: where where i ≥ α , i ≥ j and δ i is defined as:

The numerical scheme
In this section, we use spectral collocation methods with the well known finite difference method to approximate the solution of the time-fractional diffusion partial differential equation [19] where c D α t denotes the Caputo fractional derivative of order α with respect to t, k 1 and k 2 are constant parameters, and h (x) , A (t) , B (t) and g (x, t) are known functions.
To start constructing our approximation scheme for two variable function u(x, t) wherex ∈ [0, 1] and t ∈ [t 0 , T ], denote by h = x the grid size in x-direction, x i = i.x for i = 0, . . ., n x , n x is a positive integer, the function u(x n , t) and its derivatives at x = x n , are discretized and expanded as: where L k (t) denotes Lucas polynomial of order k and c n k are unknown coefficients to be determined, the first and second derivatives have the forms: and the Caputo fractional derivative of order α with respect to t for u (x, t) at x = x n , as defined in (2.7) with initial and boundary conditions Equation (3.6) and condition (3.7) are collocated together at time t i = ( i N ), N = 1, 2, . . ., n t , to generate a system of nonlinear equations with (n x + 1) × (n t + 1) unknowns, using Mathematica-9 package to solve them, approximated solutions were obtained.

Lemma 4.1 If u (x, t) is an infinitely differentiable function at the origin, then u (x n , t) can be expanded at positions x n in terms of Lucas polynomials as
Lemma 4. 3 The modified Bessel function of the first kind I μ (t) satisfies the following inequality be the well-known golden ratio.The following inequality for Lucas polynomials holds: The following are satisfied: The series converges absolutely.
after using Lemma 4.2, we obtain which led to the following after the application of Lemma 4.3: which proves part one of Theorem 4.1, To prove part two, we will consider the series ∞ i=0 c n i L i (t), and Eq.(4.6).
using Lemma 4.3 and 4.4, we obtain Now, since therefore, the series converges absolutely.
Proof By Theorem 4.1 and Eq.(4.7), we can write which can be rewritten as where (m + 1) and (m + 1, d n σ ) denote gamma and incomplete gamma functions, and inequality in (4.8) can be written as x m e −x dx, 123 since e −t < 1, ∀t > 0, then we have Theorem 4. 3 The Lucas approximation scheme when using the finite difference formula (3.6) for discretization in position variable for solving fractional diffusion partial differential equations is stable when Proof To start our method error analysis, we will apply u( into our scheme Eq. (3.6), and take the difference between them to get estimated error e m (x by using Eq.(3.5), the left hand side will be: begin with k = m + 1 and equating Lucas polynomials coefficients , (4.10) after rearrangement of similar coefficients, Eq. (4.10) becomes wheres = η α (m + 1, m + 1).From Eq. (4.11) and for k 2 > 0, we observe the following: for polynomial boundary conditions of degree less than m, c 0 m+1 will vanish.Also we have c 2 m+1 < c 1 m+1 and c 3 m+1 < c 2 m+1 , repeating this process until which leads to the Von Neumann stability condition.
which proves the desired condition.

Numerical results
In this section, we implement the proposed scheme for solving several examples of fractional diffusion partial differential equations.In this section, the performance of the proposed technique is illustrated with four examples.All the numerical experiments are executed under Mathematica 9 running in an Intel (R) Core (TM) i3-CPU @ 3.70 GHz machine.

Example 5.1 Consider the fractional diffusion boundary value problem (3.1) [19]with
(5.1)By applying the proposed scheme same as Eq.(3.6), we will obtain (5.2) with initial and boundary conditions ( Exact solution for this case u (x, t) = e x t β , with error defined as: Table 1 shows maximum absolute errors E ∞ m for Example 5.1 with β = 6 and three different values for α = 0.5, 0.7 and 0.9 at h = 0.05 for different values for m = n t and t j = j/n t , j = 1, 2, . . ., m, compared with the well-known shifted Legendre collocation method SLCM at the same values for m = n t and n x = 4 with needed CPU time for each of them.Calculated errors by our scheme.LCM indicate less time and better accuracy with actual solutions.Meanwhile, raising the value ofm to be greater than or equalβ = 6 has no significant effect on the results.
Moreover, Fig. 1 displays the approximate solution obtained by the proposed method and maximum absolute errors which indicate high accuracy with exact over the whole interval; also, Table 2 shows the maximum absolute errors at m = 8 and different values for h with estimated order of convergence using the relation Example 5.2 Consider the diffusion problem (3.1) [19]with k 2 = 1 and

Table 1
Lucas and shifted Legendre collocation method Maximum absolute errors for Example 5.  Exact solution for this case u (x, t) = e x t β , by applying the proposed scheme same as Eq.(3.6), we will obtain , with initial and boundary conditions Table 3 shows maximum the absolute errors E ∞ m for Example 5.2 with β = 4 and α = 0.5, 0.7 and 0.9, with m = 8 at different values for h, and Fig. 2 displays the approximate solution and maximum absolute errors; calculated errors in Table 3 Exact solution for this case u (x, t) = t β sin( x 2 ), by applying the proposed scheme same as Eq.(3.6), we will obtain with initial and boundary conditions     Exact solution for this case u (x, t) = x 2 (3 − 2x)t 3+α , by applying the proposed scheme same as Eq.(3.6) and unlike previous examples, we will change both n x and n t to get better accuracy.
Figure 4 displays the approximate solution and maximum absolute errors obtained by using the proposed method for Example 5.4 for α = 0.9 and m = n t = n x = 16.Errors indicate high accuracy with the actual solution.Also, Table 5 shows the maximum absolute errors for Example 5.4 with several values for m = n t = n x , calculated errors indicate high accuracy with actual solutions.
Figure 5 displays approximate solution and maximum absolute errors for Example 5.4 for α = 1 and m = 6 and n x = 16, errors indicate a very high accuracy with the actual solution.All errors obtained by our proposed method gave better performance than the method from [19].
Table 6 shows maximum absolute errors for Example 5.5 in three different cases for α i , i = 1, 2, 3 and Fig. 6 displays the approximate solution and maximum abso-

Summary and conclusion
In this paper, we construct a novel numerical method to solve fractional diffusion partial differential equations using finite difference schemes together with a spectral collocation method based on Lucas polynomials.Stability and error analysis have been proven, and results obtained by applying our scheme to different examples indicate high accuracy and convergence for various values of fractional derivatives.Moreover, we need only a limited number of collocation points to get better accuracy with excellent comparison in CPU times.
Funding Open access funding provided by The Science, Technology amp; Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).

Theorem 4 . 2
If u(x n , t) satisfies Theorem 4.1 with e m (t) = ∞ k=m+1 c n k L k (t) estimated global error, then we have

Fig. 5
Fig. 5 3D graphs for approximate solution and max absolute errors for Example 5.4 at α = 1 and m = 6 and h = 16

Table 4
Maximum absolute errors for Example 5.3 with β = 2 and m = 8

Table 5
Maximum absolute errors for Example 5.4 with n x = n t

Table 6
Maximum absolute errors for Example with m