Solving partial fractional differential equations by using the Laguerre wavelet-Adomian method

By using a nonlinear method, we try to solve partial fractional differential equations. In this way, we construct the Laguerre wavelets operational matrix of fractional integration. The method is proposed by utilizing Laguerre wavelets in conjunction with the Adomian decomposition method. We present the procedure of implementation and convergence analysis for the method. This method is tested on fractional Fisher’s equation and the singular fractional Emden–Fowler equation. We compare the results produced by the present method with some well-known results.

Many problems on the diffusion of heat and its equations in the mathematical physics and fluid dynamic are modeled by a form of the equations called Emden-Fowler equations: where φ(x, t)ψ(u) + ξ (x, t) denotes the heat source, u is the temperature, and time variable is t. Put s = 2 and ξ (x, t) = 0. Then relation (1) in one variable version reduces to and for φ(x) = 1 and ψ(u) = u n , we obtain the standard Lane-Emden equation [37,38]. Based on the singularity point at x = 0, many researchers have tried to solve these equations by using different numerical methods such as wavelets, Galerkin, or collocation [38][39][40][41][42][43][44][45][46][47]. By developing the Laguerre wavelets collocation method and using the Adomian decomposition technique, our aim is the investigation of the partial fractional differential equation with boundary conditions u(x, 0) = g(x), u(0, t) = y 1 (t), u(1, t) = y 2 (t), where 0 ≤ α < 1, C D α t is the Caputo fractional derivative, g(x), y 1 (t), y 2 (t) are some functions, F(u(x, t)) is the nonlinear term, and a(x) has singularity at the point x = 0. One can find notions of fractional calculus such as the Riemann-Liouville integral and Caputo derivative in [48].

Laguerre wavelets
On the other hand, by using dilation and translation of a map (as the mother wavelet), we can construct wavelets. For example, we can consider the family of continuous wavelets where a and b are the dilation and translation parameters. If a 0 > 1, b 0 > 0, a = a -k 0 , b = mb 0 a -k 0 and k and m are positive integers, then it reduces to the discrete wavelets ψ k,m (t) = |a 0 | k/2 ψ(a k 0 tmb 0 ) which is a wavelet basis for L 2 (R) [15]. If a 0 = 2 and b 0 = 1, then {ψ k,m (t)} k,m≥0 is an orthonormal basis [15]. It is known that the Laguerre wavelets are defined on the interval [0, 1) as (see [15]) where k ≥ 1, n = 1, 2, 3, . . . , 2 k-1 , t is the normalized time, m = 0, 1, 2, . . . , M -1, M is a fixed positive integer, L m (t) are the Laguerre polynomials of degree m which are orthogonal with respect to the weight function ω(t) = 1 on the interval [0, ∞) and satisfy the recursive relation Let u(x) ∈ L 2 (R) be a function defined over [0, 1). We say that u is expanded by Laguerre wavelets whenever If the series in (4) is truncated, then it can be written by where C and (x) are 2 k-1 M × 1 matrices given by For simplicity, we rewrite (5) as where c i = c n,m , ψ i (t) = ψ n,m (t) and i = M(n -1) + m + 1. where m = 2 k-1 M. If M = 4 and k = 2, then the Laguerre matrix is given by Similarly, the function u(x, t) ∈ L 2 ([0, 1] × [0, 1]) can be also approximated as in which U is an m × m matrix with u ij = ψ i (x), u(x, t), ψ j (t) . We use the wavelet collocation method to determine the coefficients u i,j .

Fractional integral of the Laguerre wavelets
Here, we review the Riemann-Liouville integral of the Laguerre wavelets.

Theorem 1 The fractional integral of the Laguerre wavelets on [0, 1] is given by
where Proof It is known that the Laguerre polynomials are given by where C k n = n! k!(n-k)! . Hence, for Laguerre wavelets, we have and so and so On the other hand, by calculating the integrals, we get If v = xt, then Similarly, we get Now, we apply Riemann-Liouville fractional integration of order α with respect to x for the Laguerre wavelets. Thus, we obtain This completes the proof.

Method of solution
Now, we review the method for the partial fractional differential equation. The Adomian polynomials are used to convert the nonlinear terms of the partial differential equation into a set of polynomials. No linearization process is required for the suggested method. We describe the procedure of implementation in more detail, which enables the readers to understand the method more efficiently. Consider the partial fractional differential equation with the boundary conditions where a(x) has singularity at the point x = 0 and F(u(x, t)) is the nonlinear term of the problem. By applying the Adomian decomposition method, we can express the solution of (17) as We approximate the solution of (18) by using the truncated Adomian series as follows: Moreover, the nonlinear term F(u(x, t)) in (17) is decomposed in terms of Adomian polynomials as where . . , are the Adomian polynomials. By applying (19) and (20) in (17), we obtain where 0 ≤ α < 1. Problem (17) can be decomposed into N + 1 subproblems by the principle of superposition as follows: where 0 ≤ α < 1 and i = 1, 2, . . . , N . By using the Laguerre wavelet method on (22), we approximate it as Now, apply I 2 x on (24) to obtain where p(t) and q(t) are some mappings of t, and we use the boundary conditions and (13) and (16) to get We can write (25) as + x y 2 (t)y 1 (t) + y 1 (t), and so + y 2 (t)y 1 (t) .
By substituting (28), (24) in (22), we obtain a(x) P 1 x and by integrating, we get ). From (30), (27), we have By using the collocation points and replacing ≈ with =, we obtain the matrix version of (31) in a discrete form as follows: Thus, relation (32) can be written as If we solve (33) for C 0 and substitute in (30) or (27), we obtain the solution u 0 at the collocation points. Similarly, we apply the Laguerre wavelet method on (23) by approximating higher order derivative by Laguerre wavelet series as follows: Now, by integrating I 2 x on (34), we get and so By substituting (36), (34) in (23), we obtain By applying fractional integral operator I α t to (37) and using the initial condition, we get From (38) and (35), we have By using the collocation points and replacing ≈ with =, we obtain the matrix form of (39) as follows: where is the Laguerre wavelets matrix, V 2,1,x = xP 2 x (1) and P 2 x = I 2 Relation (32) can be written as

Error analysis
Here, we are going to review the error analysis of the method by expansion of a function in terms of Laguerre wavelets.

Theorem 2 Assume that u m,m (x, t) is the Laguerre wavelets expansion of a smooth func-
tion u(x, t) ∈ . There are real numbers C 1 , C 2 , and C 3 such that  u(x, t). In this case,

Proof Consider
where ξ , ξ ∈ I k,n = [ n-1 2 k-1 , n 2 k-1 ) and ζ , ζ ∈ I n ,k = [ n -1 2 k -1 , n 2 k -1 ) (see [49]). Let = I n,k × I n ,k , we get By using Theorem 2.2.3 in [50] for error of Chebyshev interpolation nodes, we obtain , the function u(x, t) is approximated on them by using the Laguerre wavelets method as a polynomial of Mth (or Mth) degree at most with the least-square property, we get Now, choose real numbers C 1 , C 2 , and C 3 such that By replacing (43), (44), and (45) in (42), we obtain Relation (46) ensures the convergence of Laguerre wavelet approximation u m,m (x, t) for components of the Adomian series u i (x) at higher level of k and M, that is, when k and M approach infinity. According to the convergence of the Adomian method [51], N i=0 u i (x, t) converges to u(x, t) when N → ∞. According to this analysis, we conclude that the present method converges to the exact solution of (42) whenever N and k, M approach infinity. This completes the proof.
For the special case M = M and k = k , we have where C = C 1 + C 2 , C 1 = C 3 , and u m,m (x, t) is the best approximation of u(x, t).

Numerical examples
Now, using the method, we provide some illustrative examples. In the examples, exact solutions are available and a comparison is made between the approximate Laguerre technique and the exact solutions to show the efficiency of the method.
For α = 1, the exact solution of (47) is u(x, t) = 1 (1+e x-5t ) 2 . By solving (47) for k = 3 and M = 5 by the Laguerre wavelet Adomian method (LWAM), the approximate solution obtained  Table 1 Absolute errors for N = 8, k = 3, M = 5, various values of α when it goes to α = 1, and comparison of the absolute error with HPM [33] and MVIM [36] in Example 1 1.6365e-07 3.8611e-08 2.5677e-08 6.8794e-07 6.9133e-07 3.0511e + 00 6.5484e-01 by this method for N = 8 is u LWAM = 8 i=0 u i (x, t). We plotted in Fig. 1 the absolute errors for various values of N = 1, 2, . . . , 8. As can be seen, by increasing the values of N absolute errors are decreasing. Table 1 shows the comparison of absolute errors for different values of α and the methods introduced in [33,36]. Table 2 shows the comparison of absolute errors for different values of M. Also, it says that by increasing of M absolute errors are decreasing.
Example 2 Consider the fractional Fisher equation   . We solve (48) for k = 3 and M = 5 by the LWAM. The approximate solution for N = 6 is u LWAM = 6 i=0 u i (x, t). We plotted in Fig. 2 the absolute errors for various values of N = 1, 2, . . . , 6. One can check that by increasing the values of N absolute errors are decreasing. Table 3 shows the comparison of absolute errors for different values of α and the method introduced in [33,36]. Table 4 shows the comparison of absolute errors for different values of k and M. Also it shows that by increasing of k and M absolute errors are decreasing.

Conclusion
By using the Laguerre wavelets and the Adomian decomposition method, we tried to provide appropriate numerical solutions for some partial fractional differential equations. We compared our results with some other methods. Also, we gave some illustrative examples which showed that the method is an effective tool to solve the time-fractional order Fisher equations and the singular nonlinear Emden-Fowler equation. We summarize the advantages of the present methods as follows.
1) Instead of operational derivative, we used the operational integral matrix with initial conditions taken into automatically, so we did not need to choose the base to satisfy them.   2) Instead of approximating the integral operation by the use of black-pulse functions or any approximation, the fractional integral operation has been directly obtained to get a better approximation.
3) With respect to the wavelet bases used and transforming the nonlinear problem into the algebraic equations, we obtained good results by performing few calculations and resolution.

4) Operational
Laguerre wavelet matrix is sparse, so solving a system of algebraic equations obtained by using LWAM is simple and fast.