ABSTRACT
In this work we obtain a variational formulation and a priori estimates for approximate solutions of a problem involving fractional diffusion equations.
Keywords:
Galerkin method; fractional diffusion equation; a priori estimates
1 INTRODUCTION
Fractional calculus has gained much prominence in recent decades, due to its applications in different fields of science, in particular, engineering, providing several useful tools to solve differential and integral equations and other problems involving special functions of mathematical physics, in addition to their extensions and generalizations in one and more variables. Among the various applications of fractional calculus we can cite the flow of a fluid, rheology, dynamic processes in self-similar and porous structures, diffusive transport similar to diffusion, electrical networks, probability and statistics, theory of control of dynamical systems and viscoelasticity (see66 A.A. Kilbas, H.M. Srivastava & J.J. Trujillo. “Theory and Applications of Fractional Differential Equations”, volume 204. Elsevier (2006).).
Anomalous diffusion can be characterized by both Levy flights, mathematically represented by the fractional Laplacian, as well as long rests, described by the time-fractional derivative. In this case, the appropriate equation, according to Schneider and Wyss99 W.R. Schneider & W. Wyss. Fractional diffusion and wave equations. Journal of Mathematical Physics, 30(1) (1989), 134-144. and Metzler and Klafter88 R. Metzler & J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports, 339(1) (2000), 1-77., is given by
where and denotes the fractional derivative of φ of order in the Riemann-Liouville sense, that is, (see Definition 2.2). In this way, the equation (1.1) can be rewritten as the equation
where g α is the function defined in (1.4).
Let us discuss the following problem for a fractional diffusion equation
where , Ω is a smooth bounded domain of ℝn, * denotes the convolution product and g α is the Gel’fand Shilov function defined by
where Γ is the Euler gamma function. The function f belongs to L 1(0, T; L 2) and also to L ∞(0, T; L 2). Furthermore, the fractional Laplacian operator can be defined in its spectral form by (see section 2.5.1 of77 A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M.M. Meerschaert, M. Ainsworth et al. What is the fractional Laplacian? A comparative review with new results. Journal of Computational Physics, 404 (2020), 109009.):
Where , λk are eigenvalues, and e k are eigenfunctions of (-∆) with Dirichlet boundary conditions, that is,
Fractional-order diffusion equations describe anomalous diffusion phenomena, which help in the analysis of systems such as: plasma diffusion, fractal diffusion, anomalous diffusion on liquid surfaces, analysis of heart beat histograms in healthy individuals, among other physical systems (see11 P. Biler, T. Funaki & W.A. Woyczynski. Fractal Burgers equations. Journal of Differential Equations, 148(1) (1998), 9-46. and22 M.P. Bonkile, A. Awasthi, C. Lakshmi, V. Mukundan & V. Aswin. A systematic literature review of Burgers’ equation with recent advances. Pramana, 90(6) (2018), 1-21.).
For the variational formulation of the problem we will use the integral form of Problem (1.3), given by
We will give the variational formulation and prove a priori estimates for the approximate solutions of the integral equation (1.6). Those results are useful to apply the Galerkin method (see44 L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, 19(4) (1998), 7.), which consists of finding approximate solutions to the problem, projecting it into finite-dimensional subspaces, dealing with fractional-order linear differential equations with initial values.
2 PRELIMINARIES
In this section we present some definitions and notations for the present work.
Definition 2.1.Letbe a finite interval over ℝ. Riemann-Liouville fractional integrals, of orderare given by:
where Γ(α) is the gamma function and.
Definition 2.2.Riemann-Liouville fractional derivatives, of orderis given by
where [α] means the integer part of α and. We take, if.
Definition 2.3.Caputo fractional derivative of order α, on an interval, is given by
whereifand, if.
Note that the Problem (1.3) can be rewritten as
Where and . In fact, since and , we have to
We will use the following spaces L ∞(0, T; L 2(Ω)), L 2(0, T; H γ(Ω)) and L 1(0, T; L 2(Ω)), where Ω is an open on ℝn . We remember that L p (Ω) is the space of all measurable functions , with such that
Definition 2.4.Let X a Banach space. The space Lp (0, T; X) consists of all measurable functions
with
for , and
For simplicity, we sometimes denote L p (0, T; L p (Ω)) by L p (0, T; L p ). Furthermore, we denote the inner product in L 2 by (·, ·) and in H γ by .
The fractional Sobolev space H γ is a Hilbert space and is defined below.
Definition 2.5 (Definition A.5,77 A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M.M. Meerschaert, M. Ainsworth et al. What is the fractional Laplacian? A comparative review with new results. Journal of Computational Physics, 404 (2020), 109009.).For any
where (λk , e k ) are the eigenvalues and their respective eigenvectors of (-∆) with Dirichlet boundary conditions, whose norm coincides with, according toequation (1.5).
Before we present Theorem 2.1 we need the following definitions.
Definition 2.6.Let,and. We define the matrix α-exponential function by
Definition 2.7. A weighted space of continuous functions is of the form
We use the following existence and uniqueness theorem for a Cauchy problem of a fractional matrix equation with a Caputo derivative (see66 A.A. Kilbas, H.M. Srivastava & J.J. Trujillo. “Theory and Applications of Fractional Differential Equations”, volume 204. Elsevier (2006).).
Theorem 2.1 (Theorem 7.14, 6 6 A.A. Kilbas, H.M. Srivastava & J.J. Trujillo. “Theory and Applications of Fractional Differential Equations”, volume 204. Elsevier (2006). ). The following initial value problem
where and , has a single continuous solution given by
Also, we need the following result which can be found in33 L. Djilali & A. Rougirel. Galerkin method for time fractional diffusion equations. Journal of Elliptic and Parabolic Equations, 4(2) (2018), 349-368. and references therein.
Theorem 2.2.Let (H, (·, ·)) be a real Hilbert space,and. Then
Lemma 2.1 (Lemma 2.22,66 A.A. Kilbas, H.M. Srivastava & J.J. Trujillo. “Theory and Applications of Fractional Differential Equations”, volume 204. Elsevier (2006).).Letand let n be given by, ifand, if. Ifor, then
In particular, ifandor, then
Definition 2.8. We define the Sobolev space
Lemma 2.2 (Lemma 6.2, 5 5 J. Kemppainen, J. Siljander, V. Vergara & R. Zacher. Decay estimates for time-fractional and other non-local in time subdiffusion equations in ℝd. Mathematische Annalen, 366(3) (2016), 941-979. ). Let and be an open set. Let be nonnegative and nonincreasing function. Then for any and any there holds
for each.
3 MAIN RESULTS
In this section, we obtain the variational formulation and a priori estimates for the approximate solutions of Problem (1.3). Here, we perform formal calculations so that u is considered as regular as necessary.
3.1 Variational formulation
For the variational formulation we will use the integral form of Problem (1.3), given by
as long as the fractional Laplacian applied to u is continuous, where on the boundary of Ω and . Multiplying (3.1) by such that and, integrating over Ω, we have
Thus, using Fubini theorem and knowing that the fractional Laplacian is self-adjoint on L 2, in addition to having the semigroup property, we obtain
Thus, it follows from equation (3.2) that
or
where the equation (3.4) gives us the variational form of the problem. We denote by B[u, v; t]. Now, let us build the approximate solutions. For this, consider a base {v k }k orthogonal to H γ that is orthonormal to L 2(Ω).
For every natural number m, consider the vector subspace
and,
where we must determine the coefficients ( and ) such that
and
Theorem 3.3.If, then for every integer, there is a single differentiable function um , given by(3.5), satisfyingequations (3.6)and(3.7).
Proof. Suppose u m has the form equation (3.5). The proof consists in to show the existence and uniqueness of . So,
because {v j }j is orthonomal. Furthermore,
where . Define . So, from equation (3.7), we have
Let and .
We can rewrite (3.8) in the following matrix form
So, equation (3.9) can be rewritten as
Thus, by hypothesis, as , it follows that . Therefore, by Theorem 2.1, it follows the existence and uniqueness of .
3.2 A priori estimates
In this section we prove a priori estimates given by the following theorem.
Theorem 3.4.Let. If, then
If, additionally, , then
Proof. Since u m is the function defined in (3.5) and guaranteed by Theorem 3.3, we multiply equation (3.7) by and sum with j running from 1 to m, to get
We note that . In fact, looking at the expression (3.5) we can infer that
since and the sum is finite. So, by Theorem 2.2, we have
It follows from this and from equation (3.12) that
where we have used Hölder ineqality and Minkowski integral inequality. Hence,
This proves (3.10). For the proof of estimate (3.11) let us observe that
so that,
But, putting in Lemma 2.2, we have
with .
By estimating (3.10), it is immediate to see that . So (3.16) holds for u m . Therefore, (3.15) gives us
From the continuous inclusion , it follows that there is a constant such that . Therefore,
implying
Applying to (3.19), we have by Lemma 2.1
Implying
Using Minkowski integral inequality, we can estimate . In fact,
for . Then, we can write
So, applying g 1-α to (3.21), we have
which is the desired result.
4 CONCLUDING REMARKS
In this work we obtained the variational formulation and an estimate a priori of Problem (2.1), results that will help us to apply Galerkin’s method and enable us to prove the existence and uniqueness of the solution to Problem (2.1). Later, we will investigate the existence of global solutions and their stability. Also, one will be able to implement numerical simulations.
REFERENCES
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1P. Biler, T. Funaki & W.A. Woyczynski. Fractal Burgers equations. Journal of Differential Equations, 148(1) (1998), 9-46.
-
2M.P. Bonkile, A. Awasthi, C. Lakshmi, V. Mukundan & V. Aswin. A systematic literature review of Burgers’ equation with recent advances. Pramana, 90(6) (2018), 1-21.
-
3L. Djilali & A. Rougirel. Galerkin method for time fractional diffusion equations. Journal of Elliptic and Parabolic Equations, 4(2) (2018), 349-368.
-
4L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, 19(4) (1998), 7.
-
5J. Kemppainen, J. Siljander, V. Vergara & R. Zacher. Decay estimates for time-fractional and other non-local in time subdiffusion equations in ℝd Mathematische Annalen, 366(3) (2016), 941-979.
-
6A.A. Kilbas, H.M. Srivastava & J.J. Trujillo. “Theory and Applications of Fractional Differential Equations”, volume 204. Elsevier (2006).
-
7A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M.M. Meerschaert, M. Ainsworth et al. What is the fractional Laplacian? A comparative review with new results. Journal of Computational Physics, 404 (2020), 109009.
-
8R. Metzler & J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports, 339(1) (2000), 1-77.
-
9W.R. Schneider & W. Wyss. Fractional diffusion and wave equations. Journal of Mathematical Physics, 30(1) (1989), 134-144.
Publication Dates
-
Publication in this collection
14 Nov 2022 -
Date of issue
Oct-Dec 2022
History
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Received
23 Dec 2021 -
Accepted
20 May 2022