- Research
- Open Access
- Published:
A Neumann problem for a diffusion equation with n-dimensional fractional Laplacian
Advances in Difference Equations volume 2021, Article number: 255 (2021)
Abstract
We study an initial-boundary value problem for a n-dimensional stochastic diffusion equation with fractional Laplacian on \(\mathbb{R}_{+}^{n}\). In order to prove existence and uniqueness, we generalize the Fokas method to construct the Green function for the associated linear problem and then we apply a fixed point argument. Also, we present an example where the explicit solutions are given.
1 Introduction
The classical diffusion phenomenon is governed by a second order linear partial differential equation, whose Green function is given by a Gaussian probability density function and which describes the movement of energy through a medium in response to a gradient of energy. On the other hand, the diffusion processes in various systems with complex structure, such as liquid crystals, glasses, polymers, biopolymers, and proteins, usually do not follow a Gaussian density, as a consequence the phenomenon is described by a fractional partial differential equation [7]. Dipierro et al., [4] have studied the asymptotic behavior of the solutions of the time-fractional diffusion equation.
There is some previous work for the initial-boundary value problem on the first quadrant \(\mathbb{R}_{+}^{2}\) for fractional diffusion equations, where the Green function has been constructed and an integral representation of the solution was found [3, 6]. In this note, we consider the equation
where the operator \(\Delta ^{\alpha }\) is defined via the Riesz fractional derivative, for each coordinate. Let us notice that the generalization of the Laplacian most commonly used [1, 9] is different from the one we use in this work.
However, Eq. (1) is an idealized version because many aspects are missing in the modeling; such as the inhomogeneity of the medium, external sources, and measurement errors. Then a more realistic version is obtained by considering a stochastic version with additive noise. For example, Balanzario and Kaikina [2] studied the stochastic nonlinear Landau–Ginzburg equations on the half-line with Dirichlet white-noise boundary conditions, Shi and Wang [11] studied the solution for a stochastic fractional partial differential equation driven by an additive fractional space–time white noise. In Sanchez et al. [10], studied the stochastic version of (1) for the 2-dimensional case; however, the n-dimensional case on \(\mathbb{R}_{+}^{n} :=\{ {\mathbf{x}}=(x_{1},\dots , x_{n}) : x_{j}\geq 0, j=1,\dots n\}\) has not been studied. In the present work we tackle this problem via the main ideas of the Fokas method (unified transform) [5], this method is a technique for solving initial-boundary value problems for partial differential equations. Moreover, it generates integral representation formulas for solutions, where the integrals converge uniformly on the boundary.
2 Preliminaries
Let us give some known definitions and results.
Definition 1
The n-dimensional Fourier–Laplace transform is defined as follows:
where \({\mathbf{x}}\in \mathbb{R}_{+}^{n}\), \({\mathbf{k}}\in \mathbb{C}^{n}=\{ {\mathbf{k}}=(k_{1},\dots , k_{n}) : k_{j} \in \mathbb{C}, j=1,\dots n\}\) and \(\Im m(k_{j})\leq 0\), \({\mathbf{k}}\cdot {\mathbf{x}}\) is the usual inner product, and its inverse is defined by
Definition 2
The Riesz fractional operator is defined by
Here, \(\alpha \in (2,3)\), \({\mathbf{x}}_{j}\in {\mathbb{R}_{+}^{n}}\) is the vector x, where the jth coordinate is \(y_{j}\), \(j=1,\dots n\).
Note that the operator, using integration by parts, \(\mathcal{D}_{x_{j}}^{\alpha }\) can be represented in the following form [8]:
Lemma 1
If \(\Delta ^{\alpha }\), \(\alpha \in (2,3)\), is the fractional n-dimensional Laplace operator
then, for \(\Im m(k_{l})\leq 0\),
Here, \(|{\mathbf{k}}|^{\alpha }:=\sum_{l=1}^{n}|k_{l}|^{\alpha }\) and \({\mathbf{k}}_{[-l]} \in \mathbb{C}^{n}\) is the k vector, where its lth coordinate is zero.
Proof
The theorem follows from the linearity of the operator \(\Delta ^{\alpha }\) and the well-known equation
□
3 Green function
We consider a linear problem for an evolution equation with initial condition \(u_{0}\) and boundary conditions \(h_{j}\), \(j=1,\ldots , n\),
where \(\alpha \in (2,3)\), \(t>0\), \({\mathbf{x}}_{[-j]}\in \mathbb{R}_{+}^{n}\) means that the jth coordinate of x is zero, with the compatibility conditions \(h_{j}({\mathbf{x}}_{[-j,-l]},t)=h_{l}({\mathbf{x}}_{[-j,-l]},t)\) where \({\mathbf{x}}_{[-j,-l]}\in \mathbb{R}_{+}^{n}\) is such that jth and lth coordinates, \(x_{l}\) and \(x_{j}\), are equal to zero for \(j\neq l\).
Theorem 1
Let the initial data \(u_{0}({\mathbf{x}})\in {\mathbf{L}}^{1}( \mathbb{R}_{+}^{n})\) and the boundary data \(h_{j}({\mathbf{x}}_{[-j]},t) \in {\mathbf{C}}(\mathbb{R}_{+};{\mathbf{L}}^{1}( \mathbb{R}_{+}^{n}))\). Suppose that there exists some function \(u({\mathbf{x}},t)\), which satisfies (2). Then \(u({\mathbf{x}},t)\) has the following integral representation:
where the Green operators are given by
and the Green functions are
Here, \({\mathbf{k}}^{\alpha }=\sum_{l=1}^{n} k_{l}^{\alpha }\).
Proof
Applying Theorem 1 to Eq. (2), we obtain
Now, we multiply the above equation by \(e^{|{\mathbf{k}}|^{\alpha }t}\) and integrate from 0 to t,
for \(\Im m (k_{l})\leq 0\), where
Now, we initially consider 2-dimensional case. Thus, Eq. (4) is expressed as
Applying the inverse transform in (5) with respect to \(k_{1}\) and moving the contour of integration for the terms with \(g_{j}^{1}\) in the integrand, we obtain
where \(D_{1}^{+}= \{k_{1}\in \mathbb{C}: 0 \leq \Im m (k_{1}) \leq \frac{\pi }{2\alpha }|\Re e (k_{1})| \}\). Let us note the following: if we substitute \(k_{1}\) by \(-k_{1}\), the functions \(g_{j}^{1}\) from Eq. (5) are invariant. Then, making this change of variables in (5), we get
for \(\Im m (-k_{1}),\Im m (k_{2})\leq 0\). Substituting \(g_{2}^{1}\) from Eq. (7) in (6) and using the fact that
by the Cauchy theorem, we obtain the following integral representation:
Applying the inverse transform in (8) with respect to \(k_{2}\) and moving the contour of integration for the terms with \(g_{j}^{2}\) in the integrand, we obtain
where \(D_{2}^{+}= \{k_{2}\in \mathbb{C}: 0 \leq \Im m (k_{2}) \leq \frac{\pi }{2\alpha }|\Re e (k_{2})| \}\). Let us note the following: if we substitute \(k_{2}\) by \(-k_{2}\), the functions \(g_{j}^{2}\) from Eq. (8) are invariant. Then, making this change of variables in (7), we get
for \(\Im m (k_{1}),\Im m (k_{2})\geq 0\). Substituting \(g_{2}^{2}(|{\mathbf{k}}|^{\alpha },\pm {\mathbf{k}}_{[-2]},t)\) from Eq. (10) in (9) and using the fact that
by the Cauchy theorem, we obtain the following integral representation:
where \({\mathbf{r}}\in S_{2}=\{(\pm k_{1},\pm k_{2})\}\) and \({\mathbf{r}}_{[-l]}\) is such that the lth coordinate is equal to zero. In Eq. (11) we have, after interchanging the integration order, integrals of the form
and
We notice that all the integrals above are absolutely integrable, then using the Fubini theorem, after some simplifications, we arrive from Eq. (11) at the following equation:
where the Green operators are given by
and the Green functions are
where \({\mathbf{k}}^{\alpha }=k_{1}^{\alpha }+k_{2}^{\alpha }\). Now, following the previous arguments we can tackle the n-dimensional case. This can be achieved, via mathematical induction over n, passing from Eq. (4) to Eq. (12), through the steps that we describe in the 2-dimensional case. Analogous to Eq. (11), we obtain an integral representation for u,
where \({\mathbf{r}}\in S_{n}=\{(\pm k_{1},\pm k_{2}, \dots , \pm k_{n})\}\) and \({\mathbf{r}}_{[-l]}\) is such that the lth coordinate is equal to zero. Interchanging the integrals in the above equation, by Fubini’s theorem, we obtain the desired result. □
4 Stochastic nonlinear problem
In order to state the problem, we define the Brownian sheet Ḃ on \(\mathbb{R}^{n}_{+}\times [0,T]\) on a complete probability space \((\Omega ,\mathcal{F},\mathcal{F}_{t},P)\), here \(\mathcal{F}\) is a σ-algebra, \(\{\mathcal{F}_{t}\}_{t\geq 0}\) is a right-continuous filtration on \((\Omega ,\mathcal{F})\) such that \(\mathcal{F}_{0}\) contains all P-negligible subsets and P is a probability measure. We consider a center Gaussian field \(B=\{B({\mathbf{x}},t)| {\mathbf{x}}\geq 0, t\geq 0\}\) with covariance function given by
We suppose that B generates a \((\mathcal{F}_{t},t\geq 0)\)-martingale measure in the sense of Walsh [12]. Let the initial condition \(u_{0}\) be \(\mathcal{F}_{0}\times \mathcal{B}(\mathbb{R}^{n}_{+})\) measurable, where \(\mathcal{B}(\mathbb{R}^{n}_{+})\) is the Borelian σ-algebra over \(\mathbb{R}^{n}_{+}\).
Now, we consider the following initial-boundary value problem for a nonlinear equation:
where \({\mathbf{x}}\in \mathbb{R}_{+}^{n} \), \(t>0\), \(\alpha \in (2,3)\), \(\mathcal{N}\) is a Lipschitzian operator; i.e., \(|\mathcal{N}u - \mathcal{N}v| \leq C|u-v|\), \(C>0\), and the compatibility conditions \(h_{j}({\mathbf{x}}_{[-j,-l]},t)=h_{l}({\mathbf{x}}_{[-j,-l]},t)\) are satisfied. We understand the solutions for the problem (13) in the following sense: u is a solution if, for all \({\mathbf{x}}\in \mathbb{R}_{+}^{n} \) and \(t>0\), the following equation is fulfilled:
where the Green operators \(\mathcal{G}^{I}(t)\), \(\mathcal{G}^{B_{l}}(t)\) are given in Eq. (3) and the Green function is
Theorem 2
Let the initial data \(u_{0}({\mathbf{x}})\in {\mathbf{L}}^{1}( \mathbb{R}_{+}^{n})\) and the boundary data \(h_{j}({\mathbf{x}}_{[-j]},t) \in {\mathbf{C}}(\mathbb{R}_{+};{\mathbf{L}}^{1}( \mathbb{R}_{+}^{n}))\). Suppose that, for each \(T>0\), there exists a constant \(C>0\) such that, for each \({\mathbf{x}}\in \mathbb{R}_{+}^{n}\), \(t\in [0,T]\) and \(u,v \in \mathbb{R}^{n}\), \(|\mathcal{N}u-\mathcal{N}v|\leq C |u-v|\), and for some \(p\geq 1\),
Then, there exists a unique solution \(u({\mathbf{x}},t)\) to Eq. (13). Moreover, for all \(T>0\) and \(p\geq 1\),
Proof
First, we define a Picard succession:
where
Now, let us prove that \(\{u^{n}({\mathbf{x}},t)\}_{n\geq 0}\) converges in \(L^{p}(\Omega )\). Using the fact that, for all \(t\geq 0\), \(G({\mathbf{x}},t)\) from Eq. (15) is a probability density function with respect to x, we obtain, for \(n\geq 2\),
and by (16) and Burkholder’s inequality we have
Then, by Gronwall’s lemma we obtain
Hence, \(\{u^{n}({\mathbf{x}},t)\}_{n\geq 0}\) is a Cauchy sequence in \(L^{p}(\Omega )\). Let
Thus,
Taking \(n\rightarrow \infty \) in \(L^{p}(\Omega )\) at both sides of (17) shows that \(u({\mathbf{x}},t)\) satisfies the problem (2). Finally, we have to prove the uniqueness of the solution. Let u and v be the two solutions of problem (2), then
Therefore, Gronwall’s lemma yields
□
5 Example
In this section, we consider an example for the case \(n=2\), with the initial condition
and the boundary conditions, for \(l=1,2\),
In Fig. 1, we present the plot of the solution \(u({\mathbf{x}}, t)\) for \(t=0.02, 0.1,0.5, 1\), and \(\alpha =2.5\).
Availability of data and materials
Not applicable.
References
Abatangelo, N., Valdinoci, E.: Getting acquainted with the fractional Laplacian. In: Contemporary Research in Elliptic PDEs and Related Topics. Springer INdAM Ser., vol. 33, pp. 1–105. Springer, Cham (2019)
Balanzario, E.P., Kaikina, E.I.: Regularity analysis for stochastic complex Landau-Ginzburg equation with Dirichlet white-noise boundary conditions. SIAM J. Math. Anal. 52(4), 3376–3396 (2020)
Bona, J.L., Luo, L.: Generalized Korteweg–de Vries equation in a quarter plane. Contemp. Math. 221, 59–125 (1999)
Dipierro, S., Pellacci, B., Valdinoci, E., Verzini, G.: Time fractional equations with reaction terms: fundamental solutions and asymptotics. Discrete Contin. Dyn. Syst. 41(1), 257–275 (2021)
Fokas, A.: A Unified Approach to Boundary Value Problems. SIAM, Philadelphia (2008)
Mantzavinos, D., Fokas, A.S.: The unified transform for the heat equation: II. Non-separable boundary conditions in two dimensions. Eur. J. Appl. Math. 26, 887–916 (2015)
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2001)
Pozrikidis, C.: The Fractional Laplacian. CRC Press, Boca Raton (2016)
Ros-Oton, X., Valdinoci, E.: The Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains. Adv. Math. 288, 732–790 (2016)
Sanchez-Ortiz, J., Ariza-Hernandez, F.J., Arciga-Alejandre, M.P., Garcia-Murcia, E.: Stochastic diffusion equation with fractional Laplacian on the first quadrant. Fract. Calc. Appl. Anal. 22(3), 795–806 (2019)
Shi, K., Wang, Y.: On a stochastic fractional partial differential equation with a fractional noise. Stochastics 84(1), 21–36 (2012)
Walsh, J.B.: An introduction to stochastic partial differential equations. Lect. Notes Math. 1180, 265–439 (1986)
Acknowledgements
Not applicable.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Arciga-Alejandre, M.P., Sanchez-Ortiz, J., Ariza-Hernandez, F.J. et al. A Neumann problem for a diffusion equation with n-dimensional fractional Laplacian. Adv Differ Equ 2021, 255 (2021). https://doi.org/10.1186/s13662-021-03413-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-021-03413-w
MSC
- 26A33
- 31A10
- 58J32
Keywords
- Fractional Laplacian
- Fokas method
- Green function