![MathJax Logo](/templates/jsp/_style2/_tandf/pb2/images/math-jax.gif)
Abstract
In this paper, a nonlinear fractional parabolic stochastic integro-partial differential equations with nonlocal effects driven by a fractional Brownian motion is considered. In particular, first we have formulated the suitable solution form for the fractional partial differential equations with nonlocal effects driven by fractional Brownian motion using a parabolic transform. Next, the existence and uniqueness of solutions are obtained for the fractional stochastic partial differential equations without any restrictions on the characteristic forms when the Hurst parameter of the fractional Brownian motion is less than half. Further, we investigate the stability of the solution for the considered problem. The required result is established by means of standard Picard’s iteration.
Public Interest Statement
Fractional parabolic partial differential equations are found to be an effective tool to describe certain physical phenomena such as diffusion processes, visco-elasticity theories, filtration, phase transition, electromagnetism, acoustics, electrochemistry, cosmology, and biochemistry. However, no work has been reported in the literature regarding the existence and uniqueness of solutions for nonlinear fractional parabolic integro-partial differential equations with nonlocal effects driven by a fractional Brownian motion when the Hurst parameter of the fractional Brownian motion is less than half. Motivated by these facts, in this note, we studied the existence, uniqueness and stability of solutions for the fractional stochastic partial differential equations without any restrictions on the characteristic forms when the Hurst parameter of the fractional Brownian motion is less than half.
1. Introduction
Fractional differential equations has many important applications in many areas of science and engineering. Recently, many researchers have found that it describes several physical phenomena more exactly than differential equations without fractional derivative. On the other hand, the noises arise in mathematical finance, physics, telecommunication networks, hydrology, medicine etc., can be modeled by fractional Brownian motions (Baudoin, Nualart, Ouyang, & Tindel, Citation2016; Grecksch & Anh, Citation1999; Nualart & Ouknine, Citation2002; Maslowski & Nualart, Citation2003; Tindel, Tudor, & Viens, Citation2003). More and more work has been devoted to the investigation of ordinary fractional differential equations driven by fractional Brownian motions (Arthi, Park, & Jung, Citation2016; Balasubramaniam, Vembarasan, & Senthilkumar, Citation2014; Boudaoui, Caraballo, & Ouahab, Citation2016; Diop, Ezzinbi, & Mbaye, Citation2015; Hamdy, Citation2015; Ren, Wang, & Hu, Citation2017; Sathiyara & Balasubramaniam, Citation2017; Tamilalagan & Balasubramaniam, Citation2017a, Citationb). On the other hand, fractional parabolic partial differential equations are found to be an effective tool to describe certain physical phenomena such as diffusion processes, visco-elasticity theories, filtration, phase transition, electromagnetism, acoustics, electrochemistry, cosmology, and biochemistry. However, no work has been reported in the literature regarding the existence and uniqueness of solutions for nonlinear fractional parabolic integro-partial differential equations with nonlocal effects driven by a fractional Brownian motion when the Hurst parameter of the fractional Brownian motion is less than half. Motivated by these facts, in this note, we will consider the following nonlinear fractional parabolic stochastic integro-partial differential equations in the form(1.1)
(1.1)
with nonlocal initial condition(1.2)
(1.2)
where is of the form
for some
is the n-dimensional Euclidean space,
is an n-dimensional multi-index,
is a fractional Brownian motion with Hurst parameter
and E(X) denotes the expectation of a random variable X. It is well known that if
then
coincides with the classical Brownian motion B(t). For
is not a semimartingale, so one cannot use the general theory of stochastic calculus for semimartingale on
, (Caraballo, Diop, & Ndiaye, Citation2014; Decreusefond & Ustunel, Citation1999; Duncan & Nualart, Citation2009; Elliott & Van Der Hoek, Citation2003; El-Borai & El-Nadi, Citation2017; Ren, Hou, & Sakthivel, Citation2015). It should be mentioned that the kind of equations given in (1.1)–(1.2) can be used to model a variety of anomalous diffusion in continuum mechanics, particularly in connection with the investigation in turbulence. In Section 2, we shall present some properties of the stochastic solutions of the nonlocal Cauchy problem (1.1), (1.2) using a parabolic transform. In Section 3, we shall prove the existence and uniqueness of solutions for the considered stochastic equations under suitable conditions. In Section 4, we shall investigate the stability of the solution for the considered problem.
2. Parabolic transform and weak solutions
In this section, we present some basic properties and some suitable solution form for the nonlinear fractional parabolic partial differential equations with nonlocal effects driven by fractional Brownian motion using a parabolic transform. In order to obtain the required result, we impose the following conditions on the functions:
(H1) | The given function | ||||
(H2) | All the coefficients of | ||||
(H3) | The functions | ||||
(H4) | The function g is given and bounded continuous on | ||||
(H5) | The operator | ||||
(H6) | The kernel | ||||
(H7) | The kernel | ||||
(H8) | The function |
where v is given by(2.2)
(2.2) G is the fundamental solution of the parabolic partial differential equation:
The proof of formula (2.1) and the definition of the function can be found in El-Borai, El-Nadi, and El-Akabawy (Citation2010) and El-Borai, El-Nadi, and Fouad (Citation2010). The function G satisfies the following inequalities,
(2.3)
(2.3)
where , and
are positive constants. The function
is a probability density function defined on
. According to the properties of G, we can find a positive constant
such that
for all bounded continuous function f on .
Let us suppose that , where
. For every
, we define two operators
and
on the set of all bounded continuous function on
by,
(2.4)
(2.4)
According to (2.4), the inverse operator exists on the set of all bounded continuous functions on
From (2.1), one gets, formally,
(2.5)
(2.5)
If we can find the stochastic process v in a suitable space, then formulas (2.1) and (2.5) will determine the stochastic process u. Let us now try to study Equation (2.2). By a weak solution of Equation (2.1), we mean a triple of adapted processes on a filtered probability space
such that
(a) |
| ||||
(b) | The norm |
where denotes the gamma function and F(a, b, c; z) is the Gauss hyper geometric function. Define an operator
by
. The operator
is an isomorphism from the space of all square integrable functions
onto
where
is the image of
by the fractional integral operator
where
The inverse operator exists and can be defined on the set of all functions
. It is well known that there exists a Brownian motion B(t) such that the fractional Brownian motion
can be represented by
(Nualart & Ouknine, Citation2002).
Theorem 2.1
Let and v be a weak solution of Equation (2.2). If
, and
are Borel functions on
and satisfy the linear growth condition:
(2.6)
(2.6)
for all where
is a positive constant, then
almost surely for every
and
Proof
From (2.1), (2.2), (2.3), (2.5) and conditions (H4), (H6), (H7), one gets,
for some positive constant .
Thus for some positive constant , we have
(2.7)
(2.7)
From (2.6) and (2.7), one gets, for some positive constant (2.8)
(2.8)
For some positive constant , we have
(2.9)
(2.9)
Hence the required result. According to the conditions (H6), (H7) and the conditions and results of Theorem 2.1, we can find also that and
are elements of
for every
. For every
let us define an operator
from
onto
, by:
where
Using conditions (H6), (H7) and that the functions are elements of
one gets that
and
are defined and can be represented by:
where
Notice that and
are elements of
. Using condition (H4), we can see that
exists and is defined on
. Now according to Theorem 2.1 and the last discussions, the weak solution v of Equation (2.2) can be represented by
where and
. Notice that
exists according to condition (H8).
3. Existence and uniqueness of solutions
Formula (2.10) leads to the fact that two weak solutions of Equation (2.2) must have the same distributions. We can also conclude that if two weak solutions of Equation (2.2) defined on the same filtered probability space must coincide almost surely, (El-Borai & El-Said, Citation2015).
Theorem 3.1
If and
are continuous on
and satisfy a Lipschitz condition;
(3.1)
(3.1)
for all then there is a weak solution of Equation (2.1). Moreover,
.
Proof
Let us use the method of successive approximations. set,(3.2)
(3.2)
where and every
is of the form
for some
(3.3)
(3.3)
Suppose that the zero approximation . Using (2.3) and (3.1)–(3.4), one gets, for some constant
The last inequality leads to the fact that the sequence uniformly converges to a stochastic process v on
It is clear that,
(3.4)
(3.4)
From (3.5) and the fact that we get
. Using (2.1) and (2.5), we get also
. This complete the proof of the theorem, (El-Borai, Citation2002, Citation2004; El-Borai, El-Nadi, Labib, & Ahmed, Citation2004; El-Nadi, Citation2005).
4. Stability of solutions
In order to study the stability results for problem (1.1), (1.2), we shall prove that the weak solutions of the Cauchy problem (1.1), (1.2) depends continuously on the part of the initial condition . Let
be weak solutions of the equations
(3.5)
(3.5)
with initial conditions(4.1)
(4.1)
where is of the form
for some
. It is supposed that
and
are given bounded continuous functions on
Theorem 4.1
If for sufficiently small positive number then
, for some positive constant M.
Proof
We have(4.2)
(4.2)
where(4.3)
(4.3)
Using (4.3), (4.4), (4.5) and remembering that satisfy Lipschitz condition, we get
consequently
From (4.3) and (4.4), we get the required result.
5. Conclusion
In this paper, we discussed the existence, uniqueness, and stability of solutions for the fractional stochastic partial differential equations without any restrictions on the characteristic forms when the Hurst parameter of the fractional Brownian motion is less than half. Our future work will be focused on investigate the approximate controllability for Hilfer fractional stochastic partial differential equations with fractional Brownian motion and Poisson jumps.
Acknowledgements
I would like to thank the referees and the editor for their careful reading and their valuable comments.
Additional information
Funding
Notes on contributors
M.M. El-Borai
M. M. M. El-Borai is a professor of mathematics, affiliated to the Department of Mathematics, Faculty of Science, Alexandria University Egypt. His recent research interests include dynamical systems and the artificial intelligence, stochastic differential equations, optimal control, stochastic control, dynamics of robot, abstract differential equations with fractional orders, stochastic differential equations with fractional orders, and general theory of partial differential equations.
H.M. Ahmed
H. M. Ahmed is a professor of mathematics, affiliated to the Department of Physics and Engineering Mathematics, Higher institute of engineering, El Shorouk academy, Cairo, Egypt. His recent research interests include fractional stochastic differential equations, controllability of fractional differential equations, delay differential equations, fractional impulsive differential equations, and exact solution of nonlinear partial differential equations.
References
- Arthi, G., Park, J. H, & Jung, H. Y. (2016). Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion. Communications in Nonlinear Science and Numerical Simulation, 32, 145–157.
- Balasubramaniam, P., Vembarasan, V., & Senthilkumar, T. (2014). Approximate controllability of impulsive fractional integro-differential systems with nonlocal conditions in Hilbert space. Numerical Functional Analysis and Optimization, 35(2), 177–197.
- Baudoin, F., Nualart, E., Ouyang, C., & Tindel, S. (2016). On probability laws of solutions to differential systems driven by a fractional Brownian motion. Annals of Probability, 44, 2554–2590.
- Boudaoui, A., Caraballo, T., & Ouahab, A. (2016). Impulsive neutral functional differential equations driven by a fractional Brownian motion with unbounded delay. Applicable Analysis, 95, 2039–2062.
- Caraballo, T., Diop, M. A., & Ndiaye, A. A. (2014). Asymptotic behavior of neutral stochastic partial functional integro-differential equations driven by a fractional Brownian motion. Journal of Nonlinear Sciences and Applications, 7, 407–421.
- Decreusefond, L., & Ustunel, A. S. (1999). Stochastic analysis of the fractional Brownian motion. Potential Analysis, 10(2), 177–214.
- Diop, M. A., Ezzinbi, K., & Mbaye, M. M. (2015). Existence and global attractiveness of a pseudo almost periodic solution in p-th mean sense for stochastic evolution equation driven by a fractional Brownian motion. Stochastics, 87, 1061–1093.
- Duncan, T., & Nualart, D. (2009). Existence of strong solutions and uniqueness in law for stochastic differential equations driven by fractional Brownian motion. Stochastics and Dynamics, 9(3), 423–435.
- El-Borai, M. M. (2002). Some probability densities and fundamental solution of fractional evolution equations. Chaos, Solitons Fractals, 14(3), 433–440.
- El-Borai, M. M. (2004). The fundamental solutions for fractional evolution equations of parabolic type. Journal of Applied Mathematics and Stochastic Analysis, 3, 197–211.
- El-Borai, M. M., & El-Nadi, K. E.-S. (2017). On some stochastic nonlinear equations and fractional Brownian motion. Caspian Journal of Computational & Mathematical Engineering, 3, 20–32.
- El-Borai, M. M., El-Nadi, K. E.-S., & El-Akabawy, E. G. (2010). On some fractional evolution equations. Computers and Mathematics with Applications, 59, 1352–1355.
- El-Borai, M. M., El-Nadi, K. E.-S., & Fouad, H. A. (2010). On some fractional stochastic delay differential equations. Computers and Mathematics with Applications, 59, 1165–1170.
- El-Borai, M. M., El-Nadi, K. E.-S., Labib, O., & Ahmed, H. M. (2004). Volterra equations with fractional stochastic integrals. Mathematical Problems in Engineering, 5, 453–468.
- El-Borai, M. M., & El-Said, K. (2015). A prabolic transform and some stochastic ill-posed problem. British Journal of Mathematics and Computer, Science, 9, 418–426.
- Elliott, R. J., & Van Der Hoek, J. (2003). A general fractional white noise theory and applications to finance. Mathematical Finance, 13, 301–330.
- El-Nadi, K. E. (2005). On some stochastic differential equations and fractional Brownian motion. International Journal of Pure and Applied Mathematics, 24, 416–423.
- Grecksch, W., & Anh, V. V. (1999). A parabolic stochastic differential equation with fractional Brownian motion input. Statistics & Probability Letters, 41, 337–345.
- Hamdy, M. A. (2015). Controllability of impulsive neutral stochastic differential equations with fractional Brownian motion. IMA Journal of Mathematical Control and Information, 32(4), 781–794.
- Maslowski, B., & Nualart, D. (2003). Evolution equations driven by a fractional Brownian motion. Journal of Functional Analysis, 202, 277–305.
- Nualart, D., & Ouknine, Y. (2002). Regularization of differential equation by fractional noise. Stochastic Processes and their Applications, 102, 103–116.
- Ren, Y., Hou, T., & Sakthivel, Y. (2015). Non-densely defined impulsive neutral stochastic functional differential equations driven by fBm in Hilbert space with infinite delay. Frontiers of Mathematics in China, 10(2), 351–365.
- Ren, Y., Wang, J., & Hu, L. (2017). Multi-valued stochastic differential equations driven by G-Brownian motion and related stochastic control problems. Internationl Journal of Control, 90, 1132–1154.
- Sathiyara, T., & Balasubramaniam, P. (2017). Controllability of fractional higher order stochastic integrodifferential systems with fractional Brownian motion. ISA Transacations,. doi:10.1016/j.isatra.2017.11.005
- Tamilalagan, P., & Balasubramaniam, P. (2017a). Approximate controllability of fractional stochastic differential equations driven by mixed fractional Brownian motion via resolvent operators. International Journal of Control, 90(8), 1713–1727.
- Tamilalagan, P., & Balasubramaniam, P. (2017b). Moment stability via resolvent operators of fractional stochastic differential inclusions driven by fractional Brownian motion. Applied Mathematics and Computation, 305, 299–307.
- Tindel, S., Tudor, C., & Viens, F. (2003). Stochastic evolution equations with fractional Brownian motion. Probability Theory and Related Fields, 127(2), 186–204.