DE GRUYTER

DOI 10.1515/tmj-2017-0002

On the solutions of partial integrodifferential equations of

fractional order

Aruchamv Akilandeeswari1, Krishnan Balachandran1, Margarita Rivero2 and

Juan J, Trujillo3

1Department of Mathematics, Bharathiar University, Coimbatore 641046, India 2Departamento de Matemáticas, Estadística e I.O., Universidad de La Laguna, Tenerife, Spain 3Departamento de Análisis Matemático, Universidad de La Laguna, Tenerife, Spain E-mail: {akilamathematics, kb.maths.buJSgmail.com, mriveroSull.es, jtrujillSullmat.es

Abstract

The main purpose of this paper is to study the existence of solutions for the nonlinear fractional partial integrodifferential equations with Diriclilet. boundary condition. Under suitable assumption the results are established by using the Leray-Scliauder fixed point theorem and Arzela-Ascoli theorem. An example is provided to illustrate the main result.

2010 Mathematics Subject. Classification. 34A12. 45J05, 26A33, 47H10

Keywords. Existence, Partial Integrodiil'erential equations. Fractional derivatives. Fixed point theorem.

1 Introduction

Fractional calculus is the field of mathematical analysis which deals with the investigation and applications of integrals and derivatives of arbitrary order. Now fractional calculus is undergoing rapid developments with more applications in the real world. Numerous applications of fractional calculus can be found in fluid dynamics, stochastic dynamical systems, plasma physics, nonlinear control theory, image processing, nonlinear biological systems and quantum mechanics. For more details on history and applications of fractional calculus see [22]. [27] and [13] references therein.

Fractional derivatives provide more accurate models of real world problems than integer order derivatives. They also give an excellent instrument for the description of memory and properties of various materials and processes. This is the main advantage of fractional derivatives in comparison with classical integer order models. The solvability of different types of fractional differential equations have been established by Lakshmikantham et al. in [14]. Wang and Xie [31] established the existence and uniqueness of solution for fractional differential equations involving Riemann-Liouville differential operators with integral boundary conditions by employing the monotone iterative method. Agarwal et al. [1] discussed the initial value problem for a class of fractional neutral functional differential equations and obtained the existence criteria from Krasnoselskii's fixed point theorem. Momani and Odibat [20] compared the solutions of the fractional order differential equations by homotopy perturbation method and variational iteration method. Ahmed et al. [2] introduced a new concept of the coupling of nonlocal integral conditions and proved the existence and uniqueness of solutions for a coupled system of fractional differential equations. They also verified the existence results by means of Leray-Scliauder alternative and Schaufer's fixed point theorem, while uniqueness result was derived from Banach's contraction principle.

To model the process with delay, it is not sufficient to employ an ordinary or partial differential equation. An approach to resolve this problem is to use integrodifferential equations. Many-

Tbilisi Mathematical Journal 10(1) (2017), pp. 1B-2B Tbilisi Centre for Mathematical Sciences.

Received by the editors: 20 March 2015. Accepted for publication: 01 April 2016.

mathematical formulations of physical phenomena load to intogrodifforontial equations. There are few articles available in the literature for the study of fractional intogrodifforontial equations. For example. Balachandran et al. [4.5] studied the existence results for several kinds of fractional intogrodifforontial equations in a Banach space using fixed point technique. In [32]. Zhang et al. investigated the existence of nonnegativo solutions for nonlinear fractional differential equations with nonlocal fractional intogrodifforontial boundary conditions on an unbounded domain by using the Leray-Schandor nonlinear alternative theorem. The differential transform method was applied to fractional intogrodifforontial equations in [3] to solve those equations analytically. The solutions of system of fractional partial differential equations has boon found by Parthiban and Balachandran [25] by using Adomain decomposition method.

Another interesting area of research is the investigation of fractional partial differential equations. Because of their immense applications in scientific fields, fractional partial differential equations are found to be an effective tool to describe certain physical phenomena, snch as diffusion processes [10] and viscoelasticity theories [12]. In recent years, increasing number of papers by many authors from various fields of science and engineering deal with dynamical systems described by-fractional partial differential equations. Some partial differential equations of fractional order type like one-dimonsional time-fractional diffusion-wavo equation were used for modeling relevant physical processes (see [26]). Regarding fractional partial differential equations. Luchko [18] used the Fourier transform method of the variable separation to construct a formal solution and under certain condition he showed that the formal solution is the generalized solution of the initial-boundary value problem. To prove the uniqueness he used the maximum principle for generalized time fractional diffusion equation [17]. By applying the energy inequality. Onssaeif and Bouziani [24] proved the existence and uniqueness of solution for parabolic fractional differential equations in a functional weighted Sobolev space with integral conditions. Joice Nirniala and Balachandran [28] determined the solution of time fractional telegraph equation by means of Adomain decomposition method and analysed the efficiency of this method. Using measure of noncompactness and Monch's fixed point theorem, the existence of solutions is studied by Guo and Zhang [9] for a class of impulsive partial hyperbolic differential equations. In this paper, we extend the results of [23] to fractional order partial intogrodifforontial equation.

2 Preliminaries

In this section, we introduce some notations and basic facts of fractional calculus. Let Q C R and C(J, R) is the Banach space of all continuous functions from J = [0, T] into R. Let r(-) denote the gamma function. For any positive integer 0 < a < 1, the Riemann Liouville derivative and Caputo derivative are defined as follows:

Definition 2.1. [ ] The partial Riemann-Liouville fractional integral operator of order a > 0 with respect to t of a function f (x,t) is defined by

Iaf (x,t) = r(a j(t - s)a-1f (x, s) ds.

function f (x,t) with respect to t of the form

Daf (x,t) = , 1 , d if ^ ds. f ( , ) r(1 - a) dtj (t - s)a

Definition 2.3. [ ] The Caputo partial fractional derivative of order a > 0 with respect to t of f(x, t)

C Df (x,t) = -r"V ds.

JK ' ' r(1 - a) J (t - s)a ds

To know more properties above fractional operators and historical aspects of they refer the books [19] and [28]. For more details on the geometric and physical interpretation for fractional derivatives of Caputo types see [6]. There has been a significant development in ordinary and partial differential equations involving both Riemann-Lionville and Caputo fractional derivatives in the past few years, for instance, see the papers of Gojji and Jafari [8]. Furati and Tatar [7]. The

Riemann Lionvillo and Caputo fractional derivatives are linked by the following relationship.

(x,t) = Df (x,t) - .

About the called Caputo derivative we must remark here that Lionvillo in [15] and [16] was the first that introduced formally the called fractional Caputo derivative of order 2 with the objective to solve certain integral equation connected with the known Tantochrone problem.

In this paper, we consider the fractional partial integrodifferential equation of the form

cDau(x, t) = a(t)Au(x, t) + f(t,u(x,t), J g(t,s,u(x,s)) ds\ , t £ J, (2.1)

where 0 < a < 1 ^d the nonlinear functions g : J x J x R ^ R mid f : J x R x R ^ R. The initial and boundary conditions are

u(x, 0) = <^(x), x £ Q,

u(x, t) = 0, (x,t) £ dQ x J.

where ^(x) £ L1(R). In order to establish our result assume the following conditions.

(Hi) f (t, u^ u2) is continuous with respect to u^ u2, Lebesgue measurable with respect to t and satisfies

J $(x)f (t,ui,u2)dx / J$(x)ui(x,t) dx f $(x)u2(x, t) dx\ Q < f(t, Q Q *

/ $(x) dx \ ' f $(x) dx ' / $(x) dx

where $(x) is an eigenfunction.

(H2) There exists an integrable function mi(i) : J ^ [0, to) such that

II f (t,ui,U2) ||< mi(t) ^ |H|,

where m1(t) > 0 and ( /(m1(s)) 0 ds I < /1; for some ft G (0, a). ^ 0 '

(H3) g(t, s, u) is continuous with respect to u, Lebesgue measurable with respect to t and also satisfies the inequality

f $(x)g(t, s, u) dx / f §(x)u(x,t) dxs

< g ^ s

/$(x) dx " ' ' /$(x) dx r Q Q

(H4) There exists an integrable function m2(t,s) : J x J ^ [0, to) such that

11 g(t s,u) y< m2(t s) ||u|.

(H5) a(t) is continuous on J and for ft as in (H2), ( /(a(s))0 ds I < /2.

(H6) There exists an integrable function m(t, s) = m1(t)m2(t, s) such that ( /(m(s, t))0 ds I < /3,

0 < ft < a.

It is easy to show that the initial value problem (2.1) is equivalent to the following equation

u(x,t) = y(x) +--—- (t — s)a-1[a(s)Au(x,s) + f (s, u(x, s), v(x, s))] ds, (2-2)

r(a) J

where v(x, s) = f g(s, t, u(x, t)) dr, for t G J.

3 Existence Results

Consider the following eigenvalue problem

Au + Au = 0, (x, t) G Q x J, 1 , .

u = 0, (x, t) G dQ x J, J '

is well known by [ ]. Thus, for x G Q the smallest eigenvalue A1 of the problem ( ) is positive and the corresponding eigenfunction $(x) > 0. Now we define the function U(t) as

f u(x, t)$(x) dx

U(t) = Q sA-. (3-2)

J $(x) dx

Theorem 3.1. Assume that there exists aft G (0, a) for some a > 0 such that (H1)-(H6) holds. For any constant b > 0, suppose that

r = min < T,

r(a)b fa — ft

(||U (0)|| + b)(A1/1 + /2 + /sUl — ft

Then there exists at least one solution for the initial value problem ( ) on Q x [0, r]. Proof. First we have to prove the initial value problem (2.1) has a solution if and only if the equation

t t u(t) = U(0) — rAa) /(t — s)a-1U(s) ds + rO) |(t — s)a-1f (s, U(s), V(s)) ds, (3.4)

where V(t) = / g(t, s, U(t)), has a solution.

Step 1. The proof of sufficiency is similar to that of Lemma 3.1 [23]. To prove the necessary part.

u(x, t) u(x, t)

sides of equation ( ) by $(x) and integrating with respect to x G Q, we get

I $(x)u(x,t) dx = I $(xWx) dx + —^ I $(x) / (t — s)a-1a(s)Au(x, s) ds dx J J r(a) j Jo

+ i $(x) / (t — s)a-1f (s,u(x, s),v(x, s)) ds dx

r(a) J Jo

Using Green's formula and assumption (H1), we get

t t U(t) < U(0) — ray /(t — s)a-1a(s)U(s) ds + |(t — s)a-1f (s, U(s), V(s)) ds. (3.5)

Let K = {U : U G C(J, R), || U(t) — U(0) ||< b}. Define an operator F : C(J,R) ^ C(J, R) as

t t FU(t) = U(0) — rAa) /(t — s)a-1a(s)U(s) ds + |(t — s)a-1f (s, U(s), V(s)) ds. (3.6)

Clearly U(0) G K. This means that K is nonempty. From our construction of K, we can say that K is closed and bounded. Now for any U1, U2 G K and for any a1, a2 > 0 such that a1 + a2 = 1,

|| a1U1 + a2U2 — U(0) || < «1 || U1 — U(0) || +a2 || U2 — U(0) ||

< «1b + 02b = b.

Thus a1U1 + a2U2 £ K. Therefore K is nonempty closed convex set. Next we have to prove the operator F maps K into itself.

FU(t) - FU(0) || =

t t Al J(t - s)a-1a(s)U(s) ds+f(L/(t - s)a-1f (s, U(s), V(s)) ds

r(a) J v 7 ww r(a)

t t < rAa) ("U(0)1 + b)/(t - s)a-1||a(s)|| ds + |(t - s)a-1||f (s, U(s), V(s))|| ds.

Then by using Holder inequality and (He), we arrive

/ t \1-P / t \ P

|| FU(t) - FU(0) || < rAa) ("U(0)|| + b) N ((t - s)a-1)^ dsj [[ ||a(s)||1 ds

+ rR / m1 (s)(t - s)a-1 (|U(s)|| + ||V(s)||) ds

ft \ 1-P / t \ P

< pAo) (|U(0)| + b) f|((t - s)a-1)^ dsj U ||a(s)||1 ds

f t \ 1-p f t \ p + rO)(||U(0)|| + b) (/ ((t - s)a-1) ^ ds j if (m1 (s)) * ds

f t \ 1-P f t \ P

+ r(ay (|U(0)| + b) (/ ((t - s)a-1)^ ds j y (m(s, t))T ds

< (||U(0)|| + b) A1l1 / i-^\1-P ra-p + (||U(0)|| + b) ¿2 / i-^\1-P ra-p

r(a) ya - Py r(a) ya - P

(||U(0)|| + b) z3 /1 - ^1-P

r(a) ya - P /

= (||U(0)|| + b) (A1Z1 + ¿2 + ¿3) f WA1-P ra-p r(a) Va - P/

Therefore F maps K into itself. Now define a sequence {Uk (t)} in K such that

U0(t) = U(0) and Uk+1(t) = Uk(t), k = 0,1, 2,... Since K is closed, there exists a sub sequence {Uki (t)} of Uk(t) mid U (t) £ K such that

lim Uki (t) = U(t).

Then Lebesgue's dominated convergence theorem yields that

t t U(t) = U(0) — jAO) /(t — s)a-1a(s)U(s) ds + fl) /(t — s)a-1f (s, U(s), y(s)) ds,

where y(t) = / g(t, s, U(t)). Next we claim that F is completely continuous.

Step 2. For that first we prove T : K ^ K is continuous. Let {Um(t)} be a converging sequence in K to U(t). Then for any e > 0, let

r(a)£ f a — ft^ ^„a-,3

||Um(t) — U (t)|< 2A1/1 U—ft) r

By assumption (H1),

f(t,Um(t),^ g(t,s,Um(r))ds) —^ f(t,U(t),^ g(t, s,, U(t))ds),

for each t G [0, r] and since

f(t,Um(t),^s g(t,s,Um(t)) ds) — f(t,U(t),^s g(t, s, U(t)) ds)

r(a)e fa — ft

2ra V 1 — ft

we have

A / / 1 — ft \ 1 / 1 — ft ^

||FUm(t) — FU (t)|| < -1-1 -ft ||Um(t) — U (t)|| + 1

r(a) ya — ft/ r(a) ya — ft

f(t,Um(s),^ g(s,r,Um(r))dr) — f(t,U(s),jT g(s, t, U(r)) dr)

Taking limit m ^ to, the right hand side of the above inequality tends to zero. Therefore F is continuous.

A1/1 + /2 + /3,, i Jl — ^1

|| FU(t) || < ||U(0)|| + 'r(a2^ "3 (||U(0)|| + b)(^Oz^J r < ||U (0)|| + b.

FK F K

family.

Step 4. Now let U G ^^d t1,t2 G J. Then if 0 < t1 < t2 < r, % the assumptions (H1) — (H6)

wo obtain

FU(t1) - FU(t2) || < rAa) (|U(0)|| + b)J ((t2 - s)a-1 - (t1 - s)a-1) ||a(s)|| ds

+ rAa) («U(0)| + b^(t2 - s)a-1|a(s)| ds

J ((t2 - s)a-1 - (t1 - s)a-1) f (s, U(s), V(s)) ds

|(t2 - s)a-1f (s, U(s), V(s))ds

< A1^1 I I ff+ „\a-1 U ^a-h

((t2 - s)a-1 - (t1 - s)a-1) ^ ds

+ jA(oy (|U(0)|| + b) I I((t2 - s)a-1)^ ds

+ r(0) dU(0)|| + b) [ / ((t2 - s)a-1 - (t1 - s)a-1)^ ds

+r(ay (|U(0)|| + b) [ / ((t2 - s)a-1)^ ds

+ r(0) (||U (0)| + b) [J ((t2 - s)a-1 - (t1 - s)a-1)^ ds

+ r(0) (||U(0)| + b) I y((t2 - s)a-1)^ ds

The right hand side is independent of U £ K. Sinee 0 < P < a < 1, the right hand side of the above inequality goes to zero as t1 ^ i^^us, F maps K into an equieontinuous family of functions.

Example

Consider the partial fractional intgrodifferential equation

CD2u(x,t) = t2Au(x,t) + t + u(x,t) +--k su(x,s)ds, (x,t) G Q x J (3.7)

1 +t2 J

with the initial condition

u(x, 0) = uo, x G Q

and the boundary condition

u(x,t) = 0, (x,t) G dQ x J,

t t where J = [0,1] Mid Q = [0, n/2]. Here a(t) = t2, J g(t, s, u(x, s)) ds = 2 J su(x, s) ds and

f (t,u(x,t), g(t, s,u(x, s)) ds) = t + u(x,t) +--^ su(x,s)ds. (3-8)

Jo 1+12 J

Since the eigenfunctions of the Laplacian operator are sin mx and cos mx where A = m2, we note that the assumptions (H1)-(H6) of Theorem are satisfied for some ft G (0,1/2). Hence the problem (3.7) has a solution.

Acknowledgment

The last two authors are thankful to FEDER funds and to project MTM2013-41704-P from the goverment of Spain, for the partial support.

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