Oscillation of a time fractional partial differential equation

Electronic Journal of Qualitative Theory of Differential Equations 2014, No. 15, 1–10; http://www.math.u-szeged.hu/ejqtde/ Oscillation of a time fractional partial differential equation P. PrakashB1,S. Harikrishnan1,J. J. Nieto2,3 and J.-H. Kim4 1Department of Mathematics, Periyar University, Salem - 636 011, India 2Departamento de Análisis Matemático, Facultad de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain. 3Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia. 4Department of Mathematics, Yonsei University, Seoul 120-749, South Korea Received 18 December 2013, appeared 25 March 2014 Communicated by Paul Eloe Abstract. We consider a time fractional partial differential equation subject to the Neumann boundary condition. Several sufficient conditions are established for oscillation of solutions of such equation by using the integral averaging method and a generalized Riccati technique. The main results are illustrated by examples. Keywords: oscillation, fractional derivative, fractional differential equation. 2010 Mathematics Subject Classification: 35B05, 35R11, 34K37. 1 Introduction In recent years differential equations with fractional order derivatives have attracted many researchers because of their applications in many areas of science and engineering. The need for fractional order differential equations originates in part from the fact that many phenomena cannot be modeled by differential equations with integer derivatives. Analytical and numerical techniques have been developed to study such equations. The fractional calculus has allowed the operations of integration and differentiation to be applied. Recently, the theory of fractional differential equations and their applications have been attracting more and more attention in the literature [1,3,4,11,12,16–18,20,24,25,27]. Fractional differential equations are generalizations of classical differential equations of integer order and have gained considerable importance due to their various applications in viscoelasticity, rheology, dynamical processes in self-similar and porous structures, diffusive transport akin to diffusion, electroanalytical chemistry, optics and signal processing, control theory, electrical networks, probability and statistics and economics, etc. Nowadays the interest in the study of fractional-order differential equations lies in the fact that fractional-order models are more accurate than integer-order ones, that is, there are more degrees of freedom in the fractional-order models. Fractional-order differential equations are BCorresponding author. Email: [email protected] 2P. Prakash, S. Harikrishnan, J. J. Nieto and J.-H. Kim also better in the description of hereditary properties of various materials and processes than integer-order differential equations. It is found that various applications can be elegantly modeled with the help of the fractional differential equations [6,10,15]. Also fractional differential and integral equations provide in some cases more accurate models of systems under consideration. The study of oscillation theory for various equations like ordinary and partial differential equations, difference equation, dynamics equation on time scales and fractional differential equations is an interesting area of research and much effort has been made to establish oscillation criteria for these equations [7,9,14,19,21,22,26,28]. Recently the research on fractional differential equation is a hot topic and only very few publications paid the attention to oscillation of fractional differential equation; see for example [2,5,7,13,23]. However, to the best of our knowledge, very little is known regarding the oscillatory behavior of fractional differential equations. But the study of oscillatory behavior of fractional partial differential equation is initiated in this paper. To develop the qualitative properties of fractional partial differential equations, it is of great interest to study the oscillatory behavior of fractional partial differential equation. In this paper, we establish several oscillation criteria for fractional partial differential equation by applying a generalized Riccati transformation technique and certain parameter functions. These results are considered essentially new. We also provide two examples to illustrate the results. In this paper, we consider the time fractional partial differential equation of the form ∂ ∂tr(t)Dα +,tu(x,t)+q(x,t)fZt 0(t−ν)−αu(x,ν)dν=a(t)∆u(x,t), (x,t)∈G=Ω×R+, (1.1) with the Neumann boundary condition ∂u(x,t) ∂N=0, (x,t)∈∂Ω×R+, (1.2) where α∈(0,1)is a constant, Dα +,tuis the Riemann–Liouville fractional derivative of order α of uwith respect of t,Ωis a bounded domain in Rnwith piecewise smooth boundary ∂Ω,∆is the Laplacian operator and Nis the unit exterior normal vector to ∂Ω. Throughout this paper, we assume that the following conditions hold: (A1)r(t)∈C1([0, ∞);[0, ∞)),a∈C([0, ∞);R+); (A2)q(x,t)∈C(G;[0, ∞)) and min x∈Ωq(x,t) = Q(t); (A3)f:R→Ris a continuous function such that f(u)/u≥µfor certain constant µ>0 and for all u6=0. By a solution of equation (1.1) we mean a function u(x,t)∈C1+α(Ω×[0, ∞)) such that Rt 0(t−ν)−αu(x,ν)dν∈C1(G;R),Dα +,tu(x,t)∈C1(G;R)and satisfies (1.1) on G. A solution uof (1.1) is said to be oscillatory in Gif it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory. Oscillation of a time fractional partial differential equation 3 2 Preliminaries In this section, we give the definitions of fractional derivatives and integrals and a lemma which are useful throughout this paper. There are several kinds of definitions of fractional derivatives and integrals. In this paper, we use the Riemann–Liouville left-sided definition on the half-axis R+. The following notations tions will be used for our convenience v(t) = ZΩu(x,t)dx. Definition 2.1. The Riemann–Liouville fractional partial derivative of order 0 <α<1 with respect to tof a function u(x,t)is given by (Dα +,tu)(x,t):=∂ ∂t 1 Γ(1−α)Zt 0(t−ν)−αu(x,ν)dν(2.1) provided the right hand side is pointwise defined on R+where Γis the gamma function. Definition 2.2. The Riemann–Liouville fractional integral of order α>0 of a function y:R+→ Ron the half-axis R+is given by (Iα +y)(t):=1 Γ(α)Zt 0(t−ν)α−1y(ν)dνfor t>0 (2.2) provided the right hand side is pointwise defined on R+. Definition 2.3. The Riemann–Liouville fractional derivative of order α>0 of a function y:R+→Ron the half-axis R+is given by (Dα +y)(t):=ddαe dtdαeIdαe−α +y(t) =1 Γ(dαe−α) ddαe dtdαeZt 0(t−ν)dαe−α−1y(ν)dνfor t>0 (2.3) provided the right hand side is pointwise defined on R+where dαeis the ceiling function of α. Lemma 2.4. Let y be a solution of (1.1)and G(t):=Zt 0(t−ν)−αy(ν)dνfor α∈(0,1)and t >0. (2.4) Then G0(t) = Γ(1−α)(Dα +y)(t). (2.5) Proof. From (2.3) and (2.4), for α∈(0,1)and t>0, we obtain G0(t) = Γ(1−α)1 Γ(1−α) d dt Zt 0(t−ν)−αy(ν)dν =Γ(1−α)"1 Γ(dαe−α) ddαe dtdαeZt 0(t−ν)dαe−α−1y(ν)dν# =Γ(1−α)(Dα +y)(t). The proof is complete. 4P. Prakash, S. Harikrishnan, J. J. Nieto and J.-H. Kim 3 Main results Theorem 3.1. If the fractional differential inequality d dt [r(t)Dα +v(t)]+Q(t)f(G(t)) ≤0 (3.1) has no eventually positive solution, then every solution of (1.1)and (1.2)is oscillatory in G. Proof. Suppose that uis a nonoscillatory solution of (1.1) and (1.2). Without loss of generality we may assume that u(x,t)>0 in G×[t0,∞)for some t0>0. Integrating (1.1) over Ω, we obtain d dt r(t)ZΩ (Dα +,tu)(x,t)dx+ZΩq(x,t)fZt 0(t−ν)−αu(x,ν)dνdx =a(t)ZΩ ∆u(x,t)dx. (3.2) Using Green’s formula, it is obvious that ZΩ ∆u(x,t)dx ≤0, t≥t1. (3.3) By using Jensen’s inequality and (A2), we have ZΩq(x,t)fZt 0(t−ν)−αu(x,ν)dνdx ≥Q(t)fZΩZt 0(t−ν)−αu(x,ν)dνdx =Q(t)fZt 0(t−ν)−αZΩu(x,ν)dxdν. (3.4) Combining (3.2)–(3.4) and using definitions, we have d dt [r(t)Dα +v(t)]+Q(t)f(G(t))≤0. (3.5) Therefore v(t)is an eventually positive solution of (3.1). This contradicts the hypothesis and completes the proof. Theorem 3.2. Suppose that the conditions (A1)–(A3)and Z∞ t0 1 r(t)dt =∞(3.6) hold. Furthermore, assume that there exists a positive function c ∈C1[t0,∞)such that limsup t→∞Zt t1µc(s)Q(s)−1 4 (c0(s))2r(s) c(s)Γ(1−α)ds =∞. (3.7) Then every solution of (3.1)is oscillatory. Oscillation of a time fractional partial differential equation 5 Proof. Suppose that v(t)is a nonoscillatory solution of (3.1). Without loss of generality we may assume that vis an eventually positive solution of (3.1). Then there exists t1≥t0such that v(t)>0 and G(t)>0 for t≥t1. Then it is obvious that [r(t)Dα +(v(t))]0≤ −Q(t)f(G(t)) <0, t≥t0. (3.8) Thus Dα +v(t)≥0 or Dα +v(t)<0, t≥t1for some t1≥t0. We now claim that (Dα +v(t)) ≥0 for t≥t1. Suppose not, then (Dα +v(t)) <0 and there exists T≥t1such that (Dα +v(T)) <0. Since [r(t)(Dα +v(t))]0<0 for t≥t1, it is clear that r(t)(Dα +v(t))e<r(T)(Dα +v(T)) for t≥T. Therefore, from (2.4), we have G0(t) Γ(1−α)= (Dα +v(t)) ≤r(T)(Dα +v(T)) r(t). Integrating the above inequality from Tto t, we have G(t)−G(T) Γ(1−α)=r(T)(Dα +v(T)) Zt T 1 r(s)ds G(t) = G(T)−Γ(1−α)r(T)(Dα +v(T)) Zt T 1 r(s)ds Letting t→∞, we get limt→∞G(t)≤ −∞which is a contradiction. Hence (Dα +v(t)) ≥0 for t≥t1holds. Define the function wby the generalized Riccati substitution w(t) = c(t)r(t)(Dα +v(t)) G(t)for t≥t1(3.9) Then we have w(t)>0 for t≥t1. From (A3), (2.4), (3.1) and (3.9), it follows that w0(t) = c0(t)r(t)(Dα +v(t)) G(t)+c(t)(r(t)(Dα +v(t)))0 G(t)−G0(t)r(t)(Dα +v(t)) G2(t) ≤c0(t) c(t)w(t)−c(t)Q(t)f(G(t)) G(t)−c(t)r(t)Γ(1−α)(Dα +v(t))2 G2(t) ≤c0(t) c(t)w(t)−µc(t)Q(t)−Γ(1−α) c(t)r(t)w2(t) =−µc(t)Q(t)− sΓ(1−α) c(t)r(t)w(t)−1 2sc(t)r(t) Γ(1−α) c0(t) c(t)!2 +1 4 (c0(t))2r(t) c(t)Γ(1−α) ≤ − µc(t)Q(t) + 1 4 (c0(t))2r(t) c(t)Γ(1−α). (3.10) Integrating both sides from t1to t, we have w(t)≤w(t1)−Zt t1µc(s)Q(s)−1 4 (c0(s))2r(s) c(s)Γ(1−α)ds. (3.11) Letting t→∞, we get limt→∞w(t)≤ −∞which contradicts (3.7) and completes the proof. Corollary 3.3. Let assumption (3.7)in Theorem 3.2 be replaced by limsup t→∞Zt t1 c(s)Q(s)ds =∞(3.12) and limsup t→∞Zt t1 r(s)(c0(s))2 c(s)ds <∞. (3.13) Then every solution of (3.1)oscillates. 6P. Prakash, S. Harikrishnan, J. J. Nieto and J.-H. Kim From Theorem 3.2, by choosing the function cappropriately, we obtain different sufficient conditions for oscillation of (3.1) and if we define a function cby c(t) = 1 and c(t) = t, we have the following oscillation results. Corollary 3.4. Suppose that (3.6)holds. If limsup t→∞Zt t1 Q(s)ds =∞, (3.14) then every solution of (3.1)oscillates. Corollary 3.5. Suppose that (3.6)holds. If limsup t→∞Zt t1µsQ(s)−1 4 r(s) sΓ(1−α)ds =∞, (3.15) then every solution of (3.1)oscillates. For the following theorem, we introduce a class of functions R. Let D0={(t,s):t>s≥t0},D={(t,s):t≥s≥t0}. The function H∈C(D,R)is said to belong to the class R, if (i) H(t,t) = 0, for t≥t0,H(t,s)>0, for (t,s)∈D0; (ii) Hhas a continuous and non-positive partial derivative ∂H(t,s) ∂son D0with respect to s. We assume that ξ(t)for t≥t0are given continuous functions such that ξ(t)≥0 and differentiable and define θ(t) = c0(t) c(t)+2Γ(1−α)ξ(t),χ(t) = c(t)[r(t)ξ(t)]0−Γ(1−α)c(t)r(t)ξ2(t). Theorem 3.6. Suppose that the conditions (A1)–(A4)and (3.6)hold. Furthermore assume that there exists H ∈ R such that limsup t→∞ 1 H(t,t1)Zt t1(µc(s)q(s)−χ(s))H(t,s)−1 4 c(s)r(s)h2(t,s) Γ(1−α)H(t,s)ds =∞. (3.16) Then every solution of (3.1)is oscillatory. Proof. Suppose that v(t)is a nonoscillatory solution of (3.1). Without loss of generality we may assume that vis an eventually positive solution of (3.1). Then there exists t1≥t0such that v(t)>0 and G(t)>0 for t≥t1. Proceeding as in the proof of Theorem 3.2, we obtain (Dα +v(t)) ≥0 for t≥t1. Now we define the Riccati substitution wby w(t) = c(t)r(t)(Dα +v)(t) G(t)+r(t)ξ(t), (3.17) Then we have w0(t) = c0(t)r(t)(Dα +v)(t) G(t)+r(t)ξ(t) +c(t)(r(t)(Dα +v)(t))0 G(t)−r(t)G0(t)(Dα +v)(t) G2(t)+ (r(t)ξ(t))0 ≤c0(t) c(t)w(t) + c(t)[r(t)ξ(t)]0−µc(t)Q(t)−Γ(1−α)c(t) r(t)w(t) c(t)−r(t)ξ(t)2 . (3.18) Oscillation of a time fractional partial differential equation 7 Let A=w(t) c(t),B=r(t)ξ(t). By applying the inequality [8], A(1+α)/α−(A−B)(1+α)/α≤B1/α1+1 αA−1 αB, for α=odd odd ≥1, we see that w(t) c(t)−r(t)ξ(t)2 =w(t) c(t)2 + [r(t)ξ(t)]2−2r(t)ξ(t) c(t)w(t). (3.19) Substituting (3.19) into (3.18), we have w0(t)≤c0(t) c(t)+2Γ(1−α)ξ(t)w(t)−Γ(1−α) c(t)r(t)w2(t)−µc(t)Q(t) +c(t)[r(t)ξ(t)]0−Γ(1−α)c(t)r(t)ξ2(t) ≤θ(t)w(t) + χ(t)−µc(t)Q(t)−Γ(1−α) c(t)r(t)w2(t). Multiplying both sides by H(t,s)and integrating from t1to t, for t≥t1, we have Zt t1 [µc(s)Q(s)−χ(s)]H(t,s)ds ≤ − Zt t1 H(t,s)w0(s)ds +Zt t1 H(t,s)θw(s)ds −Zt t1 Γ(1−α) c(s)r(s)w2(s)H(t,s)ds. (3.20) Using the integration by parts formula, we get −Zt t1 H(t,s)w0(s)ds =−[H(t,s)w(s)]t t1+Zt t1 H0 s(t,s)w(s)ds <H(t,t1)w(t1) + Zt t1 H0 s(t,s)w(s)ds. (3.21) Substituting (3.21) into (3.20), we have Zt t1 [µc(s)Q(s)−χ(s)]H(t,s)ds ≤H(t,t1)w(t1) + Zt t1[H0 s(t,s) + H(t,s)θ(s)]w(s)−Γ(1−α)H(t,s) c(s)r(s)w2(s)ds ≤H(t,t1)w(t1) + Zt t1h(t,s)w(s)−Γ(1−α)H(t,s) c(s)r(s)w2(s)ds ≤H(t,t1)w(t1) + Zt t1"sΓ(1−α)H(t,s) c(s)r(s)w(s)−1 2sc(s)r(s) Γ(1−α)H(t,s)h(t,s)#2 ds +1 4Zt t1 c(s)r(s)h2(t,s) Γ(1−α)H(t,s)ds ≤H(t,t1)w(t1) + 1 4Zt t1 c(s)r(s)h2(t,s) Γ(1−α)H(t,s)ds, which yields Zt t1[µc(s)Q(s)−χ(s)]H(t,s)−1 4Zt t1 c(s)r(s)h2(t,s) Γ(1−α)H(t,s)ds ≤H(t,t1)w(t1). 8P. Prakash, S. Harikrishnan, J. J. Nieto and J.-H. Kim Since 0 <H(t,s)≤H(t,t1)for t>s≤t1, we have 0 <H(t,s) H(t,t1)≤1 for t>s≤t1. Hence we have 1 H(t,t1)Zt t1(µc(s)q(s)−χ(s))H(t,s)−1 4 c(s)r(s)h2(t,s) Γ(1−α)H(t,s)ds ≤w(t1). Letting t→∞, we have limsup t→∞ 1 H(t,t1)Zt t1(µc(s)q(s)−χ(s))H(t,s)−1 4 c(s)r(s)h2(t,s) Γ(1−α)H(t,s)ds ≤w(t1) which contradicts (3.16) and completes the proof. In Theorem 3.6, if we choose H(t,s) = (t−s)λ,t≥s≥t1, where λ>1 is a constant, then we obtain the following corollaries. Corollary 3.7. Under the conditions of Theorem 3.6, if limsup t→∞ 1 (t−t1)λZt t1(µc(s)Q(s)−χ(s))(t−s)λ−1 4 c(s)r(s)((t−s)θ(s)−λ) Γ(1−α)(t−s)ds <∞, then every solution of (3.1)is oscillatory. 4 Examples Example 4.1. Consider the time-fractional partial differential equation ∂ ∂t(Dα +,tu(x,t)) + ex t2Zt 0(t−ν)−αu(x,ν)dν=et 4∆u(x,t),(x,t)∈(0, π)×(0, ∞), (4.1) with the boundary conditions ux(0, t) = ux(π,t) = 0, where α∈(0,1), In (4.1), r(t) = 1, Q(t) = minx∈Ωq(x,t) = minx∈(0,π)ex t2=1 t2,a(t) = et 4and f(u) = u. Take t0>0 and µ=1. Thus all the conditions of the theorem (3.6) hold. Therefore every solution of (4.1) is oscillatory. Example 4.2. Consider the time-fractional partial differential equation ∂ ∂t(t2Dα +,tu(x,t)) + 2 t2exp Zt 0(t−ν)−αu(x,ν)dν·Zt 0(t−ν)−αu(x,ν)dν =t 2∆u(x,t),(x,t)∈(0, π)×(0, ∞), (4.2) with the boundary conditions ux(0, t) = ux(π,t) = 0, where α∈(0,1), In (4.2), r(t) = t2,Q(t) = minx∈Ωq(x,t) = minx∈(0,π)2 t2=2 t2,a(t) = t 2and f(u) = euu. Take t0>0 and µ=1. Thus all the conditions of the theorem (3.6) hold. Therefore every solution of (4.2) is oscillatory. Acknowledgements This work is supported by University Grants Commission (India) and partially supported by Ministerio de Economía y Competitividad (Spain), project MTM2010-15314, and co-financed by the European Community fund FEDER. Oscillation of a time fractional partial differential equation 9 References [1] S. ABBAS, M. BENCHOHRA, J. J. NIETO, Global attractivity of solutions for nonlinear fractional order Riemann–Liouville Volterra–Stieltjes partial integral equations, Electron. J. Qual. Theory Differ. Equ. 2012, No. 81, 1–15. 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