HIGHER ORDER MULTI-TERM TIME-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS INVOLVING CAPUTO-FABRIZIO DERIVATIVE

Transcription

1 Electronic Journal of Differential Equations, Vol ), No. 243, pp.. ISSN: URL: or HIGHER ORDER MULTI-TERM TIME-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS INVOLVING CAPUTO-FABRIZIO DERIVATIVE ERKINJON KARIMOV, SARDOR PIRNAFASOV Communicated by Mokhtar Kirane Abstract. In this work we discuss higher order multi-term partial differential equation PDE) with the Caputo-Fabrizio fractional derivative in time. Using method of separation of variables, we reduce fractional order partial differential equation to the integer order. We represent explicit solution of formulated problem in particular case by Fourier series.. Introduction Consideration of new fractional derivative with non-singular kernel was initiated by Caputo and Fabrizio in their work []. Motivation came from application. Precisely, new fractional derivatives can better describe material heterogeneities and structures with different scales. Special role of spatial fractional derivative in the study of the macroscopic behaviors of some materials, related with nonlocal interactions, which are prevalent in determining the properties of the material was also highlighted. In their next work [2], authors represented some applications of the introduced fractional derivative. Nieto and Losada [3] studied some properties of this fractional derivative naming it as Caputo-Fabrizio CF) derivative. They introduced fractional integral associated with the CF derivative, applying it to the solution of linear and nonlinear differential equations involving CF derivative. Later, many authors showed interest in the CF derivative and as a result, several applications were discovered. For instance, in groundwater modeling, in electrical circuits, in controlling the wave movement, in nonlinear Fisher s reaction-diffusion equation, in modeling of a mass-spring-damper system, etc. We note also some recent works related with CF derivative [4, 5, 6, 7, 8, 9]. Different methods were applied for solving differential equations involving CF derivative. Namely, Laplace transform, reduction to integral equations, and reduction to integer order differential equations. Last two methods were used in the works [, ]. In this paper, we aim to show an algorithm to reduce initial value problem IVP) for multi-term fractional DE with CF derivative to the IVP for integer order DE and 2 Mathematics Subject Classification. 33E2. Key words and phrases. Fractional differential equations; separation of variables; Caputo-Fabrizio derivative. c 27 Texas State University. Submitted August, 27. Published October 5, 27.

2 2 E. KARIMOV, S. PIRNAFASOV EJDE-27/243 using this result to prove a unique solvability of a boundary-value problem BVP) for PDE involving CF derivative on time-variable. First, we give preliminary information on CF derivative and then we formulate our main problem. Representing formal solution of the formulated problem by infinite series, in particular case, we prove uniform convergence of that infinite series. 2. Preliminaries The fractional derivative of order < < ) in CF sense [2] is defined as CF D atgt) = This operator is well defined on the space a g s)e t s) ds. 2.) W, = { gt) L a, ) : gt) g a s))e t s) L a, t) L a, ) }, whose norm, for is given by gt) W, = a gt)dt + a g s s) e t s) dsdt, where g a t) = gt), t a, g a t) =, < t < a [2]. The following equality is shown in [4], CF D +n at gt) = CF D at CF D n atgt)). 3. Formulation of a problem and formal solution Consider the time-fractional PDE k λ n CF Dt +n ut, x) u xx t, x) = ft, x) 3.) n= in a domain Ω = {t, x) : < t < q, < x < }. Here λ n are given real numbers, ft, x) is a given function, k N, q R +. Problem. Find a solution of 3.) satisfying the following conditions: ut, x) C 2 n ut, x) Ω), t n W,, q), 3.2) i ut, x) ut, ) = ut, ) =, t= t i = C i, i =,, 2,..., k, 3.3) where C i are any real numbers. 3.. Solution of higher order multi-term fractional ordinary differential equations. We expand ut, x) as the Fourier series ut, x) = T m t) sin mπx, 3.4) m= where T m t) are the Fourier coefficients of ut, x). Substituting this representation in 3.) and considering the initial conditions 3.3), we get the following IVP with respect to time-variable: k λ n CF Dt +n T m t) + mπ) 2 T m t) = f m t), 3.5) n= T i) m ) = C i, i =,, 2,..., k

3 EJDE-27/243 HIGHER ORDER PDE WITH CAPUTO-FABRIZIO DERIVATIVE 3 where f m t) are Fourier coefficients of ft, x). Based on definition 2.) and initial conditions 3.3), after integrating by parts, we rewrite the fractional derivatives as follows: CF DtT m t) = T mt) ) 2 CF D + t T m t) = T mt) ) 2 T mt) + + T m) ) 2 e t T m) e t, CF Dt +2 T m t) = T mt) 3 ) 4 T m s)e t s) ds T m) e 2 ) 3 ) 2 T mt) + T m s)e 2 ) 3 T mt) T m s)e t s) ds 2 T m ) ) 3 e + T m) ) 2 e t T m) e t. t, t s) ds t Continuing this procedure, we find for n the formula CF D +n t T m t) = { n i= + ) t n+ ) i [ T n i) m t) T m n i) )e t] T m s)e t s) ds }. 3.6) We substitute 3.6) into 3.5) and deduce k n= λ n ) { n i= + ) t n+ = f m t). ) i [ T n i) m t) T m n i) )e t] } T m s)e t s) ds + mπ) 2 T m t) We multiply this equality by )e t : k { n λ n n= i= + = )e ) n+ t f m t). ) i [ T n i) m t)e t T m n i) ) ] } T m s)e s ds + mπ) 2 )T m t)e t 3.7)

4 4 E. KARIMOV, S. PIRNAFASOV EJDE-27/243 Introducing new function T m t) = T m t)e t, we rewrite some items of 3.7): T mt)e t = T m n) t)e t = T mt)e t = T mt) T m t), T mt) 2 T mt) + )2 Tm t),... n ) n j n! ) n j j) T m t). j!n j)! j= ) We note that T m t) = T m t). Considering 3.8), from 3.7) we deduce 3.8) k { n λ n n= i= ) i n i j= n i)! ) n i j j!n i j)! [ j) j) T m t) T m ) ] + ) t n+ T m s)ds = )e t f m t). } + mπ) 2 ) T m t) 3.9) Differentiating 3.9) once by t, we will get k + )th order DE. Using its general solution and applying initial conditions, one can get explicit form of functions T m t), consequently formal solution of the formulated problem is represented by infinite series 3.4). Imposing certain conditions to the given functions, we prove uniform convergence of infinite series, which will complete the proof of the unique solvability of the formulated problem. In the next section we show the complete steps in a particular case. We note that even this particular case was not considered before. 4. Particular case In this subsection we consider the case k = 2, to show the complete steps. In this case, after differentiating 3.9) once with respect to t, we get the following third order ordinary DE, where T m t) = T m t)e t, T m t) + A T mt) + A 2 T m t) + A 3 Tm = g m t) 4.) A 2 = 3 A 3 = A = 3 + λ λ 2, ) 2 λ 2λ + + mπ)2 ), λ 2 ) 2λ λ λ 2, ) 3 + g m t) = [f m t) + )f m t)]e t. 4.2)

5 EJDE-27/243 HIGHER ORDER PDE WITH CAPUTO-FABRIZIO DERIVATIVE General solution. The characteristic equation of 4.) is whose discriminant is µ 3 + A µ 2 + A 2 µ + A 3 =, m = 4A 3 A 3 + A 2 A 2 2 4A A A 2 A 3 27A 2 3. According to the general theory, form of solutions depends on the sign of the discriminant. Below we will give explicit forms of solutions case by case. Case m >. The characteristic equation has 3 different real roots µ, µ 2, µ 3 ), hence based on general solution we can find explicit form of T m t) as T m t) = C e µ )t + C 2 e µ2 )t + C 3 e µ3 )t + µ 2 µ 2 3 µ2 2 µ 3 µ µ µ2 µ 3 + µ µ 2 2 µ2 µ 2) [ e µ gm )t µ 3 µ 2 ) z) + )g mz) ) e + e µ2 )t µ µ 3 ) g m z) + )g mz))e + e µ3 )t µ 2 µ ) g m z) + )g mz))e µ)z dz µ2)z dz µ3)z dz ]. 4.3) Case m <. The characteristic equation has one real µ ) and two complexconjugate roots µ 2 = µ 2 ± iµ 22 ). Therefore, T m t) has the form )t T m t) = C e µ )t + C 2 cos µ 22 t + C 3 sin µ 22 t)e µ2 [ + µ µ2 2 2µ µ 2 + µ 2 e µ )t g m z) + )g mz))e µ)z dz + e µ2 )t cos µ 22 t µ sin µ 22 z µ 22 µ 2 sin µ 22 z µ 22 cos µ 22 z) g m z) + )g mz)) e + e µ2 )t sin µ 22 t µ 2 cos µ 22 z µ 22 sin µ 22 z µ 22 ] µ cos µ 22 z) g m z) + )g mz)) e µ2)z dz. µ2)z dz Case m =. We have have 2 sub-cases: a) Three real roots, two of which are equal, third one is different µ = µ 2, µ 3 ): T m t) = C e µ )t + C 2 te µ )t + C 3 e µ3 )t [ + µ 2 2µ e µ )t µ 3 z µ z)g m z) µ 3 + µ 3 ) + )g mz))e µ)z dz + te µ )t µ µ 3 ) g m z) + )g mz))e + e µ3 )t g m z) + )g mz))e µ3)z dz µ)z dz ] ; 4.4)

6 6 E. KARIMOV, S. PIRNAFASOV EJDE-27/243 b) all 3 real roots are the same µ = µ 2 = µ 3 ): T m t) = C e µ )t + C 2 te µ )t + C 3 t 2 e µ + 2 eµ )t z 2 g m z) + )g mz))e te µ )t + 2 t2 e µ )t zg m z) + )g mz))e )t g m z) + )g mz))e µ)z dz µ)z dz µ)z dz. 4.5) Here C j j =, 3) are any constants, which will be defined using initial conditions Convergence part. We consider the case m > in details. Found solution we satisfy to the initial conditions 3.5). Without losing generality, we assume that C i = i =, 2). Regarding the C j we will get the following algebraic system of equations where C + C 2 + C 3 = d, C µ ) + C 2µ 2 ) + C 3µ 3 ) = d 2, C µ )2 + C 2 µ 2 )2 + C 3 µ 3 )2 = d 3, d [ = µ 2 µ 2 3 µ2 2 µ 3 µ µ µ2 µ 3 + µ µ 2 2 µ2 µ µ 3 µ 2 ) gm z) 2) + )g mz) ) dz t= + µ µ 3 ) gm z) + )g mz) ) dz t= + µ 2 µ ) gm z) + )g mz) ) dz ] t=, 4.6) d 2 = µ 2 µ 2 3 µ2 2 µ 3 µ µ µ2 µ 3 + µ µ 2 2 µ2 µ 2) [µ )µ 3 µ 2 ) gm z) + )g mz) ) dz t= + µ 3 µ 2 )g m ) + )g m)) + µ 2 )µ µ 3 ) gm z) + )g mz) ) dz t= + µ µ 3 )g m ) + )g m)) + µ 3 )µ 2 µ ) 4.7) gm z) + )g mz) ) dz t= ] + µ 2 µ )g m ) + )g m)),

7 EJDE-27/243 HIGHER ORDER PDE WITH CAPUTO-FABRIZIO DERIVATIVE 7 d 3 = µ 2 µ 2 3 µ2 2 µ 3 µ µ µ2 µ 3 + µ µ 2 2 µ2 µ 2) [ µ )2 µ 3 µ 2 ) gm z) + )g mz) ) dz t= + µ )µ 3 µ 2 )g m ) + )g m)) + µ 3 µ 2 )g m) + )g m)) + µ 2 )2 µ µ 3 ) g m z) + )g mz)) dz t= + µ 2 )µ µ 3 ) g m ) + )g m)) + µ µ 3 )g m) + )g m)) + µ 3 )2 µ 2 µ ) g m z) + )g mz)) dz t= + µ 3 )µ 2 µ )g m ) + )g m)) ] + µ 2 µ )g m) + )g m)). 4.8) Solving this system, we get d 2 µ C = d ) + d 3 µ 2 )µ ) µ 2 d µ )2 )µ 2 ) d 2µ 2 ) + d 2µ ) d 3 [µ µ 2 µ 2 µ 3 µ 3 µ µ )2][ µ 2 )µ ) µ 2 )2]) d µ )µ 2 ) + d 2µ 2 ) + d 2µ µ µ 2 µ 2 µ 3 µ 3 µ µ )2 d 2 µ C 2 = ) + d 3 µ 2 )µ ) µ 2 d )µ )2 ) ) d 3 )µ 2 ) + d 2µ 2 ) + d 2µ ) d 3 [µ µ 2 µ 2 µ 3 µ 3 µ µ )2][ µ 2 )µ ) µ 2 )2]) C 3 = d )µ )µ 2 ) + d 2µ 2 ) + d 2µ ) d 3 µ µ 2 µ 2 µ 3 µ 3 µ µ )2 In general, we can write C j M d + M 2 d 2 + M 3 d 3. ) )

8 8 E. KARIMOV, S. PIRNAFASOV EJDE-27/243 Hence, we need the following estimations in order to provide convergence of used series: d M4 mπ) 4 f 4,z) + )f 4,z))e µ)z dz t= + M 5 f 4, z) + )f 4, z))e µ2)z dz t= + M 6 f 4, z) + )f 4, z))e µ3)z dz t= mπ) 4 M 7 d 2 M8 mπ) 4 e µ )t f 4, z) + )f 4, z))e µ)z dz t= + M 9 e µ2 )t f 4, z) + )f 4, z))e µ2)z dz t= + M e µ3 )t f 4, z) + )f 4, z))e µ3)z dz t= + M f 4, z) + )f 4, z)) mπ) 4 M 2 d 3 M3 mπ) 4 e µ )t f 4, z) + )f 4, z))e µ)z dz t= + M 4 e µ2 )t f 4, z) + )f 4, z))e µ2)z dz t= + M 5 e µ3 )t f 4, z) + )f 4, z))e µ3)z dz t= + M 6 f 4, z) + )f 4, z)) + M 7 f 4, z) + )f 4,2 z)) mπ) 4 M 8 Here M i i =, 8) are positive constants, f m t) = f mt) = mπ) 4 f mt) = mπ) 4 ft, x) sin mπx dx = mπ) 4 f 4,t), 4 t x 4 ft, x)) sin mπx dx = mπ) 4 f 4,t), 4 2 t x 4 ft, x)) sin mπx dx = mπ) 4 f 4,2t), f 4, t) = f 4, t) = 4 ft, x) x 4 5 ft, x) t x 4 sin mπx dx, sin mπx dx,

9 EJDE-27/243 HIGHER ORDER PDE WITH CAPUTO-FABRIZIO DERIVATIVE 9 f 4,2 t) = 6 ft, x) t 2 x 4 sin mπx dx. We note that for above-given estimations, we need to impose the following conditions to the given function ft, x): f t t= =, ft, ) = ft, ) =, Based on estimations 4.6)-4.8), we obtain and considering 4.3), finally we get 2 f t 2 3 f t= =, t 3 t= =, 2 ft, ) x 2 = 2 ft, ) x 2 =. C j M 9 mπ) 4 T m M 2 mπ) 4. Taking 3.4) into account, on can easily deduce that The required estimate can be deduced easily, as well. ut, x) M 2 mπ) 4, u xxt, x) M 22 mπ) 2. CF D tut, x) M 23 mπ) 4 Theorem 4.. If ft, x) C 2 Ω), 3 ft,x) t 3 CΩ) and 3 ft,x) t 3 4.9) is continuous up to t =, 6 ft,x) t 2 x 4 L, ) together with 4.9), then problem 3.)-3.3), when k = 2 has a unique solution represented by 3.4), where T m t) are defined by 4.3)-4.5) depending on the sign of m. Remark 4.2. Similar result can be obtained for general case, as well. For this, one needs to differentiate 3.9) once by t and write explicit form of k + )th order ordinary DE. Remark 4.3. We note that used algorithm allows us to investigate fractional spectral problems such as CF D + t T t) + µt t) =, T ) =, T ) =, reducing it to the second order usual spectral problem. Using above-given algorithm, we obtain the following second order spectral problem T t) + µ 2 ) T t) + )2 T t) = T ) =, T ) =,

10 E. KARIMOV, S. PIRNAFASOV EJDE-27/243 eigenvalues and corresponding eigenfunctions of which have a form µ n = 2 ± 2 )2 nπ) 2, n N, 2 T n t) = e µn 2 sin nπx, n N. Remark 4.4. We can apply this approach for studying the more general equation k λ n t, x) CF Dt +n ut, x) CF D + ut, x) = ft, x), n= where λ n t, x) might have singularity as well. Remark 4.5. We are able to consider another kind of fractional derivative without singularity with th order < < ) such D tt t) = G) involving it in the fractional DE t T s)kt, s, )ds, λ D tt t) + λ 2 T t) = ft). In this case, our kernel should satisfy to the condition Kt, s, ) Ks, t, ) = [ ][ ] λ G)Ks, s, ) + λ 2 λ G)Kt, t, ) + λ 2. s t Acknowledgments. The authors are very grateful to the anonymous referees for their helpful comments. References [] M. Caputo, M. Fabrizio; A new definition of fractional derivative without singular kernel, Progr. Frac. Differ. Appl., 25, 3. [2] M. Caputo, M. Fabrizio; Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels, Progr. Frac. Differ. Appl. 2, 26,. [3] J. Losada, J. J. Nieto; Properties of a New Fractional Derivative without Singular Kernel, Progr. Frac. Differ. Appl., 25, [4] J. F. Gómez-Aguilar et al; Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel, Entropy, 7, 25, [5] V. F. Morales-Delgado et al; On the solutions of fractional order of evolution equations, Eur. Phys. J. Plus, 27, 32:47 DOI:.4/epjp/i [6] F. Gao et al; Fractional Maxwell fluid with fractional derivative without singular kernel, Thermal Science, 2, 26, S87 S877. [7] Farhad Ali et al; Application of Caputo-Fabrizio derivatives to MHD free convection flow of generalized Walters-B fluid model, Eur. Phys. J. Plus, 26, 3: 377 DOI.4/epjp/i x [8] J. F. Gómez-Aguilar et al; Modeling diffusive transport with a fractional derivative without singular kernel, Physica A, 447, 26, [9] J. Hristov; Transient heat diffusion with a non-singular fading memory, Thermal Science, 2, 26, [] N. Al-Salti, E. Karimov, S. Kerbal; Boundary-value problem for fractional heat equation involving Caputo-Fabrizio derivative, New Trend in Mathematical Sciences, 4, 26, [] N. Al-Salti, E. Karimov, K. Sadarangani; On a Differential Equation with Caputo-Fabrizio Fractional Derivative of Order < β 2 and Application to Mass-Spring-Damper System, Progr. Fract. Differ. Appl., 2, 26,

11 EJDE-27/243 HIGHER ORDER PDE WITH CAPUTO-FABRIZIO DERIVATIVE Erkinjon Karimov Institute of Mathematics named after V. I. Romanovsky, Uzbekistan Academy of Sciences, Tashkent 25, Uzbekistan address: Sardor Pirnafasov Institute of Mathematics named after V. I. Romanovsky, Uzbekistan Academy of Sciences, Tashkent 25, Uzbekistan address: