Discretized fractional substantial calculus
aa r X i v : . [ m a t h . NA ] O c t DISCRETIZED FRACTIONAL SUBSTANTIAL CALCULUS ∗ MINGHUA CHEN † AND
WEIHUA DENG ‡ Abstract.
This paper discusses the properties and the numerical discretizations of the fractionalsubstantial integral I νs f ( x ) = 1Γ( ν ) Z xa ( x − τ ) ν − e − σ ( x − τ ) f ( τ ) dτ, ν > , and the fractional substantial derivative D µs f ( x ) = D ms [ I νs f ( x )] , ν = m − µ, where D s = ∂∂x + σ = D + σ , σ can be a constant or a function without related to x , say σ ( y ); and m is the smallest integer that exceeds µ . The Fourier transform method and fractional linear multistepmethod are used to analyze the properties or derive the discretized schemes. And the convergencesof the presented discretized schemes with the global truncation error O ( h p ) ( p = 1 , , , ,
5) aretheoretically proved and numerically verified.
Key words. fractional substantial calculus, fractional linear multistep methods, fourier trans-form, stability and convergence
AMS subject classifications.
1. Introduction.
Anomalous diffusion processes are usually characterized bythe nonlinear time dependance of the mean squared displacement, i.e., h z ( t ) i ∼ t α .When 0 < α <
1, it is called subdiffusion; 1 < α corresponds to superdiffusion,and α = 1 to normal diffusion. A versatile framework for describing the anomalousdiffusion is the continuous time random walks (CTRWs), which is governed by thewaiting time probability density function (PDF) and jump length PDF. When thewaiting time PDF and/or jump length PDF are power-law, and the two PDFs areindependent, the transport equations can be derived, namely fractional Fokker-Planckand Klein-Kramers equations [9]. The time fractional Fokker-Planck equation can wellcharacterize the subdiffusion, and the space fractional Fokker-Planck equation candepict the L´evy flight. The L´evy flight has a diverging mean squared displacement,and can just be applied to rather exotic physical processes [13].L´evy walk gives another proper dynamical description for the superdiffusion(roughly speaking, now the particle has finite physical speed), and the PDFs of waitingtime and jump length are spatiotemporal coupling [13]. Friedrich and his co-workersdiscuss the CTRW model with position-velocity coupling PDF [5]. Carmi and Barkaiuse the CTRW model with functional of path and position coupling PDF [1]. Basedon the CTRW models with coupling PDFs, they all derive the deterministic equa-tions; and mathematically an important operator, fractional substantial derivative, isintroduced [1, 2, 5, 13, 14].With the wide applications of the fractional substantial derivative, it seems to beurgent to mathematically analyze its properties and numerically provide its effectivediscretizations. This paper focuses on these two topics. The fractional substantial ∗ This work was supported by the National Natural Science Foundation of China under GrantNo. 11271173. † School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China. ‡ Corresponding author (Email: [email protected]). School of Mathematics and Statistics,Lanzhou University, Lanzhou 730000, P. R. China.1
M. H. CHEN AND W. H. DENG derivative is defined by [1, 5] D νs f ( x ) = 1Γ( ν ) (cid:20) ∂∂x + σ (cid:21) Z x ( x − τ ) ν − e − σ ( x − τ ) f ( τ ) dτ, < ν < , where σ can be a constant or a function not related to x , say, σ ( y ). In this paper,we extend the order of fractional substantial derivative ν ∈ (0 ,
1) to ν >
0. First, weintroduce the fractional substantial integral.
Definition 1.1.
Let ν > , f ( x ) be piecewise continuous on ( a, ∞ ) and integrableon any finite subinterval [ a, ∞ ) ; and let σ be a constant or a function without relatedto x . Then the fractional substantial integral of f of order ν is defined as (1.1) I νs f ( x ) = 1Γ( ν ) Z xa ( x − τ ) ν − e − σ ( x − τ ) f ( τ ) dτ, x > a. Definition 1.2.
Let µ > , f ( x ) be (m-1)-times continuously differentiable on ( a, ∞ ) and its m-times derivative be integrable on any finite subinterval of [ a, ∞ ) ,where m is the smallest integer that exceeds µ ; and let σ be a constant or a functionwithout related to x . Then the fractional substantial derivative of f of order µ isdefined as (1.2) D µs f ( x ) = D ms [ I m − µs f ( x )] , where (1.3) D ms = (cid:18) ∂∂x + σ (cid:19) m = ( D + σ ) m = ( D + σ )( D + σ ) · · · ( D + σ ) . When σ = 0, obviously, the fractional substantial integral and derivative reduceto the Riemann-Liouville fractional integral and derivative, respectively.In the following, using Fourier transform methods and fractional linear multistepmethods, respectively, we derive the p -th order ( p ≤
5) approximations of the α -thfractional substantial derivative ( α >
0) or fractional substantial integral ( α <
0) bythe corresponding coefficients of the generating functions κ p,α ( ζ ), with(1.4) κ p,α ( ζ ) = p X i =1 i (cid:0) − e − σh ζ (cid:1) i ! α , where h is the uniform stepsize. We rewrite (1.4) as a tabular, see Table 1.1.For σ = 0, formula (1 .
4) reduces to the fractional Lubich’s methods [8]. For σ = 0 , α = 1, the scheme reduces to the classical ( p + 1)-point backward differenceformula [7].The outline of this paper is as follows. In Section 2, we give some properties of thefractional substantial calculus. In Sections 3 and 4, using Fourier transform methodand fractional linear multistep method, respectively, we derive the convergence of thediscretized schemes of the fractional substantial calculus. And the convergence withthe global truncation error O ( h p ) ( p = 1 , , , ,
5) are numerically verified in Section5. Finally, we conclude the paper with some remarks in the last section.
ISCRETIZED FRACTIONAL SUBSTANTIAL CALCULUS p -th order approximationof α -th fractional substantial derivative. p κ p,α ( ζ )1 (cid:16) − e − σh ζ (cid:17) α (cid:16) / − e − σh ζ + 1 / e − σh ζ ) (cid:17) α (cid:16) / − e − σh ζ + 3 / e − σh ζ ) − / e − σh ζ ) (cid:17) α (cid:16) / − e − σh ζ + 3( e − σh ζ ) − / e − σh ζ ) + 1 / e − σh ζ ) (cid:17) α (cid:16) / − e − σh ζ + 5( e − σh ζ ) − / e − σh ζ ) + 5 / e − σh ζ ) − / e − σh ζ ) (cid:17) α
2. Properties for the fractional substantial calculus.
Let us now considersome properties of the fractional substantial calculus.
Lemma 2.1.
Let f ( x ) be continuous on [ a, ∞ ) , and ν > , then for all x ≥ a , lim ν → I νs f ( x ) = f ( x ) . Hence we can put I s f ( x ) = f ( x ) . Proof . If f ( x ) has continuous derivative for x ≥ a , then using integration by partsto (1.1), there exists I νs f ( x ) = − ν + 1) Z xa e − σ ( x − τ ) f ( τ ) d ( x − τ ) ν = ( x − a ) ν e − σ ( x − a ) f ( a )Γ( ν + 1) + 1Γ( ν + 1) Z xa ( x − τ ) ν e − σ ( x − τ ) D s f ( τ ) dτ, where D s is defined by (1.3). So we getlim ν → I νs f ( x )= e − σ ( x − a ) f ( a ) + σ Z xa e − σ ( x − τ ) f ( τ ) dτ + Z xa e − σ ( x − τ ) df ( τ ) = f ( x ) . If f ( x ) is only continuous for x ≥ a , the similar arguments can be performed as[12, p. 66-67], we omit it here. Lemma 2.2.
Let f ( x ) be continuous on [ a, ∞ ) and µ, ν > , then for all x ≥ a , I νs [ I µs f ( x )] = I µ + νs f ( x ) = I µs [ I νs f ( x )] . Proof . I νs [ I µs f ( x )] = 1Γ( ν ) Z xa ( x − τ ) ν − e − σ ( x − τ ) [ I µs f ( τ )] dτ = 1Γ( µ )Γ( ν ) Z xa ( x − τ ) ν − e − σ ( x − τ ) dτ Z τa ( τ − ξ ) µ − e − σ ( τ − ξ ) f ( ξ ) dξ = 1Γ( µ )Γ( ν ) Z xa e − σ ( x − ξ ) f ( ξ ) dξ Z xξ ( x − τ ) ν − ( τ − ξ ) µ − dτ = I µ + νs f ( x ) , M. H. CHEN AND W. H. DENG where the integral Z xξ ( x − τ ) ν − ( τ − ξ ) µ − dτ = Γ( µ )Γ( ν )Γ( µ + ν ) ( x − ξ ) µ + ν − . Lemma 2.3.
Let f ( x ) be (m-1)-times continuously differentiable on ( a, ∞ ) andits m-times derivative be integrable on any finite subinterval of [ a, ∞ ) and ν > ,where m is the smallest integer that exceeds ν . Then for all x ≥ a , D νs [ I νs f ( x )] = f ( x ) . Proof . Let us first consider the case of integer ν = m ≥ D ms [ I ms f ( x )] = D ms (cid:20) m − Z xa ( x − τ ) m − e − σ ( x − τ ) f ( τ ) dτ (cid:21) = D s Z xa e − σ ( x − τ ) f ( τ ) dτ = D s [ I s f ( x )] = f ( x ) . For m − < ν < m , from Lemma 2.2, there exists I ms = I m − νs [ I νs f ( x )] . Thus, using (1.2) and above equation, we obtain D νs [ I νs f ( x )] = D ms { I m − νs [ I νs f ( x )] } = D ms [ I ms f ( x )] = f ( x ) . Lemma 2.4.
Let f ( x ) be (r-1)-times continuously differentiable on ( a, ∞ ) andits r-times derivative be integrable on any finite subinterval of [ a, ∞ ) , where r =max( m, n ) , m and n are positive integers. Denoting that m − ν = n − µ, µ > , ν > , then for all x ≥ a , D ns [ I µs f ( x )] = D ms [ I νs f ( x )] . Proof . If m = n , the lemma is trivial. Supposing that n > m and γ = n − m > µ = ν + γ >
0. Then according to Lemmas 2.2 and 2.3, we obtain D γs [ I ν + γs f ( x )] = D γs [ I γs I νs f ( x )] = I νs f ( x ) . Letting D ms perform on both sides of the above equation leads to D m + γs [ I ν + γs f ( x )] = D ms [ I νs f ( x )] , that is D ns [ I µs f ( x )] = D ms [ I νs f ( x )] . ISCRETIZED FRACTIONAL SUBSTANTIAL CALCULUS Lemma 2.5.
Let f ( x ) be continuously differentiable on [ a, ∞ ) , and ν > . Thenfor all x ≥ a , (2.1) I ν +1 s [ D s f ( x )] = I νs f ( x ) − f ( a )Γ( ν + 1) ( x − a ) ν e − σ ( x − a ) ; and (2.2) D s [ I νs f ( x )] = I νs [ D s f ( x )] + f ( a )Γ( ν ) ( x − a ) ν − e − σ ( x − a ) . Proof . Using integration by parts, it is easy to get I νs f ( x ) = f ( a )Γ( ν + 1) ( x − a ) ν e − σ ( x − a ) + 1Γ( ν + 1) Z xa ( x − τ ) ν e − σ ( x − τ ) [ D s f ( τ )] dτ where D s is defined by (1.3). Thus we obtain (2.1).Next we prove (2.2). From (2.1), it leads to D s [ I νs f ( x )]= D s (cid:26) I ν +1 s [ D s f ( x )] + f ( a )Γ( ν + 1) ( x − a ) ν e − σ ( x − a ) (cid:27) = I νs [ D s f ( x )] + f ( a )Γ( ν + 1) ( D + σ ) h ( x − a ) ν e − σ ( x − a ) i = I νs [ D s f ( x )] + f ( a )Γ( ν ) ( x − a ) ν − e − σ ( x − a ) . Hence, we get (2.2).
Lemma 2.6.
Let f ( x ) be (m-1)-times continuously differentiable on ( a, ∞ ) andits m-times derivative be integrable on any finite subinterval of [ a, ∞ ) , µ > , ν > ;and m is the smallest integer that exceeds µ . Then for all x ≥ a , (2.3) I νs f ( x ) = I m + νs [ D ms f ( x )] + m − X k =0 D ks f ( a )( x − a ) k + ν e − σ ( x − a ) Γ( k + ν + 1) ; and D µs f ( x ) = I m − µs [ D ms f ( x )] + m − X k =0 D ks f ( a )( x − a ) k − µ e − σ ( x − a ) Γ( k − µ + 1)= C D µs f ( x ) + m − X k =0 D ks f ( a )( x − a ) k − µ e − σ ( x − a ) Γ( k − µ + 1) , (2.4) where C D µs f ( x ) = I m − µs [ D ms f ( x )] can be similarly called Caputo fractional substantialderivative [12]. In particular, from (2.3) and (2.4), we can extend the definitions of I νs and D µs , i.e., µ, ν can belong to R instead of being limited to R + , then for any real α , there exists (2.5) I αs = D − αs . M. H. CHEN AND W. H. DENG
Proof . Replacing ν by ν + 1 and f by D s f in (2.1), we obtain I ν +1 s [ D s f ( x )] = I ν +2 s [ D s f ( x )] + D s f ( a )Γ( ν + 2) ( x − a ) ν +1 e − σ ( x − a ) . Thus, according to the above equation and (2.1), there exists I νs f ( x ) = I ν +1 s [ D s f ( x )] + f ( a )Γ( ν + 1) ( x − a ) ν e − σ ( x − a ) = I ν +2 s [ D s f ( x )] + D s f ( a )Γ( ν + 2) ( x − a ) ν +1 e − σ ( x − a ) + f ( a )Γ( ν + 1) ( x − a ) ν e − σ ( x − a ) = I ( ν + m ) s [ D ms f ( x )] + m − X k =0 D ks f ( a )( x − a ) ν + k e − σ ( x − a ) Γ( ν + k + 1) . To prove (2.4), letting D s perform on both sides of (2.2) leads to D s [ I νs f ( x )] = D s { I νs [ D s f ( x )] } + f ( a )Γ( ν −
1) ( x − a ) ν − e − σ ( x − a ) , and replacing f with D s f in (2.2) yields D s { I νs [ D s f ( x )] } = I νs [ D s f ( x )] + D s f ( a )Γ( ν ) ( x − a ) ν − e − σ ( x − a ) . Therefore, there exists D s [ I νs f ( x )] = I νs [ D s f ( x )]+ D s f ( a )Γ( ν ) ( x − a ) ν − e − σ ( x − a ) + f ( a )Γ( ν −
1) ( x − a ) ν − e − σ ( x − a ) . Repeating the procedure m − D ms [ I νs f ( x )] = I νs [ D ms f ( x )] + m − X k =0 D ks f ( a )( x − a ) ν + k − m e − σ ( x − a ) Γ( ν + k − m + 1) . Taking ν = m − µ , then Eq. (2.6) can be rewritten as D µs f ( x ) = D ms [ I νs f ( x )] = I m − µs [ D ms f ( x )] + m − X k =0 D ks f ( a )( x − a ) k − µ e − σ ( x − a ) Γ( k − µ + 1) . From (2.3) and (2.4), it yields that I αs = D − αs for any real α . Lemma 2.7.
Let f ( x ) be (m-1)-times continuously differentiable on ( a, ∞ ) andits m-times derivative be integrable on any finite subinterval of [ a, ∞ ) and ν > ,where m is the smallest integer that exceeds ν . Then for all x > a , I νs [ D νs f ( x )] = f ( x ) − m X j =1 [ D ν − js f ( x )] x = a ( x − a ) ν − j e − σ ( x − a ) Γ( ν − j + 1) . Proof . On the one hand, there exists I νs [ D νs f ( x )] = D s (cid:26) ν + 1) Z xa ( x − τ ) ν e − σ ( x − τ ) [ D νs f ( τ )] dτ (cid:27) . (2.7) ISCRETIZED FRACTIONAL SUBSTANTIAL CALCULUS ν + 1) Z xa ( x − τ ) ν e − σ ( x − τ ) D νs f ( τ ) dτ = 1Γ( ν + 1) Z xa ( x − τ ) ν e − σ ( x − τ ) D ms [ I m − νs f ( τ )] dτ = 1Γ( ν ) Z xa ( x − τ ) ν − e − σ ( x − τ ) D m − s [ I m − νs f ( τ )] dτ − ( x − a ) ν e − σ ( x − a ) Γ( ν + 1) (cid:8) D m − s [ I m − νs f ( x )] (cid:9) x = a = 1Γ( ν − m + 1) Z xa ( x − τ ) ν − m e − σ ( x − τ ) [ I m − νs f ( τ )] dτ − m X j =1 (cid:8) D m − js [ I m − νs f ( x )] (cid:9) x = a ( x − a ) ν − j +1 e − σ ( x − a ) Γ( ν − j + 2)= I ν − m +1 s [ I m − νs f ( τ )] − m X j =1 (cid:2) D ν − js f ( x ) (cid:3) x = a ( x − a ) ν − j +1 e − σ ( x − a ) Γ( ν − j + 2)= I s f ( τ ) − m X j =1 (cid:2) D ν − js f ( x ) (cid:3) x = a ( x − a ) ν − j +1 e − σ ( x − a ) Γ( ν − j + 2) . (2.8)Combining 2.7 and 2.8, we obtain I νs [ D νs f ( x )] = f ( x ) − m X j =1 [ D ν − js f ( x )] x = a ( x − a ) ν − j e − σ ( x − a ) Γ( ν − j + 1) . Lemma 2.8.
Let f ( x ) be (m-1)-times continuously differentiable on ( a, ∞ ) and itsm-times derivative be integrable on any finite subinterval of [ a, ∞ ) and µ > , ν > ,where m is the smallest integer that exceeds µ . Then for all x > a , D µs [ D − νs f ( x )] = D µ − νs f ( x ) . Proof . Two cases must be considered: µ > ν ≥ ν ≥ µ ≥ µ > ν ≥
0: taking 0 ≤ n − ≤ µ − ν < n , n is an integer and using0 ≤ m − ≤ µ < m , then from (2.5) and (1.2) and Lemmas 2.2 and 2.4, we have D µs [ D − νs f ( x )] = D µs [ I νs f ( x )] = D ms (cid:8) I m − µs [ I νs f ( x )] (cid:9) = D ms (cid:8) I m − µ + νs f ( x ) (cid:9) = D ns (cid:8) I n − µ + νs f ( x ) (cid:9) = D µ − νs f ( x ) . Case ν ≥ µ ≥
0: according to Lemmas 2.2 and 2.3, we obtain D µs [ I νs f ( x )] = D µs [ I µs I ν − µs f ( x )] = D µ − νs f ( x ) . Lemma 2.9.
Let f ( x ) be (m-1)-times continuously differentiable on ( a, ∞ ) and itsm-times derivative be integrable on any finite subinterval of [ a, ∞ ) and µ > , ν > ,where m is the smallest integer that exceeds ν . Then for all x > a , D − µs [ D νs f ( x )] = D ν − µs f ( x ) − m X j =1 [ D ν − js f ( x )] x = a ( x − a ) µ − j e − σ ( x − a ) Γ( µ − j + 1) . M. H. CHEN AND W. H. DENG
Proof . If ν ≤ µ , there exists D − µs = D ν − µs D − νs by Lemma 2.2; and if ν ≥ µ , therealso exists D − µs = D ν − µs D − νs by Lemma 2.8. Therefore, using Lemma 2.7 we have D − µs [ D νs f ( x )] = D ν − µs { D − νs [ D νs f ( x )] } = D ν − µs f ( x ) − m X j =1 [ D ν − js f ( x )] x = a ( x − a ) ν − j e − σ ( x − a ) Γ( ν − j + 1) = D ν − µs f ( x ) − m X j =1 [ D ν − js f ( x )] x = a ( x − a ) µ − j e − σ ( x − a ) Γ( µ − j + 1) , where we use the following formula(2.9) D µs [ e − σ ( x − a ) ( x − a ) ν ] = Γ( ν + 1)Γ( ν + 1 − µ ) ( x − a ) ν − µ e − σ ( x − a ) , which can be similarly proven as the way in [12, p. 56]. Lemma 2.10.
Let µ > , ν > and f ( x ) be (r-1)-times continuously differen-tiable on ( a, ∞ ) and its r-times derivative be integrable on any finite subinterval of [ a, ∞ ) , where r = max( m, n ) , m and n is the smallest integer that exceeds µ and ν ,respectively. Then for all x > a , D µs [ D νs f ( x )] = D µ + νs f ( x ) − n X j =1 [ D ν − js f ( x )] x = a ( x − a ) − µ − j e − σ ( x − a ) Γ( − µ − j + 1) . Proof . Similar to the well-known property of integer-order derivatives: d m dx m (cid:18) d n f ( x ) dx n (cid:19) = d n dx n (cid:18) d m f ( x ) dx m (cid:19) = d m + n f ( x ) dx m + n , it is easy to check that D ms [ D ns f ( x )] = D ns [ D ms f ( x )] = D m + ns f ( x ) . Therefore, according to (1.2), the above equation, and Lemma 2.8, there exists D ns (cid:2) D m − αs f ( x ) (cid:3) = D n + ms [ I αs f ( x )] = D n + m − αs f ( x ) , for α ∈ (0 , , and denoting that γ = m − α , it leads to(2.10) D ns [ D γs f ( x )] = D n + γs f ( x ) . According to (1.2), Lemma 2.9, and (2.10), we obtain D µs [ D νs f ( x )] = D ms n D − ( m − µ ) s [ D νs f ( x )] o = D ms D µ + ν − ms f ( x ) − n X j =1 [ D ν − js f ( x )] x = a ( x − a ) m − µ − j e − σ ( x − a ) Γ( m − µ − j + 1) = D µ + νs f ( x ) − n X j =1 [ D ν − js f ( x )] x = a ( x − a ) − µ − j e − σ ( x − a ) Γ( − µ − j + 1) . ISCRETIZED FRACTIONAL SUBSTANTIAL CALCULUS
Remark 2.1. If D µs f ( x ) exists and is integrable, then the fractional substantialderivative D νs f ( x ) also exists and is integrable for < ν < µ . Remark 2.2.
Let f ( x ) be (m-1)-times continuously differentiable on ( a, ∞ ) andits m-times derivative be integrable on any finite subinterval of [ a, ∞ ) , then for all x ≥ a , [ D µs f ( x )] x = a = 0 , m − ≤ µ < m if and only if D ( j ) s = 0 , for j = 0 , , . . . , m − .
3. Discretizations of fractional substantial calculus and its convergence;Fourier transform methods.
In this section, we derive the discretization schemesof fractional substantial calculus and prove their convergence by Fourier transformmethod.
Lemma 3.1.
Let ν > , f ( x ) ∈ L q ( R ) , q ≥ , and (3.1) I νs f ( x ) = 1Γ( ν ) Z x −∞ ( x − τ ) ν − e − σ ( x − τ ) f ( τ ) dτ, then F ( I νs f ( x )) = ( σ − iω ) − ν b f ( ω ) , where F denotes Fourier transform operator and b f ( ω ) = F ( f ) , i.e., b f ( ω ) = Z R e iωx f ( x ) dx. Proof . Taking the fractional substantial integral (1.1) with the lower terminal a = −∞ , Eq. (1.1) reduces to (3.1).Let us start with the Laplace transform of the function h ( x ) = x ν − Γ( ν ) e − σx , i.e., 1Γ( ν ) Z ∞ x ν − e − ( σ + s ) x dx = ( σ + s ) − ν , (3.2)where we use the well-known Laplace transform of the function x ν − L { x ν − ; s } = Z ∞ x ν − e − sx dx = Γ( ν ) s − ν . It follows from the Dirichlet theorem [4, p. 564] that the integral (3.2) converges if ν >
0. Taking s = − iω , where ω is real, we immediately have the Fourier transformof the function h + ( x ) = ( x ν − Γ( ν ) e − σx , x > , x ≤ , M. H. CHEN AND W. H. DENG in the form F ( h + ( x )) = Z ∞−∞ h + ( x ) e iωx dx = 1Γ( ν ) Z ∞ x ν − e − ( σ − iω ) x dx = ( σ − iω ) − ν . Since I νs f ( x ) = 1Γ( ν ) Z x −∞ ( x − τ ) ν − e − σ ( x − τ ) f ( τ ) dτ = x ν − e − σx Γ( ν ) ∗ f ( x ) = h ( x ) ∗ f ( x ) , where the asterisk means the convolution, then we have F ( I νs f ( x )) = F ( h ( x ) ∗ f ( x )) = F ( h ( x )) · F ( f ( x )) = ( σ − iω ) − ν b f ( ω ) . Lemma 3.2.
Let ν > , f ∈ C m − ( a, ∞ ) and its m-times derivative be integrableon any finite subinterval of [ a, ∞ ) . Denoting that (3.3) D νs f ( x ) = D ms [ I m − νs f ( x )] , where m is the smallest integer that exceeds ν and D ms and I m − νs are defined by (1.3)and (3.1), respectively. Then F ( D νs f ( x )) = ( σ − iω ) ν b f ( ω ) . Proof . Taking the lower terminal a = −∞ and using (2.4), we obtain D νs f ( x ) = D ms [ I m − νs f ( x )] = I m − νs [ D ms f ( x )] . Then from Lemma 3.1, there exists F ( D νs f ( x )) = ( σ − iω ) ν − m F ( D ms f ( x )) = ( σ − iω ) ν b f ( ω ) , where F ( D ms f ( x )) = ( σ − iω ) m b f ( ω ) can be proven by the mathematical induction.In the following, we do the expansions to (1.4) to get the formulas of the coef-ficients when p = 1 , , , ,
5; and we prove that the operators have their respectivedesired convergent order by the technique of Fourier transform.First, taking p = 1 and h be the uniform space stepsize, then from (1.4), we have κ ,α ( ζ ) = (1 − ζe σh ) α = ∞ X m =0 e − mσh ( − m (cid:18) αm (cid:19) ζ m = ∞ X m =0 g ,αm ζ m , with the recursively formula(3.4) g ,α = 1 , g ,αm = e − σh (cid:18) − α + 1 m (cid:19) g ,αm − , m ≥ , where σ is defined in Definition 1.1.Similar to the way performed in [3], it is easy to compute(3.5) κ p,α ( ζ ) = p X i =1 i (cid:18) − ζe σh (cid:19) i ! α = ∞ X m =0 g p,αm ζ m , p = 1 , , , , , ISCRETIZED FRACTIONAL SUBSTANTIAL CALCULUS g ,αm given in (3.4); and g p,αm = e − σmh l p,αm , p = 1 , , , , , where l ,αm , l ,αm , l ,αm , l ,αm and l ,αm are defined by (2.2), (2.4), (2.6), (2.8) and (2.10)in [3], respectively. And it implies that to get the coefficients g p,αm , we only need tocompute the coefficients l p,αm . Theorem 3.3. (Case p = 1 ) Let f , D α +1 s f ( x ) with α > and their Fouriertransforms belong to L ( R ) , and denote that A ,α f ( x ) = 1 h α ∞ X m =0 g ,αm f ( x − mh ) , where D α +1 s and g ,αm is defined by (3.3) and (3.4), respectively, Then D αs f ( x ) = A ,α f ( x ) + O ( h ) . Proof . Using Fourier transform, we obtain F ( A ,α f )( ω ) = 1 h α ∞ X m =0 g ,αm F ( f ( x − mh )) ( ω )= 1 h α ∞ X m =0 g ,αm (cid:0) e iωh (cid:1) m b f ( ω )= 1 h α (cid:18) − e iωh e σh (cid:19) α b f ( ω )= ( σ − iω ) α (cid:18) − e − ( σ − iω ) h ( σ − iω ) h (cid:19) α b f ( ω )= ( σ − iω ) α (cid:18) − e − z z (cid:19) α b f ( ω ) , with z = ( σ − iω ) h . It is easy to check that (cid:18) − e − z z (cid:19) α = 1 − α z + 3 α + α z − α + α z + O ( z ) . Therefore, from Lemma 3.2, there exists F ( A ,α f )( ω ) = F ( D αs f ) + b φ ( ω ) , where b φ ( ω ) = ( σ − iω ) α (cid:0) − α z + O ( z ) (cid:1) b f ( ω ), z = ( σ − iω ) h . Then | b φ ( ω ) | ≤ e c · | ( σ − iω ) α +1 b f ( ω ) | · h. With the condition F [ D α +1 s f ( x )] ∈ L ( R ), it leads to |D αs f ( x ) − A ,α f ( x ) | = | φ ( x ) | ≤ π Z R | b φ ( ω ) | dx ≤ c ||F [ D α +1 s f ]( ω ) || L · h = O ( h ) . M. H. CHEN AND W. H. DENG
Theorem 3.4.
Let f , D α + ps f ( x ) ( p = 2 , , , ) with α > and their Fouriertransforms belong to L ( R ) , and denote that A p,α f ( x ) = 1 h α ∞ X m =0 g p,αm f ( x − mh ) , where g p,αm is defined by (3.5). Then D αs f ( x ) = A p,α f ( x ) + O ( h p ) , p = 2 , , , . Proof . Using the ideas of the proof of Theorem 3.3 and Lemmas 2.3-2.7 of [3], wecan similarly prove this theorem; the details are omitted here.
Remark 3.1.
Theorems 3.3-3.4 still hold for the fractional substantial integraloperators I αs ; in fact, comparing Lemmas 3.1 with 3.2 gives us the intuition. All the above schemes are applicable to finite domain, say, ( a, b ), after performingzero extensions to the functions considered. Let f ( x ) be the zero extended functionfrom the finite domain ( a, b ), and satisfy the requirements of the above correspondingtheorems. Taking p = 1 , , , , e A p,α f ( x ) = 1 h α [ x − ah ] X m =0 g p,αm f ( x − mh ) , α > , (3.6)with g p,αm given in (3.5). Then D αs f ( x ) = e A p,α f ( x ) + O ( h p ) , α > , (3.7)where D αs is defined by (1.2). Thus the approximation operator of (3.6) can bedescribed as e A p,α f ( x i ) = 1 h α i X m =0 g p,αm f ( x i − m ) , α > , and the fractional substantial derivative has p -th order approximations D αs f ( x i ) = h − α i X m =0 g p,αm f ( x i − m ) + O ( h p ) , α > , (3.8)Similarly, the fractional substantial integral has p -th order approximations I αs f ( x i ) = h α i X m =0 g p, − αm f ( x i − m ) + O ( h p ) , α > . (3.9)
4. Discretizations of fractional substantial calculus and its convergence;fractional linear multistep methods.
Essentially the results given this section arethe generalizations of the ones for fractional calculus provided in [8] to fractional sub-stantial calculus; some of them are not straightforward, so we restate and prove them.In particular, comparing with Section 3, by adding some terms at the neighborhood
ISCRETIZED FRACTIONAL SUBSTANTIAL CALCULUS a = 0 (that is notessential, a can be any given constant but not infinity). Then the fractional substantialintegral (1.1) and fractional substantial derivative (1.2), respectively, reduce to(4.1) I αs f ( x ) = 1Γ( α ) Z x ( x − τ ) α − e − σ ( x − τ ) f ( τ ) dτ, and(4.2) D αs f ( x ) = D ms [ I m − αs f ( x )] , where m is the smallest integer that exceeds α .For σ = 0, the fractional substantial integral (4.1) and fractional substantialderivative (4.2), respectively, reduce to the Riemann-Liouville fractional integral(4.3) I α f ( x ) = 1Γ( α ) Z x ( x − τ ) α − f ( τ ) dτ, and Riemann-Liouville fractional derivative [6, 10, 11](4.4) D α f ( x ) = d m dx m m − α ) Z x ( x − τ ) m − α − f ( τ ) dτ, m − < α < m. Using the homogeneity and the convolution structure of I α in (4.3):( I α f )( x ) = x α ( I α f ( tx ))(1) and I α f = 1Γ( α ) t α − ∗ f, Lubich gets the following important property [8](4.5) (cid:0) E αh t β − (cid:1) ( x ) = x α + β − (cid:16) E αh/x t β − (cid:17) (1) , with(4.6) E αh = Ω αh − I α and Ω αh f ( x ) = h α n X j =0 ω αn − j f ( jh ) , ( x = nh ) , where ω αn denotes the convolution quadrature weights. So Lubich obtains the followingconvolution quadratures to approximation the Riemann-Liouville fractional integral(4.7) I αh f ( x ) = h α n X j =0 ω αn − j f ( jh ) + h α r X j =1 ω αn,j f ( jh ) , ( x = nh ) , α > , where ω αn,j denotes the starting quadrature weights. The added term h α P rj =1 ω αn,j f ( jh )is mainly for keeping the accuracy when relaxing the requirement of the regularity of f ( x ).For D αh f ( x ) or I − αh f ( x ) in (4.7) with α >
0, taking D ( j ) f (0) = 0, j = 0 , , . . . , m − m − < α < m , then it yields the convolution structure of D α in (4.4): D α f ( x ) = d m dx m (cid:20) m − α ) Z x ( x − τ ) m − α − f ( τ ) dτ (cid:21) = 1Γ( m − α ) Z x ( x − τ ) m − α − ( d m f ( τ ) /dτ m ) dτ = 1Γ( m − α ) x m − α − ∗ d m f ( x ) dx m , M. H. CHEN AND W. H. DENG and the homogeneity of D α :( D α f )( x ) = x m − α ( D α f ( tx ))(1) , m − < α < m. Therefore, we also obtain following property (cid:0) E − αh t β − (cid:1) ( x ) = x − α + β − (cid:16) E − αh/x t β − (cid:17) (1) , β > m, where E − αh = Ω − αh − D α and Ω − αh f ( x ) = h − α n X j =0 ω − αn − j f ( jh ) , ( x = nh ) . So similar to the discussions in [8], we can also get the following scheme to approximatethe Riemann-Liouville fractional derivative(4.8) D αh f ( x ) = h − α n X j =0 ω − αn − j f ( jh ) + h − α r X j =1 ω − αn,j f ( jh ) , ( x = nh ) , α > . Form (4.7) and (4.8), there exists(4.9) I αh f ( x ) = h α n X j =0 ω n − j f ( jh ) + h α r X j =1 ω n,j f ( jh ) , ( x = nh ) , α ∈ R , where α > ω n = ω αn , ω n,j = ω αn,j ) and α < ω n = ω − αn , ω n,j = ω − αn,j ).In this section, we mainly focus on the discretized fractional substantial calculus;for simplicity, the following notations are used: E αs,h = Ω αs,h − I αs , where Ω αs,h f ( x ) = h α n X j =0 κ n − j f ( jh ) , ( x = nh ) , α ∈ R , (4.10)and it is easy to get the following properties for α ∈ R : (cid:0) I αs [ e − σt f ( t )] (cid:1) ( x ) = e − σx ( I α [ f ( t )]) ( x ); (cid:0) E αs,h [ e − σt f ( t )] (cid:1) ( x ) = e − σx ( E αh [ f ( t )]) ( x ) , (4.11)where α > α < I αs,h f ( x ) = h α n X j =0 κ n − j f ( jh ) + h α r X j =1 κ n,j f ( jh )) , ( x = nh ) , α ∈ R , (4.12)where(4.13) κ j = e − jσh ω j , ω j is defined by (4.9) , and κ j and κ n,j also denote the convolution quadrature weights and the startingquadrature weights, respectively. ISCRETIZED FRACTIONAL SUBSTANTIAL CALCULUS κ = ( κ n ) ∞ (or ω = ( ω n ) ∞ ) and take [8] κ ( ζ ) = ∞ X n =0 κ n ζ n , (cid:16) or ω ( ζ ) = ∞ X n =0 ω n ζ n (cid:17) , to be its generating power series. Definition 4.1.
A convolution quadrature κ is stable (for I αs ) if κ n = O ( n α − ) . Definition 4.2.
A convolution quadrature κ is consistent of order p (for I αs ) if h α κ (cid:0) e σh e − h (cid:1) = 1 + O ( h p ) . Definition 4.3.
A convolution quadrature κ is convergent of order p (to I αs ) if (4.14) ( E αs,h [ e − σt t β − ])(1) = O ( h β ) + O ( h p ) for all β ∈ C , β = 0 , − , − , · · · . Lemma 4.4. If ( E αs,h [ e − σt t k − ])(1) = O ( h k ) + O ( h p ) for k = 1 , , , . . . , then κ is consistent of order p . Moreover, κ is consistent of order p if and only if ω isconsistent of order p .Proof . According to (4.11), we have( E αs,h [ e − σt t k − ])(1) = e − σ ( E αh t k − )(1) , and it leads to ( E αh t k − )(1) = O ( h k ) + O ( h p ) , for k = 1 , , , . . . . Then from Lemma 3.1 of [8], we obtain h α ω ( e − h ) = 1 + O ( h p ) , with ω ( ζ ) = ∞ X n =0 ω n ζ n . Using (4.13), there exists(4.15) κ ( ζ ) = ∞ X n =0 κ n ζ n = ∞ X n =0 e − nσh ω n ζ n = ω (cid:18) ζe σh (cid:19) . Therefore h α κ (cid:0) e σh e − h (cid:1) = h α ω ( e − h ) = 1 + O ( h p ) , and it means that κ is consistent of order p if and only if ω is consistent of order p .Using (3.6) of [8] and (4.15), we get κ ( ζ ) = ω (cid:18) ζe σh (cid:19) = (cid:18) − ζe σh (cid:19) − α h c + c (cid:18) − ζe σh (cid:19) + c (cid:18) − ζe σh (cid:19) + · · · + c N − (cid:18) − ζe σh (cid:19) N − + (cid:18) − ζe σh (cid:19) N e r (cid:18) ζe σh (cid:19) i , (4.16)6 M. H. CHEN AND W. H. DENG and κ ( ζ ) = ω (cid:18) ζe σh (cid:19) = (cid:18) − ζe σh (cid:19) − α e ω (cid:18) ζe σh (cid:19) . Therefore, we can characterize consistency in terms of the coefficients c i . Lemma 4.5.
Let ∞ P i =0 γ i (1 − ζ ) i = (cid:16) − ln ζ − ζ (cid:17) − α . Then κ is consistent of order p ifand only if the coefficients c i in (4.16) satisfy c i = γ i for i = 0 , , . . . , p − . Proof . From Lemma 4.4, it implies that κ is consistent of order p if and only if ω is consistent of order p . Thus, using Lemma 3.2 of [8], the desired result is obtained.Whether the method κ is stable depends on the remainder in the expansion (4.16),and (4.16) can be rewritten as κ ( ζ ) = (cid:18) − ζe σh (cid:19) − α h c + c (cid:18) − ζe σh (cid:19) + · · · + c N − (cid:18) − ζe σh (cid:19) N − i + (cid:18) − ζe σh (cid:19) N r (cid:18) ζe σh (cid:19) , (4.17)where r ( ζ ) = (1 − ζ ) − α e r ( ζ ). Lemma 4.6. κ is stable if and only if ω is stable; and ω is stable if and only ifthe coefficients r n of r ( ζ ) in (4.17) satisfy r n = O ( n α − ) . Proof . By Lemma 3.3 of [8], we have ω is stable if and only if r n = O ( n α − ).From (4.13) and e − jσh ∈ [ e −| σ | x , e | σ | x ], j = 0 , , . . . , n, x = nh , it implies that κ isstable if and only if ω is stable. Lemma 4.7.
Convergence implies stability. Moreover, κ is convergent of order p if and only if ω is convergent of order p .Proof . According to (4.11), we have( E αs,h [ e − σt t β − ])(1) = e − σ ( E αh t β − )(1) , and it implies that κ is convergent of order p if and only if ω is convergent of order p .Hence, according to Lemma 3.4 of [8], the desired result is got. Lemma 4.8.
Let α, β ∈ C , β = 0 , − , − , · · · . If κ is stable, then the convolutionquadrature error of e − σt t β − has the asymptotic expansion as ( E αs,h [ e − σt t β − ])(1) = e − σ (cid:0) e + e h + · · · + e N − h N − + O ( h N ) + O ( h β ) (cid:1) , and the coefficients e j = e j ( α, β, c , · · · , c j ) depend analytically on α, β and the coef-ficients c , · · · , c j of (4.17).Proof . From Lemma 4.6, κ is stable if and only if ω is stable. According to (4.11)and Lemma 3.5 of [8], we get( E αs,h [ e − σt t β − ])(1) = e − σ ( E αh t β − )(1)= e − σ (cid:0) e + e h + · · · + e N − h N − + O ( h N ) + O ( h β ) (cid:1) . ISCRETIZED FRACTIONAL SUBSTANTIAL CALCULUS Lemma 4.9.
Let ℜ ( α ) > . If ( E αs,h [ e − σt t p − ])(1) = O ( h p ) , then ( E αs,h [ e − σt t β − ])(1) = O ( h p ) for all ℜ ( β ) > p .Proof . According to (4.11), it leads to( E αs,h [ e σt t p − ])(1) = e − σ ( E αh t p − )(1) = O ( h p ) . Then form Lemma 3.6 of [8], we obtain ( E αh t β − )(1) = O ( h p ) for all ℜ ( β ) > p . Using(4.11) again, there exists ( E αs,h [ e − σt t β − ])(1) = O ( h p ) for all ℜ ( β ) > p . Lemma 4.10.
Let ℜ ( α ) > . There exist e γ , e γ , · · · (independent of κ ) such thatthe following holds for stable κ : ( E αs,h [ e − σt t q − ])(1) = O ( h q ) , for q = 1 , , · · · , p, if and only if c i of (4.17) satisfy c i = e γ i , for i = 0 , , · · · , p − . Proof . From Lemma 4.6, κ is stable if and only if ω is stable. From (4.11) andLemma 3.7 of [8], there eixsts( E αs,h [ e − σt t q − ])(1) = e − σ ( E αh t q − )(1) = O ( h q ) , if and only if ( E αh t q − )(1) = O ( h q ) , for q = 1 , , · · · , p if and only if the coefficients c i of (4.17) satisfy c i = e γ i , for i = 0 , , · · · , p − . Lemma 4.11.
Let α ∈ R . κ is convergent of order p , if it is stable and consistentof order p .Proof . According to Lemmas 4.6 and 4.4, κ is stable and consistent of order p ifand only if ω is stable and consistent of order p . Then from Lemma 3.8 of [8], ω isconvergent of order p , and it leads to that κ is also convergent of order p by Lemmas4.7. Theorem 4.12. κ is stable and consistent of order p if and only if it is convergentof order p .Proof . From lemmas 4.4, 4.7 and 4.11, we obtain it. Theorem 4.13.
Let κ satisfy (4.14), and f ( x ) = x β − g ( x ) , where β = 0 , − , − , · · · ,for α ≥ and β > ⌈− α ⌉ for α < ; and g ( x ) is sufficiently differentiable. Then, thereexists a starting quadrature κ n,j , such that the approximation I αs,h f given by (4.12)satisfies I αs,h f ( x ) − I αs f ( x ) = O ( h p ) . Proof . A suitable starting quadrature can be chosen by putting I αs,h [ e − σt t q + β − ]( x ) − I αs [ e − σt t q + β − ]( x ) = 0 , q = 0 , , · · · , m − , M. H. CHEN AND W. H. DENG where m satisfies ℜ ( m + β − ≤ p < ℜ ( m + β ); then the following holds(4.18) h α m X j =1 κ n,j e − σjh ( jh ) q + β − + ( E αs,h [ e − σt t q + β − ])(1) = 0 , nh = 1 . According to (4.18), we have m X j =1 κ n,j e − σjh j q + β − = Γ( q + β )Γ( α + q + β ) e − σnh n q + α + β − − n X j =1 κ n − j e − σjh j q + β − , this gives a Vandermonde type system for κ n,j . From (4.14) and (4.18), we have m X j =1 κ n,j e − σjh j q + β − = O ( n α − );then κ n,j = O ( n α − ) . Let f ( x ) = x β − g ( x ) = e − σx x β − h ( x ), where h ( x ) = e σx g ( x ), and g ( x ) is suffi-ciently differentiable. Let β ∈ [ d, d + 1), d is an integer, then γ = β − d ∈ [0 , f ( x ) = N X q =0 D ( q + γ − s f (0)Γ( q + γ ) x q + γ − e − σx + 1Γ( N + γ ) h(cid:0) t N + γ − e − σt (cid:1) ∗ D ( N + γ ) s f i ( x ) . If ℜ ( N + γ − > p and additionally ℜ ( N − p + α + γ ) >
0, then using (4.11) andfollowing the proof of Theorem 2.4 in [8], it is easy to get I αs,h f ( x ) − I αs f ( x ) = O ( e − σx x m − p + α + γ − h p ) uniformly for bounded x. If m in (4.18) is replaced by l ( > m ) with ℜ ( l − p + α + γ − ≥
0, then thefollowing for the corresponding starting quadrature weights holds κ n,j = O ( n l − − p + α + γ − ) , and by the similar arguments performed above, we can prove that I αs,h f ( x ) − I αs f ( x ) = O ( h p ) uniformly for bounded x.
5. Numerical Results.
We use two numerical examples to confirm that thetheoretical results given in the above sections, including the fractional substantialderivatives and integrals. The first example mainly verifies the numerical stabilityand convergent order; and the second one primarily focuses on illustrating that thestarting quadrature numerically works very well for keeping the high order accuracywhen the performed function becomes less regular. And the l ∞ norm is used tomeasure the numerical errors. Example 5.1.
To numerically verify the truncation error given in Theorem 3.4in a bounded domain. We utilize the approximation (3.8) with p = 5 to simulate thefollowing equation D αs f ( x ) = Γ(6 + α )Γ(6) x e − σx , x ∈ (0 , , σ = 1 / . ISCRETIZED FRACTIONAL SUBSTANTIAL CALCULUS α <
0, the fractional operator D αs becomes fractional substantial integral op-erator; if α ∈ (0 ,
1) we take f (0) = 0; and if α ∈ (1 ,
2) let f (0) = 0, f (1) = e − σ ; theexact solution of the above equation is f ( x ) = e − σx x α .Table 5.1: The maximum errors and convergent orders for (3.8), when p = 5, σ = 1 / h α = − / α = 1 / α = 3 / Table 5.1 numerically verifies Theorem 3.4, and shows that the truncation errorsare O ( h ). Example 5.2.
To numerically confirm the result given in Sec. 4 that the startingquadrature can keep the accuracy when the performed function is not sufficientlyregular, we utilize the approximation (4.12) and (3.8) (both with p = 5), respectively,to simulate the following equation D αs f ( x ) = Γ(6 + α )Γ(6) x e − σx + Γ(1 . . − α ) x . − α e − σx , x ∈ (0 , , σ = 1 / . When α <
0, the fractional operator D αs is a fractional substantial integral operator;if α ∈ (0 ,
1) we take f (0) = 0; the exact solution of the above equation is f ( x ) = e − σx ( x α + x . ) . Table 5.2: The maximum errors and convergent orders for (4.12) and (3.8), respec-tively, when p = 5, σ = 0 . β = 1 . r = 4. Numerical scheme (4.12) Numerical scheme (3.8) h α = − . α = 0 . α = − . α = 0 . Table 5.2 numerically verifies Theorem 4.13, i.e., the scheme (4.12) can keep thehigh convergent order when the regularity requirements of the performed functionsare relaxed; but the scheme (3.8) fails.
6. Conclusions.
When studying the anomalous diffusion, CTRW is the mostwidely used model. However, if the boundary conditions and external fields are neededto consider, the equations are more convenient to include these quantities. Assumingthe probability density functions (PDFs) of the waiting time and jump lengths inCTRW model are independent, from CTRW model we can derive the correspondingfractional partial differential equations (PDEs). On the other cases, when the PDFs of0
M. H. CHEN AND W. H. DENG the CTRW model are coupled in some way, the derived PDEs usually have a fractionalsubstantial derivative/integral. Nowadays, it seems that there are less mathematicalworks for this kind of operators. This paper detailedly discusses the properties of frac-tional substantial calculus, and provide a series of high order discretization schemes,which does well preparation for numerically solving PDEs with fractional substantialcalculus.
Acknowledgements.
We thanks Eli Barkai for the fruitful discussions and let-ting us know the urgency to solve the PDE with fractional substantial derivative inphysical community.
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