On a method of solution of systems of fractional pseudo-differential equations
-
Sabir Umarov
,
Ravshan Ashurov
Abstract
This paper is devoted to the general theory of linear systems of fractional order pseudo-differential equations. Single fractional order differential and pseudo-differential equations are studied by many authors and several monographs and handbooks have been published devoted to its theory and applications. However, the state of systems of fractional order ordinary and partial or pseudo-differential equations is still far from completeness, even in the linear case. In this paper we develop a new method of solution of general systems of fractional order linear pseudo-differential equations and prove existence and uniqueness theorems in the special classes of distributions, as well as in the Sobolev spaces.
1 Introduction
In the last few decades, fractional order differential equations have proved to be an essential tool in the modeling of dynamics of various complex stochastic processes arising in anomalous diffusion in physics [17, 33, 35, 52], finance [31, 45], hydrology [6], cell biology [32], and other fields of modern science and engineering. The complexity of stochastic processes includes phenomena such as the presence of weak or strong correlations, different sub- or super-diffusive modes and jump effects.
Various versions of fractional order differential and pseudo-differential equations are studied by many authors and several books have been published (see e.g. [21, 25, 40, 44, 47, 48]). However, the state of systems of fractional order ordinary and partial differential equations is still far from completeness, even in the linear case. At the same time systems of fractional order ordinary and partial differential equations have rich applications. For example, they are used in modeling of processes in biosystems [8, 15, 43], ecology [20, 42], epidemiology [19, 53], etc.
For some nonlinear systems of fractional order ordinary differential equations numerical and analytic approximate solution methods are developed; see e.g. [1, 2, 11, 36, 49, 51]. Many applied processes can be modeled by by-linear systems of fractional differential equations, including COVID-19 pandemic [3, 14, 28, 38, 41]. The advance of fractional order modeling is it adds parameters controlling effects like memory and correlations, leading to a better analysis and prediction.
In the linear case obtaining a representation for the solution is also possible. For example, in the paper [7], the authors prove existence and uniqueness of the system
of time-fractional ordinary differential equations, where x(t) is a vector-function, A is a nonsingular matrix, and α ∈ (0, 1) is scalar, with the solution representation x(t) = Eα(tαA)x0. Here Eα(𝓩) is the matrix-valued Mittag-Leffler function of a matrix 𝓩. The paper [37] studies stability conditions for the system Dαu(t) = Au(t) of fractional order ordinary differential equations with a vector-order α, with components αj ∈ (0, 1), j = 1, …, m.
More general cases of linear systems of the form DαLu(t) = Mu(t), where L and M are linear operators from a Banach space to another Banach space, were also considered. Gordievskikh and Fedorov [13] studied the Cauchy problem for degenerate operator L, that is Ker L ≠ 0. Regular case of the invertible operator L was studied in [4, 29]. Mamchuev [34] studied the boundary value problem for the fractional order system of the form
with boundary conditions
Here Ai, i = 1, … m, and B are n × n-matrices and Dα is the Riemann-Liouville derivative. The existence and uniqueness theorem is obtained as well as a representation formula for the solution through the Green function.
An important aspect of systems with integer order derivatives is that one can reduce such a system to a first order system increasing the number of equations/unknowns. In general, this approach loses its meaning in the case of fractional order systems, though as shown in [10] in some cases the systems with distinct fractional orders can be reduced to a system with the same fractional order in each equation. However, in this case, on the one hand the orders of the original system assumed to be rational, and on the other hand the number of equations in the reduced system may increase significantly. For example, if the orders in the original system of 4 equations are
In what concerns systems of fractional order partial differential equations, many of them can be treated within the theory of fractional order operator-differential equations in Banach or topological-vector spaces [5, 27, 46, 47]. However, such systems are of single scalar order or distributed scalar order equations. They can not be of vector-order. Moreover, some important specific features of fractional order systems of partial differential equations , such as parabolicity or hyperbolicity properties, can not be captured by operator-differential equations. Kochubei [22, 23, 24] studied fractional (scalar) order generalizations of parabolic and hyperbolic systems and found the corresponding fundamental solutions. Vazquez and Mendes [50] and Pierantozzi [39] studied fractional (scalar) order systems of Dirac-like equations. Some other issues related to fractional order systems, such as stability problems, numerical solution, along with others, are considered in works [9, 12, 16, 30]. The orders of systems in these works are also scalar.
In this paper we will deal with the following general system of linear fractional vector-order pseudo-differential equations
where Dβj, j = 1, …, m, is the fractional order derivative of order 0 < βj ≤ 1 in the sense of Riemann-Liouville or Caputo, and Aj,k(D) are pseudo-differential operators with (possibly singular) symbols depending only on dual variables (for simplicity) and described later. The obtained results can be extended for wider classes of pseudo-differential operators with symbols depending on time and spatial variables and non-symmetric as well, but this level of extension is not a goal of this paper. The initial conditions depend on the form of fractional derivatives. The results also can be extended to the case when the orders (some or all) βj ∈ (1, 2] adjusting properly the initial conditions.
The paper is organized as follows. Section 2 provides some preliminary facts on pseudo-differential operators with constant singular symbols, on the functional spaces where these pseudo-differential operators act continuously, and on fractional calculus used in this paper. In Section 3 we present main results. Here we prove the existence and uniqueness theorems in the general form for systems of time-fractional pseudo-differential equations. The representation formulas for solutions are also obtained in this section.
2 Preliminaries and auxiliaries
In this section we introduce some auxiliary notations and facts. We briefly recall definitions and related basic facts on general pseudo-differential operators without smoothness and growth restrictions to symbols as well as elliptic pseudo-differential operators and the spaces of distributions where these operators act. For details we refer the reader to the book [47].
2.1 Generalized function spaces ΨG,p(ℝn), Ψ−G,q(ℝn)
Let p > 1, q > 1, p−1 + q−1 = 1 be two conjugate numbers. The generalized functions space Ψ−G,q(ℝn), which we are going to introduce is distinct from the classical spaces of generalized functions.
Let G ⊂ ℝn be an open domain and a system 𝓖 ≡
and
Let N ∈ ℕ. Denote by ΨN,p the set of functions f ∈ Lp(ℝn) satisfying the conditions:
supp F[f] ⊂ GN;
supp F[f] ∩ supp ϕj = ∅ for j > N;
pN(f) = ∥F−1κNFf∥p < ∞.
Lemma 2.1
ForN = 1, 2, …, the relations
ΨN,p ↪ ΨN+1,p,
ΨN,p ↪ Lp(ℝn)
are valid, where ↪ denote the operation of continuous embedding.
It follows from Lemma 2.1 that ΨN,p form an increasing sequence of Banach spaces. Its limit with the inductive topology we denote by ΨG,p. Thus,
The inductive limit topology of ΨG,p(ℝn) is equivalent to the following convergence. A sequence of functions fm ∈ ΨG,p(ℝn) is said to converge to an element f0 ∈ ΨG,p(ℝn) iff:
there exists a compact set K ⊂ G such that supp f̂m ⊂ K for all m ∈ ℕ;
Remark 2.1
According to the Paley-Wiener-Schwartz theorem, elements of ΨG,p(ℝn) are entire functions of exponential type which, restricted to ℝn, are in the space Lp(ℝn).
The space topologically dual to ΨG,p(ℝn), which is the projective limit of the sequence of spaces conjugate to ΨN,p, is denoted by
In other words,
Further, we denote by ΨG,p(ℝn) the m-times topological direct product
of spaces ΨG,p(ℝn). Elements of ΨG,p(ℝn) are vector-functions Φ (x) = (φ1(x), …, φm(x)), where φj(x) ∈ ΨG,p(ℝn), j = 1, …, m. The space, topologically dual to ΨG,p(ℝn), is the direct sum
Finally for a topological vector space X we denote by C(k)[[a, b]; X] the space of vector-functions g(t), t ∈ [a, b], with values in X and k times differentiable in the sense of the topology of X. Similarly, one can define the space C∞[[a, b]; X].
2.2 Pseudo-differential operators with constant symbols
Now we introduce and consider some properties of pseudo-differential operators with constant (that is not depending on the variable x) symbols defined and continuous in a domain G ⊂ ℝn. Outside of G or on its boundary the symbol a(ξ) may have singularities of arbitrary type. It is clear that the corresponding class of pseudo-differential operators are not in the frame of classic pseudo-differential operators with infinitely differentiable symbols, studied first in works by Kon-Nirenberg [26] and Hörmander [18]. For the systematic presentation of the theory of pseudo-differential operators being considered in this paper we refer the reader to [47].
For a function φ ∈ ΨG,p(ℝn) the operator A(D) corresponding to the symbol A(ξ) is defined by the formula
We note that the assumption φ ∈ ΨG,p(ℝn) is crucial in the definition of A(D) in (2.3). Generally speaking, A(D) has no sense even for functions in the space
Theorem 2.1
The space ΨG,p(ℝn) (
and
act continuously.
Remark 2.2
In the case p = 2 an arbitrary pseudo-differential operator whose symbol is continuous in G acts continuously without the additional condition for A(ξ) kN(ξ) to be a multiplier in L2 for every N ∈ ℕ.
Finally, the following theorem establishes conditions for continuous closability of the pseudo-differential operator A(D) acting in the space ΨG,p(ℝn) to Sobolev spaces
Theorem 2.2
([47]) Let 1 < p < ∞, − ∞ < s, ℓ < +∞ andμ(ℝn ∖ G) = 0. For a pseudo-differential operator
there exists a closed extension
if and only if the symbolA(ξ) satisfies the estimate
Theorems 2.1 and 2.2 can be extended to matrix pseudo-differential operators, elements of which satisfy theses theorems. Let
be the matrix pseudo-differential operator with the matrix-symbol
Namely, the following theorems are valid.
Theorem 2.3
The spaceΨG,p(ℝn) (
and
act continuously.
Theorem 2.4
Let 1 < p < ∞, − ∞ < s, ℓ < +∞ andμ(ℝn ∖ G) = 0. For a pseudo-differential operator
there exists a closed extension
if and only if each entryajk(ξ) of the symbol 𝓐(ξ) satisfies the estimate
Proofs of these statements directly follow from Theorems 2.1 and 2.2.
2.3 Fractional integrals and derivatives
Let a function f(t) be defined and measurable on an interval (a, b), a < b ≤ ∞. The fractional integral of order β > 0 of the function f is defined by
where Γ(β) is Euler’s gamma function, that is
If β = 0, then we agree that
Let m be a natural number and m − 1 ≤ β < m. Then the fractional derivative of order β of a function f in the sense of Riemann–Liouville is defined as
provided the expression on the right exists. One can write
This operator is the left-inverse to the fractional integration operator
To explore a domain of
The operator form of
Proposition 2.1
([44])) Letf ∈ Cλ[a, b], 0 < λ ≤ 1. Then for anyβ < λ the fractional derivative
whereψ ∈ Cλ−β[a, b].
Let m be a natural number and m − 1 ≤ β < m. Then the fractional derivative of order β of a function f in the sense of Caputo is defined as
provided the integral on the right exists.
The operator form of the fractional derivative
which is well defined, for instance, in the class of m-times differentiable functions defined on an interval [a, b), b > a. It follows from definition (2.13) that if 0 < β < 1, then
The operator form of
Let a = 0. In this case we write simply
Proposition 2.2
Letβ > 0. Then the Laplace transform ofJβf(t) is
Proposition 2.3
Let m − 1 ≤ β < m, m = 1, 2, …. Then the Laplace transform of
Proposition 2.4
Let m − 1 < β ≤ m. The Laplace transform of the Caputo derivative of a functionf ∈ Cm−1[0, ∞) is
For β ∈ (0, 1] formulas (2.17) and (2.18) respectively take the forms:
We will use these formulas in the vector form:
In these formulas 𝓑 = 〈β1, …, βm〉 is a vector-order with 0 < βj ≤ 1, j = 1, …, m, and
for both operators 𝓓 = D+ and 𝓓 = D∗.
3 Main results
Consider the following system of fractional order differential equations
where 0 < βj ≤ 1, j = 1, …, m, and the operator 𝓓 on the left expresses either the Riemann-Liouville derivative D+ or the Caputo derivative D∗. We will specify the initial conditions later depending on whether 𝓓 is the Riemann-Liouville or the Caputo derivative.
With the vector-order 𝓑 = 〈β1, …, βm〉, introducing vector-functions U(t, x) = 〈u1(t, x), …, um(t, x)〉, H(t, x) = 〈h1(t, x), …, hm(t, x)〉, we can represent system (3.1) in the vector form:
where 𝔸(D) is the matrix pseudo-differential operator with the matrix-symbol 𝓐(ξ), ξ ∈ G, defined in (2.6), and
For simplicity we assume that the matrix-symbol is symmetric, ak,j(ξ) = aj,k(ξ) for all k, j = 1, …, m, and ξ ∈ G, and diagonalizable. Namely, there exists an invertible (m × m)-matrix-function M(ξ), such that
with a diagonal matrix
We denote entries of matrices M(ξ) and M−1(ξ) by μj,k(ξ), j, k = 1, …, m, and νj,k(ξ), j, k = 1, …, m, respectively.
First we derive a representation formula for the solution of the initial value problem for system (3.2) in the homogeneous case. Since initial conditions depend on the form of the fractional derivative on the left hand side of equation (3.2), the corresponding representations of solutions differ. We demonstrate the derivation in the case of Caputo fractional derivative. The case of Rieman-Liouville fractional derivative can be treated similarly.
Consider the following Cauchy problem:
where the fractional derivatives on the left are in the sense of Caputo. Applying Fourier transform we obtain a system of fractional order ordinary differential equations with a parameter ξ:
with the initial conditions
Now applying the Laplace transform in the vector form (2.22), one has
Taking into account (3.3) the letter can be rewritten in the form
where Is𝓑, Is𝓑−1 are diagonal matrices with diagonal entries sβj, sβj−1j = 1, …, m, respectively. The solution to the obtained system is
where
It follows from (3.7) and (3.8) that
Here 𝓔𝓑(Λ(ξ) t𝓑) is the diagonal matrix of the form
where Eβj(z), j = 1, …, m, are the Mittag-Leffler functions of indices β1, …, βm. Thus, the solution of problem (3.5)-(3.6) has the representation
where S(t, D) is the solution matrix pseudo-differential operator with the matrix-symbol
whose entries are
The explicit component-wise form of the solution is
Theorem 3.1
Let 𝔸 be a pseudo-differential operator with the symbol 𝓐(ξ) continuous onGand satisfying the condition(3.3). Assume thatΦ (x) ∈ ΨG,p(ℝn), H(t, x) ∈ AC[ℝ+; ΨG,p(ℝn)], and
has a unique solutionU(t, x) ∈ C∞[(0, T]; ΨG,p(ℝn)] ∩ C[[0, T]; ΨG,p(ℝn)], having the representation
whereS(t, D) is the pseudo-differential operator with the matrix-symbol 𝓢(t, ξ) defined in(3.12).
Proof
The representation (3.15) follows directly from (3.11) and from fractional Duhamel’s principle [46, 47]. Denote the first and second terms on the right of (3.15) by V(t, x) and W(t, x), respectively:
Then, in accordance with Theorem (2.3)V(t, x) ∈ ΨG,p(ℝn) for every fixed t ≥ 0, continuous on [0, T], and infinitely differentiable on (0, T) in the topology of ΨG,p(ℝn) due to the construction of the solution operator S(t, D). Further, there exists a sequence (see (2.1))
such that HN(t, x) → H(t, x) as N → ∞ in the topology of ΨG,p(ℝn). Moreover, pN(HN) = ∥HN∥p. Let
Then we have pN(WN) = ∥F−1κNF[WN]∥p ≤ T ∥WN∥p < ∞ for all N ≥ 1. It follows that WN ∈ ΨN,p(ℝn) and WN(t, x) → W (t, x), as N → ∞, in the topology of ΨN,p(ℝn) for each fixed t ∈ [0, T]. The continuity of W(t, x) on [0, T] in the variable t and its infinite differentiability on (0, T) follows from the construction of the solution operator S(t, D) in standard way. □
Theorem 3.2
Letpandq, 1 < p, q < ∞, be a conjugate pair and 𝔸 be a pseudo-differential operator with the symbol 𝓐(ξ) continuous onGand satisfying the condition(3.3). Assume thatΦ (x) ∈
has a unique solution V(t, x) ∈ C∞[(0, T];
whereS(t, −D) is the pseudo-differential operator with the matrix-symbol 𝓢(t, −ξ) defined in(3.12)[*].
Proof
We note that elements
The solution V(t, x) of the Cauchy problem (3.18) – (3.19), by definition, must satisfy the following conditions:
for an arbitrary element F(x) in the space ΨG,p(ℝn). We show that U(t, x) defined in (3.20) satisfies both conditions in (3.21) and (3.22). Indeed, to show this fact let us first assume that H(t, x) = 0 [**] for all t ≥ 0. Then (3.21) takes the form
The operator S(t, D) is constructed so that
Further, it follows from (3.12) that the symbol 𝓢(t, ξ) at t = 0 reduces to the identity matrix, since the matrix 𝓔𝓑(0) is the identity matrix. Therefore, the operator corresponding to the matrix-symbol 𝓢(0, ξ) is the identity pseudo-differential operator. Hence, V(0, x) = S(0, −D) Φ(x) = Φ(x). Thus, condition (3.22) is also verified.
In the general case, for non-zero H(t, x), the representation (3.20) is an implication of the fractional Duhamel principle [46, 47]. □
Now consider the following initial-value problem
where the fractional derivatives on the left hand side of equation (3.23) are in the sense of Riemann-Liouville. Performing similar calculations, in this case for the solution we obtain the representation
where S+(t, D) is the solution matrix pseudo-differential operator with the matrix-symbol
whose the entries are
The explicit component-wise form of the solution is
Theorem 3.3
Let 𝔸 be a pseudo-differential operator with the symbol 𝓐(ξ) continuous onGand satisfying the condition(3.3)andΦ ∈ ΨG,p(ℝn). Then for anyT > 0 Cauchy problem
has a unique solutionU(t, x) ∈ C∞[(0, T]; ΨG(ℝn)], having the representation
whereS+(t, D) is the pseudo-differential operator with the matrix-symbol 𝓢+(t, ξ) defined in(3.26).
Theorem 3.4
Let 𝔸 be a pseudo-differential operator with the symbol 𝓐(ξ) continuous onGand satisfying the condition(3.3)andΦ ∈ ΨG,p(ℝn). Then for anyT > 0 Cauchy problem
has a unique solutionU(t, x) ∈ C∞[(0, T]; ΨG(ℝn)], having the representation
where S+(t, −D) is the pseudo-differential operator with the matrix-symbol 𝓢+(t, -ξ) defined in (3.26).
The proofs of Theorems 3.3 and 3.4 are omitted, since they are similar to the proofs of Theorems 3.1 and 3.2, respectively.
The properties of the solutions of problems (3.13)-(3.14) and (3.27) essentially depend on the asymptotic behavior of the functions
which form the symbols of solution operators; see (3.12) and (3.26). It is known that for 0 < β < 2 the Mittag-Leffler function Eβ (z) has asymptotic behavior ∼ exp(z1/β), |z| → ∞, if |arg(z)| ≤ βπ/2; and Eβ (z) ∼ 1/|z|, |z| → ∞, if βπ/2 ≤ | arg(z) ≤ π. Therefore, if a symbol A (ξ) is complex-valued, then Eβ(A(ξ) tβ) may have an exponential growth as | ξ | → ∞, even though A(ξ ) has a polynomial growth at infinity.
Now suppose that the pseudo-differential operator 𝔸(D) satisfies the following ellipticity condition: the symbol 𝓐(ξ), ξ ∈ ℝn, is symmetric, satisfies the condition (3.3) with a diagonal matrix Λ(ξ), and there exists a number R0 > 0 such that for the entries λℓ(ξ), ℓ = 1, …, m, of Λ(ξ) the inequalities
where ℜ(z) is the real part of z, hold for all ξ : |ξ| ≥ R0; η > 0, rℓ ∈ ℝ, ℓ = 1, …, m, are constants. In this case we have
with some C1, C2 positive constants. In the theorem below we use the notation r = (r1, …, rm).
Theorem 3.5
Let the following conditions be verified:
the operator 𝔸 is an \elliptic pseudo-differential operator satisfying the condition(3.34);
the symbol 𝓐(ξ) of the operator 𝔸 is symmetric, continuous on ℝn, and satisfies the condition(3.3);
Φ ∈
H(t, x) ∈ AC[ℝ+;
Then for anyT > 0 Cauchy problem(3.13)-(3.14)has a unique solutionU(t, x) ∈ C∞[(0, T];
whereŜ(t, D) is the closure of the pseudo-differential operator with the matrix-symbol 𝓢(t, ξ) defined in(3.12)in the space
Proof
Let components of Φ(x) are φk ∈
Since the components of the symbol 𝓢(t, ξ) of the solution operator S(t, D) satisfy the estimate (3.34), it follows from Theorem 2.4 that there exists a unique continuous closure Ŝ(t, D) of the operator S(t D), such that
is continuous. Thus for the solution U(t, x) we have representation (3.6). The fact that U(t, x) ∈ C∞[(0, T];
Similarly one can prove the existence of a unique solution in the Sobolev spaces of the Cauchy problem (3.27)-(3.28). Below is the formulation of the corresponding theorem.
Theorem 3.6
Let the following conditions be verified:
the operator 𝔸 is an \elliptic pseudo-differential operator satisfying the condition(3.34);
the symbol 𝓐(ξ) of the operator 𝔸 is symmetric, continuous on ℝn, and satisfies the condition(3.3);
Φ ∈
H(t, x) ∈ AC[ℝ+;
Then for anyT > 0 Cauchy problem(3.27)has a unique solutionU(t, x) ∈ C∞[(0, T];
whereŜ+(t, D) is the closure of the pseudo-differential operator with the matrix-symbol 𝓢+(t, ξ) defined in(3.26)in the space
Remark 3.1
The results of Theorems 3.1 - 3.6 coincide with the known results in 1-D case, see, e.g. [47].
The results obtained in Theorems 3.1 - 3.6 can be extended to the case, when 0 < βj ≤ 1, j = 1, …, m0 and 1 < βj ≤ 2, j = m0 + 1, …, m, where 0 ≤ m0 ≤ m, with properly adjusted initial conditions.
The results also can be extended to the case of fractional distributed order differential operators (DODE) on the left hand side of the considered systems.
Example
To illustrate the theorems proved above consider the following Cauchy problem
It is not hard to see that the symbol of the operator on the right hand side of (3.37)-(3.38) is symmetric and has the representation
As is seen from (3.40) that λ1(ξ) = −ξ2 + ξ and λ2(ξ) = −ξ2 − ξ. The symbol of the solution operator S(t, D) is the matrx 𝓢(t, ξ) = {sj,k(t, ξ)}, j, k = 1, 2, with entries
Therefore, the solution U(t, x) = 〈u1(t, x), u2(t, x)〉 to Cauchy problem (3.37)-(3.39) has the representation:
In the explicit form this solution has the form
Moreover, obviously, λk(ξ) ≤ 0, k = 1, 2, for all ξ satisfying the inequality |ξ| ≥ 1. Applying Theorem 3.5 we have U(t, x) ∈ C∞[(0, T];
References
[1] O. Abdulaziz, I. Hashim, S. Momani, Solving systems of fractional differential equations by homotopy-perturbation method. Phys. Letters A372 (2008), 451–459.10.1016/j.physleta.2007.07.059Suche in Google Scholar
[2] N. Aguila-Comacho, M.A. Duarte-Mermoud, J.A. Gallegos, Lyapunov functions for fractional order systems. Commun. Nonlinear. Sci. Simulat. 19 (2014), 2951–2957.10.1016/j.cnsns.2014.01.022Suche in Google Scholar
[3] R. Almeida, Analysis of fractional SEIR model with treatment. Applied Mathematical Letters84 (2018), 56–62.10.1016/j.aml.2018.04.015Suche in Google Scholar
[4] K. Balachandran, S. Kiruthika, Existence of solutions of abstract fractional integro-differential equations of Sobolev type. Computers and Math. with Appl. 64 (2012), 3406–3413.10.1016/j.camwa.2011.12.051Suche in Google Scholar
[5] E. Bazhlekova, The abstract Cauchy problem for the fractional evolution equation. Fract. Calc. Appl. Anal. 1, No 3 (1998), 255–270.Suche in Google Scholar
[6] D. Benson, M. Meerschaert, J. Revielle, Fractional calculus in hydrology modeling: A numerical perspective. Advances in Water Resources51 (2013), 479–497.10.1016/j.advwatres.2012.04.005Suche in Google Scholar PubMed PubMed Central
[7] V. Daftardar-Gejji, A. Babakhani, Analysis of a system of fractional differential equations. J. of Math. Anal. and Appl. 293 (2004), 511–522.10.1016/j.jmaa.2004.01.013Suche in Google Scholar
[8] S. Das, P.K. Gupta, A mathematical model on fractional Lotka-Volterra equations. J. of Theoretical Biology277 (1) 2011, 1–6.10.1016/j.jtbi.2011.01.034Suche in Google Scholar PubMed
[9] B. Datsko, V. Gafeychuk, I. Podlubny, Solitary traveling auto-waves in fractional reaction-diffusion systems. Commun. Nonlinear Sci. Numer. Simulat. 23 (2015), 378–387.10.1016/j.cnsns.2014.10.028Suche in Google Scholar
[10] W. Deng, Ch. Li, Q. Guo, Analysis of fractional differential equations with multi-orders. Fractals15, No 2 (2007), 173–182.10.1142/S0218348X07003472Suche in Google Scholar
[11] V.S. Ertürk, S. Momani, Solving systems of fractional differential equations using differential transform method. J. Comput. Appl. Math. 215 (2008), 142–151; 10.1016/j.cam.2007.03.029.Suche in Google Scholar
[12] V. Gafiychuk, B. Datsko, Mathematical Modeling of different types of instabilities in time fractional reaction-diffusion systems. Computers and Math. with Appl. 59 (2010), 1101–1107.10.1016/j.camwa.2009.05.013Suche in Google Scholar
[13] D.M. Gordievskikh, V.E. Fedorov, Solutions for initial boundary value problems for some degenerate equations of fractional order with respect to the time. The Bull. of Irkutsk State University. Ser. Mathematics12 (2015), 12–22.Suche in Google Scholar
[14] E.F.D. Goufo, R. Maritz and J. Munganga, Some properties of Kermack-McKendrick epidemic model with fractional derivative and nonlinear incidence. Adv. Difference Equ. 2014, No 1 (2014), Art. ID 278, 9 pp.10.1186/1687-1847-2014-278Suche in Google Scholar
[15] Ch. Guo, Sh. Fang, Stability and approximate analytic solutions of the fractional Lotka-Volterra equations for three competitors. Adv. Difference Equations219 (2016), 1–14.10.1186/s13662-016-0943-ySuche in Google Scholar
[16] A.S. Hendy, J.E. Macias-Diaz, A.J. Serna-Reyes. On the solution of hyperbolic two-dimensional systems via discrete variational schemes of high order of accuracy. J. of Comput. and Appl. Math. 354 (2019), 612–622.10.1016/j.cam.2018.10.059Suche in Google Scholar
[17] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific (2000).10.1142/3779Suche in Google Scholar
[18] L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-Differential Operators. Springer-Verlag, Berlin (1985).Suche in Google Scholar
[19] R. Islam, A. Pease, D. Medina, T. Oraby, Integer versus fractional order SEIR deterministic and stochastic models of measles. Intern. J. of Environmental Res. and Public Health17, No 6 (2020), 1–19.10.3390/ijerph17062014Suche in Google Scholar PubMed PubMed Central
[20] N.A. Khan, O.A. Razzaq, S.P. Mondal, Q. Rubbab, Fractional order ecological system for complexities of interacting species with harvesting threshold in imprecise environment. Adv. in Difference Equations405 (2019), 1–34.10.1186/s13662-019-2331-xSuche in Google Scholar
[21] A.A. Kilbas, H.M. Srivastava, J.J. Trijillo, Theory and Applications of Fractional Differential Equations. Elsevier Science (2006).Suche in Google Scholar
[22] A.N. Kochubei, Fractional-parabolic systems. Potential Analysis37 (2012), 1–30.10.1007/s11118-011-9243-zSuche in Google Scholar
[23] A.N. Kochubei, Fractional-hyperbolic systems. Fract. Calc. Appl. Anal. 16, No 4 (2013), 860–873; 10.2478/s13540-013-0053-4; https://www.degruyter.com/view/journals/fca/16/4/fca.16.issue-4.xml.Suche in Google Scholar
[24] A.N. Kochubei, Fractional-parabolic equations and systems. Cauchy problem. Handbook of Fractional Calculus and Applications, DeGruyter (2019), 145–158; Fractional-hyperbolic equations and systems. Cauchy problem. Handbook of Fractional Calculus and Applications, De Gruyter (2019), 197–222.Suche in Google Scholar
[25] A. Kochubei, Yu. Luchko (Eds), Handbook of Fractional Calculus with Applications. Volume 2: Fractional Differential Equations. De Gruyter (2019).10.1515/9783110571660Suche in Google Scholar
[26] A. Kohn, L. Nirenberg, An algebra of psudo-differential operators. Commun. on Pure and Appl. Math. 18 (1965), 269–305.10.1002/cpa.3160180121Suche in Google Scholar
[27] V.A. Kostin, The Cauchy problem for an abstract differential equation with fractional derivatives. Russ. Dokl. Math. 46 (1993), 316–319.Suche in Google Scholar
[28] K. Koziol, R. Stanislawski, G. Bialic, Fractional-order SIR epidemic model for transmission prediction of COVID-19 disease. Appl. Sci. 10 (2020), 1–9; 10.3390/app10238316.Suche in Google Scholar
[29] F. Li, J. Liang, H.-K. Xu, Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions. J. of Math. Anal. and Appl. 391 (2012), 510–525.10.1016/j.jmaa.2012.02.057Suche in Google Scholar
[30] Y. Luchko, M. Rivero, J.J. Trijillo, M.P.Velasco, Fractional models, non-locality, and complex systems. Computers and Math. with Appl. 59 (2010), 1048–1056.10.1016/j.camwa.2009.05.018Suche in Google Scholar
[31] T. Machado, A. Lopes, Relative fractional dynamics of stock markets. Nonlin. Dynamics86, No 3 (2016), 1613–1619.10.1007/s11071-016-2980-1Suche in Google Scholar
[32] R. Magin, Fractional calculus in bioengineering. Critical Reviews in Biomedical Engineering32, No 1 (2004), 1–104.10.1615/CritRevBiomedEng.v32.10Suche in Google Scholar
[33] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press (2010).10.1142/p614Suche in Google Scholar
[34] M.O. Mamchuev, Boundary value problem for a multidinensional system of equations with Riemann–Liouville fractional derivatives. Sib. Elektron. Mat. Izv. 16 (2019), 732–747; 10.33048/semi.2019.16.049.Suche in Google Scholar
[35] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, No 1 (2000), 1–77.10.1016/S0370-1573(00)00070-3Suche in Google Scholar
[36] S.K. Ntouyas, M. Obaid, A coupled system of fractional differential equations with nonlocal integral boundary conditions. Adv. in Difference Equations2012 (2012), Art. ID 130, 8 pp.10.1186/1687-1847-2012-130Suche in Google Scholar
[37] Z. Odibat, Analytic study on linear systems of fractional differential equations. Computers and Math. with Appl. 59 (2010), 1171–1183.10.1016/j.camwa.2009.06.035Suche in Google Scholar
[38] I. Owusu-Mensah, L. Akinuemi, B. Oduro, O.S. Iyiola, A fractional order approach to modeling and simulations of the novel COVID-19. Adv. in Difference Equations2020 (2020), Art. ID 683, 21 pp.; 10.1186/s13662-020-03141-7.Suche in Google Scholar
[39] T. Pierantozzi, Fractional evolution Dirac-like equations: Some properties and a discrete Von Neumann-type analysis. J. Comp. Appl. Math. 224 (2009), 284–295.10.1016/j.cam.2008.05.032Suche in Google Scholar
[40] I. Podlubny, Fractional Differential Equations. Academic Press (1998).Suche in Google Scholar
[41] K. Rajagopal, N. Hasanzadeh, F. Parastesh, I.I. Hamarash, S. Jafari, I Hussain, A fractional-order model for the novel coronavirus (COVID-19) outbreak. Nonlin. Dynamics101 (2020), 701–718.10.1007/s11071-020-05757-6Suche in Google Scholar
[42] S. Rana, S. Bhattacharya, J. Pal, G. N’Guérékata, J. Chattopadhyay, Paradox of enrichment: A fractional differential approach with memory. Phys. A: Stat. Mech. and its Appl. 392, No 17) (2013), 3610–3621; 10.1016/j.physa.2013.03.061.Suche in Google Scholar
[43] F. Rihan, Numerical modeling of fractional-order biological systems. Abstract and Appl. Anal. 2013 (2013), 1–13.10.1155/2013/816803Suche in Google Scholar
[44] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers (1993).Suche in Google Scholar
[45] E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance. Phys. A: Stat. Mech. and its Appl. 284, No 1-4 (2000), 376–384.10.1016/S0378-4371(00)00255-7Suche in Google Scholar
[46] S. Umarov, On fractional Duhamel’s principle and its applications. J. Diff. Equations252, No 10 (2012), 5217–5234.10.1016/j.jde.2012.01.029Suche in Google Scholar
[47] S. Umarov, Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols. Springer (2015).10.1007/978-3-319-20771-1Suche in Google Scholar
[48] S. Umarov, M. Hahn, K. Kobayashi, Beyond the Triangle: Brownian Motion, Ito Calculus and Fokker-Planck equations - Fractional Generalizations. World Scientific (2018).10.1142/10734Suche in Google Scholar
[49] C. Vargas-de-León. Volterra-type Lyapunov functions for fractional-order epidemic systems. Coomun. Nonlin. Sci. Numer. Simulat. 24 (2015), 75–85.10.1016/j.cnsns.2014.12.013Suche in Google Scholar
[50] L. Vazquez, R.V. Mendes, Fractionally coupled solutions of the diffusion equation. Appl. Math. Comp. 141 (2003), 125–130.10.1016/S0096-3003(02)00326-0Suche in Google Scholar
[51] G. Wang, R.P. Agarwal, A. Cabada, Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations. Appl. Math. Letters25, No 6 (2012), 1019–1024.10.1016/j.aml.2011.09.078Suche in Google Scholar
[52] B. West, Physics of \Fractal Operators. Springer (2003).10.1007/978-0-387-21746-8Suche in Google Scholar
[53] A. Zeb, G. Zaman, M.I. Chohan, Sh. Momani, V.S. Erturk, Analytic numeric solution for SIRC epidemic model in fractional order. Asian J. of Math. and Appl. 2013 (2013), 1–19.Suche in Google Scholar
© 2021 Diogenes Co., Sofia
Artikel in diesem Heft
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–volume 24–1–2021)
- Survey Paper
- Renormalization group and fractional calculus methods in a complex world: A review
- On the asymptotics of wright functions of the second kind
- Research Paper
- On fractional heat equation
- Completely monotone multinomial mittag-leffler type functions and diffusion equations with multiple time-derivatives
- A fractional generalization of the dirichlet distribution and related distributions
- Should i stay or should i go? zero-size jumps in random walks for lévy flights
- Well-posedness for weak and strong solutions of non-homogeneous initial boundary value problems for fractional diffusion equations
- Error analysis of nonlinear time fractional mobile/immobile advection-diffusion equation with weakly singular solutions
- Exact stability and instability regions for two-dimensional linear autonomous multi-order systems of fractional-order differential equations
- On a method of solution of systems of fractional pseudo-differential equations
- Monte carlo estimation of the solution of fractional partial differential equations
- Stability analysis for discrete time abstract fractional differential equations
- Short Paper
- Existence of local solutions for fractional difference equations with left focal boundary conditions
Artikel in diesem Heft
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–volume 24–1–2021)
- Survey Paper
- Renormalization group and fractional calculus methods in a complex world: A review
- On the asymptotics of wright functions of the second kind
- Research Paper
- On fractional heat equation
- Completely monotone multinomial mittag-leffler type functions and diffusion equations with multiple time-derivatives
- A fractional generalization of the dirichlet distribution and related distributions
- Should i stay or should i go? zero-size jumps in random walks for lévy flights
- Well-posedness for weak and strong solutions of non-homogeneous initial boundary value problems for fractional diffusion equations
- Error analysis of nonlinear time fractional mobile/immobile advection-diffusion equation with weakly singular solutions
- Exact stability and instability regions for two-dimensional linear autonomous multi-order systems of fractional-order differential equations
- On a method of solution of systems of fractional pseudo-differential equations
- Monte carlo estimation of the solution of fractional partial differential equations
- Stability analysis for discrete time abstract fractional differential equations
- Short Paper
- Existence of local solutions for fractional difference equations with left focal boundary conditions
