Efficient Galerkin solution of stochastic fractional differential equations using second kind Chebyshev wavelets

abstract: Stochastic fractional differential equations (SFDEs) have been used for modeling many physical problems in the fields of turbulance, heterogeneous, flows and matrials, viscoelasticity and electromagnetic theory. In this paper, an efficient wavelet Galerkin method based on the second kind Chebyshev wavelets are proposed for approximate solution of SFDEs. In this approach, operational matrices of the second kind Chebyshev wavelets are used for reducing SFDEs to a linear system of algebraic equations that can be solved easily. Convergence and error analysis of the proposed method is considered. Some numerical examples are performed to confirm the applicability and efficiency of the proposed method.


Introduction
Recently, stochastic analysis has been an interesting research area in mathematics, fluid mechanics, geophysics, biology, chemistry, epidemiology, microelectronics, physics, economics, and finance [1,2,3].The behavior of dynamical systems in u(s)k 1 (s, t)ds + t 0 u(s)k 2 (s, t)dB(s), t ∈ [0, 1] , ( with these initial conditions where u(t), f (t) and k i (s, t), i = 1, 2 are the stochastic processes defined on the same probability space (Ω, F, P ), and u(t) is unknown.Also B(t) is a Brownian motion process and t 0 k 2 (s, t)u(s)dB(s) is the Itô integral.Many phenomena in science that have been modeled by fractional differential equations have some uncertainty, so for deriving a more accurate solution, we need the solution of SFDEs [13,40,41,42].For deriving an approximate solution of SFDEs (1.1) we first derive some operational matrices for the second kind Chebyshev wavelets.Then, these operational matrices along with second kind Chebyshev wavelet are used to obtain approximate solution.
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The reminder of the paper is organized as follows: In section 2 some preliminary definitions of stochastic calculus, fractional calculus and Block Pulse Functions (BPFs) are reviewed.Section 3 is devoted to the basic definitions of the second kind Chebyshev wavelets and their properties.In section 4 general procedures for forming operational matrices of the second kind Chebyshev wavelets are explained.In section 5 a wavelet Galerkin method based on the second kind Chebyshev wavelets and their operational matrices are proposed for solving SFDEs.Numerical examples are included in section 6.Finally, a conclusion is given in section 7.

Preliminary definitions
In this section we review some necessary definitions and mathematical preliminaries about stochastic calculus, fractional calculus and BPFs which are required for establishing our results in the next sections [1,2].

Stochastic calculus
Definition 2.1.(Brownian motion) A real-valued stochastic process B(t), t ∈ [0, T ] is called Brownian motion, if it satisfies the following properties: (i) The process has independent increments for 0 ≤ t 0 ≤ t 1 ≤ ... ≤ t n ≤ T , (ii) For all t ≥ 0, B(t+h)−B(t) is a normal distribution with mean 0 and variance h, (iii) The function t → B(t) is a continuous function of t.Definition 2.2.Let {N t } t≥0 be an increasing family of σ-algebras of subsets of Ω.
where B denotes the Borel algebra on [0, ∞) and F is the σ -algebra on Ω. (ii) f is adapted to F t , where F t is the σ -algebra generated by the random variables B(s), s ≤ t.
where, ϕ n is a sequence of elementary functions such that For more details about stochastic calculus and integration please see [1,2,43].

Fractional calculus
Fractional order calculus is a branch of calculus which deal with integration and differentiation operators of non-integer order.Among the several formulations of the generalized derivative, the Riemann-Liouville and Caputo definition are most commonly used.Here we give some necessary definitions and mathematical preliminaries of the fractional calculus which are required for establishing our results [20].
Definition 2.5.A real function f (t), t > 0, is said to be in the space C µ , µ ∈ R if there exists a real number p > µ and a function f , and it is said to be in the space Definition 2.6.The Riemann-Liouville fractional integration of order α ≥ 0 of a function f ∈ C µ , µ ≥ −1, is defined as The Riemann-Liouville fractional operator J α has the following properties: Definition 2.7.Riemann-Liouville fractional derivative of order α > 0 is defined as The Riemann-Liouville derivatives have certain disadvantages when trying to model real-world phenomena with fractional differential equations.Therefore, a modified fractional differential operator D α * was proposed by Caputo [20].
Definition 2.8.The fractional derivative of order α > 0 in the Caputo sense is defined as where n is an integer, t > 0, and f ∈ C n 1 .
Some useful relation between the Riemann-Liouvill and Caputo fractional operators is given by the following expression: For more details about fractional calculus please see [20].

Block pulse functions
BPFs have been studied by many authors and applied for solving different problems.In this section we recall definition and some properties of the block pulse functions [11,27].
The m-set of BPFs are defined as in which t ∈ [0, T ), i = 1, 2, ..., m and h = T m .The set of BPFs are disjoint with each other in the interval [0, T ) and where δ ij is the Kronecker delta.The set of BPFs defined in the interval [0, T ) are orthogonal with each other, that is If m → ∞ the set of BPFs is a complete basis for L 2 [0, T ), so an arbitrary real bounded function f (t), which is square integrable in the interval [0, T ), can be expanded into a block pulse series as where Rewritting Eq. (2.7) in the vector form we have in which ) can be expanded with respect to BPFs such as where Φ(t) is the m-dimensional BPFs vectors and K is the m × m BPFs coefficient matrix with (i, j)-th element and h 1 = T1 m and h 2 = T2 m .Let Φ(t) be the BPFs vector, then we have and For an m-vector F we have where F is an m × m matrix, and F = diag(F ).Also, it is easy to show that for an m × m matrix A where Â = (a 11 , a 22 , ..., a mm ) is an m-vector.

Second kind Chebyshev wavelets
Wavelets constitute a family of functions constructed from dilation and translation of a single function ψ called the mother wavelet.When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets [10,32,35,36] The second kind Chebyshev wavelets ψ nm (x) = ψ(k, n, m, x) are defined on the interval [0, 1) by Efficient Galerkin solution

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where U m (t) is the second kind Chebyshev polynomials of degree m, given by [44] U m (t) = sin ((m + 1)θ) sin(θ) , t = cos(θ) The second kind Chebyshev wavelets {ψ nm (t)|n = 0, 1, . . ., . By using the orthonormality of the second kind Chebyshev wavelets, any square inegrable function f (t) defined over [0, 1) can be expanded in terms of the second kind Chebyshev wavelets as where c mn = (f (t), ψ mn (t)) w nk and (., .)w nk denotes the inner product on L 2 w nk [0, 1] .If the infinite series in (3.4) is truncated, then it can be written as By changing indices in the vectors Ψ(t) and C the series (3.5) can be rewritten as where and ) can be expanded into second kind Chebyshev wavelets basis as where K = [k ij ] and k ij = ψ i (s), u(s, t), ψ j (t) w nk w nk .

Second kind Chebyshev wavelets and BPFs
In this section we will review the relation between the second kind Chebyshev wavelets and BPFs.It is worth mention that here we set T = 1 in definition of BPFs.
Theorem 3.1.Let Ψ(t) and Φ(t) be the m-dimensional second kind Chebyshev wavelets and BPFs vector respectively, the vector Ψ(t) can be expanded by BPFs vector Φ(t) as where Q is an m × m block matrix and Proof: Let ψ i (t), i = 1, 2, ..., m be the i-th element of second kind Chebyshev wavelets vector.Expanding ψ i (t) into an m-term vector of BPFs, we have taking the collocation points η j = 2j−1 2 m and evaluating relation (3.13) we get and this prove the desired result.✷ The following remarks are consequence of relations (2.16), (2.17) and Theorem 3.1.Remark 3.2.For an m-vector F we have in which F is an m × m matrix as where F = diag Q T F .
Remark 3.3.Let A be an arbitrary m × m matrix, then for the second kind Chebyshev wavelets vector Ψ(t) we have where ÂT = U Q −1 and U is an m-vector that its elements are diagonal entries of matrix Q T AQ.
Efficient Galerkin solution 203 4. Operational matrices for second kind Chebyshev wavelets In this section some operational matrices for the second kind Chebyshev wavelets vector Ψ(t) are derived.Next theorems provide general procedures for forming these matrices.First, we remind some useful results for BPFs [11].
Lemma 4.1.[11] Let Φ(t) be the m-dimensional BPFs vector defined in (2.10), then integration of this vector can be derived as where P is called the operational matrix of integration for BPFs and is given by Lemma 4.2.
[45] Let Φ(t) be the m-dimensional BPFs vector defined in (2.10), then integration of this vector can be derived as where P α is called the operational matrix of integration for BPFs and is given by Lemma 4.3.[11] Let Φ(t) be the m-dimensional BPFs vector defined in (2.10), the Itô integral of this vector can be derived as where P s is called the stochastic operational matrix of BPFs and is given by Fakhrodin Mohammadi Now we are ready to derive operational matrices of stochastic and fractional integration for the second kind Chebyshev wavelets.Theorem 4.4.Suppose Ψ(t) be the m-dimensional second kind Chebyshev wavelets vector defined in (3.8), the integral of this vector can be derived as where Q is introduced in (3.11) and P is the operational matrix of integration for BPFs derived in (4.2).
Proof: Let Ψ(t) be the second kind Chebyshev wavelets vector, by using Theorem 3.1 and Lemma 4.2 we have now Theorem 3.1 gives by using this identity we obtain the desired result.✷ Theorem 4.5.Let Ψ(t) be the m-dimensional second kind Chebyshev wavelets vector defined in (3.8), the operational matrix of the fractional order integration for Ψ(t) can be derived as where Λ α is called the operational matrix of second kind Chebyshev wavelets, Q is the matrix introduced in (3.11) and F α is the operational matrix of fractional integration for BPFs derived in (4.4).
Proof: By using Theorem 3.1 we have so, the second kind Chebyshev wavelet operational matrix of the fractional order integration P α is given by and this completes the proof.✷ Efficient Galerkin solution

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Theorem 4.6.Suppose Ψ(t) be the m-dimensional second kind Chebyshev wavelets vector defined in (3.8), the Itô integral of this vector can be derived as where Λ s is called stochastic operational matrix for second kind Chebyshev wavelets, Q is introduced in (3.11) and P s is the stochastic operational matrix of integration for BPFs derived in (4.6).
Proof: Let Ψ(t) be the second kind Chebyshev wavelets vector, by using Theorem 3.1 and Lemma 4.3 we have now Theorem 3.1 result and this complete the proof.✷

Description of the numerical method
Here we present a wavelet Galerkin method based on the second kind Chebyshev wavelets and their operational matrices for solving SFDEs (1.1).For this purpose, and by using the relation of the fractional derivative and integral, the solution u(t) can be derived as now functions u(t), f (t) and k i (s, t), i = 1, 2, can be expanded in term of the second kind Chebyshev wavelets as ) where C and F are second kind Chebyshev wavelets coefficients vectors, and K i , i = 1, 2, are second kind Chebyshev wavelets coefficient matrices defined in Eqs.(3.8) and (3.10).Substituting above approximations in Eq. (5.1), we get where are m-vectors.As this equation is hold for all t ∈ [0, 1), the standard Galerkin method results (5.5) The vectors C 1 and C 2 are linear functions of vector C, so Eq.(5.5) is a linear system of algebraic equations for unknown vector C. Solving this linear system we obtain vector C, which can be used to approximate solution of SFDE (1.1) by substituting in Eq. ( 5.3).

Convergence analysis
The aim of this section is to analyze the proposed the second kind Chebyshev wavelets numerical scheme for solving SFDEs.∞ m=0 c mn ψ mn (x) be its infinite second kind Chebyshev wavelets expansion, then this means the second kind Chebyshev wavelets series converges uniformly to f (x) and Efficient Galerkin solution

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Proof: From definition of coefficient c mn we have by substituting 2 k+1 t − 2n − 1 = cos(θ) in (6.4) we get using integration by part two times we obtain where since n ≤ 2 k − 1, we obtain ✷ Theorem 6.2.Let f (x) be a continuous function defined on [0, 1), with second derivatives f ′′ (x) bounded by L, then we have the following accuracy estimation , where .
Proof: We have now by considering the relation (6.1) we achive the desired result.✷ Now we state the main result of this section which investigate the convergency of the proposed method for the approximate solution of SFDE (1.1).Hereafter e n (t) is error function of the second kind Chebyshev wavelets approximate solution u n (t) and .denotes L 2 norm in [0, 1] defined by . Theorem 6.3.Suppose u(t) is the exact solution of (1.1) and u n (t), k 1n (s, t), k 2n (s, t) are the second kind Chebyshev wavelets approximate solution for u(t), k 1 (s, t), k 2 (s, t) respectively.Also assume that then lim n→∞ e n (t) = 0.
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Proof: Let e n (t) = u(t) − u n (t), from (5.1) we get consequently, we can write So we get Theorem 6.1 shows that second kind Chebyshev wavelets expansion of any squar intgrable function converges uniformly.So, for any ε > 0 there exist n such that Fakhrodin Mohammadi therefore from (6.15) we get and the proof is complete.✷

Numerical results
In this section, we implement the proposed algorithm in section 5 for solving SFDEs.In all examples the algorithms are performed by Maple 17 with 20 digits precision.
Example 7.1.Consider the following SFDE subject to the initial condition u(0) = 0.The exact solution of this SFDE is unknown.Here we use the wavelet Galerkin method proposed in section 5 to solve it.Example 7.2.As the second example consider the following SFDE subject to the initial condition u(0) = 0.The exact solution of this SFDE is unknown.The approximate solution obtained by the proposed method for various values of t and α are listed in Table 2. Fig. 2 plots the approximate solution for different values of α with m = 128.subject to the initial condition u(0) = 0.The exact solution of this SFDE is unknown.Here the proposed wavelet Galerkin method is used for deriving numerical solution of it.The approximate solution for diffrent values of α and t with m = 128 is listed in Table 3.Moreover, Fig. 3 shows the approximate solutions for different values of α and m = 128.0.1 0.0088179999 0.0088039620 0.0087980641 0.3 0.0851554626 0.0867376737 0.0875707189 0.5 0.4390176500 0.4566864654 0.4713103382 0.7 0.4950304749 0.4876362989 0.4832995027 0.9 0.9761532307 0.8973556536 0.8564481929

Conclusion
Many phenomena in science that have been modeled by fractional differential equations have some uncertainty, so for deriving a more accurate solution, we have to solve a SFDEs.In this paper, we proposed a Galerkin scheme based on the second kind Chebyshev wavelets for solving SFDEs.In this scheme, we used the operational matrices of fractional and stochastic integration for the second kind Chebyshev wavelets.The main advantage of this method is to reduce the SFDEs into a problem consisting of a system of algebraic equations.The reduction is based on the operational matrices and the Galerkin method.The efficiency and applicability of the suggested scheme is confirmed on some examples.

Table 1
lists the approximate solution for different values of t and α with m = 128.Moreover, Fig.1shows the approximate solutions obtained for different values of α.

Table 1 :
Numerical results for different values of t, α and m = 128.

Table 2 :
Numerical results for different values of t and α.

Table 3 :
Numerical results for different values of t and α.