Wavelet collocation method based on Legendre polynomials and its application in solving the stochastic fractional integro-differential equations

doi:10.1016/j.jocs.2021.101342

Journal of Computational Science

Volume 51, April 2021, 101342

https://doi.org/10.1016/j.jocs.2021.101342 Get rights and content

Highlights

•
A wavelet collocation method based on Legendre polynomials has been proposed.
•
A stochastic and deterministic operational matrix of integration have been derived for Legendre wavelet.
•
Numerical experiments are conducted to validate the theoretical findings.
•
Concerning the real-world application, a stock market model has been simulated and the results are demonstrated.

Abstract

This work is an extended version of the ICCS 2020 conference paper [1]. The paper aims to present an efficient numerical method to quantify the uncertainty in the solution of stochastic fractional integro-differential equations. The numerical method presented here is a wavelet collocation method based on Legendre polynomials, and their deterministic and stochastic operational matrix of integration. The operational matrices are used to convert the stochastic fractional integro-differential equation to a linear system of algebraic equations. The accuracy and efficiency of the proposed method are validated through numerical experiments. Moreover, the results are compared with the numerical methods based on the Gaussian radial basis function (GA RBF) and thin plate splines radial basis function (TBS RBF) to show the superiority of the proposed method. Finally, concerning the real-world application, a stock market model has been simulated and the results are demonstrated.

Introduction

In the last few decades, fractional calculus has attracted the attention of numerous mathematicians, yet additionally, a few analysts in different regions like physics, finance, biology, and engineering (see [2], [3], [4], [5], [6], [7], [8], [9] and references therein). From the various research papers, we can observe that several real-world phenomena can be better described by a mathematical model involving fractional derivatives. This is due to two reasons: first, we can choose any real derivative/integral operator. Second, as a fractional-order derivative is a non-local operator, we can model systems with long term memory. Also, we know that most real-world data or experimental data are noisy, therefore, stochastic fractional differential equations are more suitable for real-world problems.

A stochastic fractional integro-differential equation (SFIDE), where order of derivative is non-integer, is a generalization of the fractional Fokker–Planck equation which describes the random walk of a particle [10]. This model has the following form $\begin{matrix} D_{0, t}^{α} u (t) & = f (t) + \int_{0}^{t} u (s) k_{1} (s, t) ds + \int_{0}^{t} u (s) k_{2} (s, t) dW (s), t \in (0, 1), \\ u (0) & = u_{0}, \end{matrix}$ where $D_{0, t}^{α},$ 0 < α < 1, denotes the Caputo fractional derivative, W(s), s ∈ [0, 1) is the standard Wiener process and the integral with respect to it is the It $\hat{o}$ integral. Presence of the It $\hat{o}$ integral in Eq. (1.1) causes randomness in the solution and hence it becomes non-deterministic. In this paper, we develop a novel approach to quantify this uncertain behavior in the numerical solution.

In recent decade, the need to obtain the numerical solution of SFIDE has increased significantly. However, in literature, only a handful of papers are available that actually discuss about the numerical solution of SFIDE. We now review some research articles that motivated us to structure a numerical method for SFIDE (1.1).

•
Maleknejad et al. [11] considered a stochastic Volterra integral equation based on the operational matrix method.
•
Mirzaee and Samadyar [12] constructed a numerical method based on Bernstein polynomials to solve the SFIDE.
•
Taheri et al. [13] considered a spectral collocation method based on shifted Legendre polynomials.
•
Again, Mirzaee and Samadyar [14] considered SFIDE and presented a collocation method based on radial basis functions.

In [14], author presented a numerical method based on radial basis function. This method needs smaller value of shape parameter for higher accuracy which increases the condition number of coefficient matrix and as a result the method become unstable. In [11], [12], [13], authors presented a numerical method based on orthogonal functions. Wavelets have numerous applications in approximation theory and construction of wavelets based on orthogonal functions are easy to generate orthonormal basis of L²[0, 1] with help of orthogonal functions. Legendre wavelets use Legendre polynomials as their basis functions and they have good interpolating properties and give better accuracy for smaller number of collocation points (see [15] and references therein). Apart from the Legendre wavelets, researchers are also using Haar wavelets, Gegenbauer wavelets, and Bernoulli wavelets for solving fractional differential equations (see [16], [17], [18], [19]).

Motivated by above work and discussion, in this paper, a new scheme is derived based on Legendre wavelet collocation method and block pulse function involving the operational matrix for solving SFIDE (1.1). In Section 2, we give basic definition of fractional calculus and construction of Legendre wavelet based on Multi-resolution analysis. Then in Section 3, operational matrix of fractional order integration and integration operational matrix are derived. The proposed scheme for the SFIDE is discussed in Section 4. The convergence analysis and error estimate of the proposed scheme are discussed in Section 5. Section 6 provides numerical experiments performed to showcase the effectiveness of the approach. In Section 7, we present various applications of SFIDE. Finally, Section 8 gives the brief conclusion.

Section snippets

Preliminaries

In this section, we discuss the mathematical preliminaries of fractional calculus and construction of wavelet which is required for subsequent development.

Definition 2.1

[20]

The left Riemann–Liouville fractional integral of order α ≥ 0 of a function f(t), t ∈ (a, b) is defined as follows $\begin{matrix} _{a} I_{t}^{α} f (t) & = \frac{1}{Γ (α)} \int_{a}^{t} {(t - s)}^{α - 1} f (s) ds, \\ _{a} I_{t}^{0} f (t) & = f (t) . \end{matrix}$ Similar to integer order integration, the left Riemann–Liouville fractional integral operator is a linear operator $_{a} I_{t}^{α} (λ f (t) + μ g (t)) = λ_{a} I_{t}^{α} f (t) + μ_{a} I_{t}^{α} g (t),$ where λ and μ are constants.

Definition 2.2

Legendre wavelet matrix and block pulse operational matrix

Let $t_{i} = \frac{2 i - 1}{2^{J} M}, i = 1, 2, \dots, 2^{J - 1} M$ be the collocation points and we denote the Legendre wavelet matrix as $ϕ_{2^{J - 1} M \times 2^{J - 1} M}$ and define it as the combination of $ϕ_{k, m}^{J} (t_{i})$ at the collocation points (t_i) as $ϕ_{2^{J - 1} M \times 2^{J - 1} M} = (\begin{matrix} ϕ_{1, 0}^{J} (t_{1}) & ϕ_{1, 0}^{J} (t_{2}) & \dots & ϕ_{1, 0}^{J} (t_{2^{J - 1} M}) \\ ϕ_{2, 0}^{J} (t_{1}) & ϕ_{2, 0}^{J} (t_{2}) & \dots & ϕ_{2, 0}^{J} (t_{2^{J - 1} M}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ ϕ_{2^{J - 1}, M - 1}^{J} (t_{1}) & ϕ_{2^{J - 1}, M - 1}^{J} (t_{2}) & \dots & ϕ_{2^{J - 1}, M - 1}^{J} (t_{2^{J - 1} M}) \end{matrix})$ For example, Legendre wavelet matrix for J = 2 and M = 2 is $ϕ_{4 \times 4} = (\begin{matrix} 1.1442 & 1.4142 & 0 & 0 \\ - 1.2247 & 1.2247 & 0 & 0 \\ 0 & 0 & 1.4142 & 1.4142 \\ 0 & 0 & - 1.2247 & 1.2247 \end{matrix})$

Description of numerical method

Here we present the wavelet collocation method based on the Legendre wavelets for solving SFIDE (1.1). We use the relation between the fractional derivative and integral to obtain the solution u(t) derived as follows

•
Let $D_{0, t}^{α} u (t) \approx C^{T} Φ (t)$ , this implies that $u (t) \approx C^{T} < ce : \inf > 0 < / ce : \inf > I_{t}^{α} Φ (t) + u_{0} .$
•
Let k₁(s, t) ∈ L²([0, 1) × [0, 1)). It can be expanded with respect to Legendre wavelet as $k_{1} (s, t) \approx Φ^{T} (s) K_{1} Φ (t) = Φ^{T} (t) K_{1}^{T} Φ (s),$
where $K_{1} = {(k_{1})}_{ij}, i = 1, 2, \dots, n, j = 1, 2, \dots, n$ is the n × n Legendre wavelets coefficient matrix

Error analysis

In this section, we investigate the convergence of the proposed method discussed in Section 4. Define $e (t) = u (t) - P_{V_{M}^{J}} u (t)$ be the projection error and $E ∥ e ∥_{L^{2} [0, 1]} = \frac{1}{N} \sum_{l = 1}^{N} ∥ e_{l} ∥_{L^{2} [0, 1]},$ where $E$ denotes mathematical expectation, N is the number of simulation and e_l denotes projection error in lth simulation. If u(t) is any deterministic function then Eq. (5.1) becomes $E ∥ e ∥_{L^{2} [0, 1]} = ∥ e ∥_{L^{2} [0, 1]} .$ Similarly for bivariate function define $e^{'} (s, t) = k (s, t) - P_{V_{M}^{J, 2}} k (s, t)$ be the projection error and $E ∥ e^{'} ∥_{L^{2} ([0, 1] \times [0, 1])}$

Numerical experiments

To illustrate the proposed method discussed in Section 4, we consider the following four examples. Let u_num(t_i, l) denotes the approximate solution of lth simulation at t_i and u_exact(t_i, l) denotes the exact solution of lth simulation at t_i. If the exact solution of SFIDE (1.1) is known then maximum absolute error and l_2,Δt error are defined as $E ∥ e ∥_{\infty} = \frac{1}{N} \sum_{l = 1}^{N} {max}_{1 \leq i \leq m} | u_{exact} (t_{i}, l) - u_{num} (t_{i}, l) |,$ $E ∥ e ∥_{2, Δ t} = \frac{1}{N} \sum_{l = 1}^{N} \sqrt{\sum_{i = 1}^{m} | u_{exact} (t_{i}, l) - u_{num} (t_{i}, l) |^{2} Δ t}$ respectively, where $Δ t = \frac{1}{n}$ and N is total number of

Applications

In this section, we present some special cases of the proposed model and their applications in real life examples. SFIDE (1.1) have many practical applications in scientific field such as physics, finance and biology etc. When α = 0, f(t) = 0, k₁(s, t) = μ and k₂(s, t) = σ, the proposed model reduces in the following form $u (t) = u_{0} + \int_{0}^{t} μ u (s) ds + \int_{0}^{t} σ u (s) dW (s),$ or, $du (t) = μ u (t) dt + σ u (t) dW (t) .$ The stochastic differential equation (7.1) is called geometric Brownian motion (GBM) model, which is used to model

Conclusion

Most of the existing numerical methods for the SFIDE (1.1) are based on orthogonal polynomials. We know that the polynomial methods do not always improve accuracy if we are going to a higher degree approximation. To overcome this problem, we develop a new and efficient method for the SFIDE, which is the Legendre wavelet collocation method. To reduce the complexity of actual computation, we use block pulse functions, which are useful and straightforward in accounting for computing operational

Authors’ contribution

Abhishek Kumar Singh: writing-original draft, investigation, formal analysis, validation, visualization. Mani Mehra: writing-review & editing, supervision, methodology, conceptualization.

Conflict of interest

None declared.

Declaration of Competing Interest

The authors report no declarations of interest.

Acknowledgements

The First author acknowledges the support provided by University Grants Commission (UGC), India. The second author acknowledges the support provided by the Department of Science and Technology, India, under the grant number SERB/F/11946/2018-2019.

Abhishek Kumar Singh received his master of technology degree from Indian Institute of Technology Madras, India, in 2017. Currently, he is a third year PhD student at the Department of Mathematics, Indian Institute of Technology Delhi, India.

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Mani Mehra earned her M.Sc. degree in Applied Mathematics from Indian Institute of Technology Roorkee, India, July 2000 and Ph.D. degree in Mathematics from the Indian Institute of Technology Kanpur, April 2005. From 2005 to 2007, she was a postdoctoral fellow at the Department of Mathematics, McMaster University, Canada. She has been with the Department of Mathematics, IIT Delhi, since 2008. Her research interest includes wavelet methods for PDEs, numerical methods.

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Wavelet collocation method based on Legendre polynomials and its application in solving the stochastic fractional integro-differential equations

Highlights

Abstract

Introduction

Section snippets

Preliminaries

Legendre wavelet matrix and block pulse operational matrix

Description of numerical method

Error analysis

Numerical experiments

Applications

Conclusion

Authors’ contribution

Conflict of interest

Declaration of Competing Interest

Acknowledgements

J. Comput. Appl. Math.

Appl. Numer. Math.

J. Math. Anal. Appl.

Appl. Math. Modell.

J. Comput. Appl. Math.

Comput. Math. Appl.

Math. Comput. Modell.

Optik

J. Comput. Appl. Math.

Eng. Anal. Boundary Elem.

Appl. Math. Modell.

Math. Comput. Simul.

Appl. Math. Comput.

Comput. Math. Appl.

Comput. Math. Appl.