Wavelet collocation method based on Legendre polynomials and its application in solving the stochastic fractional integro-differential equations
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 formwhere 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 integral. Presence of the It 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 followsSimilar to integer order integration, the left Riemann–Liouville fractional integral operator is a linear operatorwhere λ and μ are constants. Definition 2.2
Legendre wavelet matrix and block pulse operational matrix
Let be the collocation points and we denote the Legendre wavelet matrix as and define it as the combination of at the collocation points (ti) asFor example, Legendre wavelet matrix for J = 2 and M = 2 is
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 , this implies that
- •
Let k1(s, t) ∈ L2([0, 1) × [0, 1)). It can be expanded with respect to Legendre wavelet as
where 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 be the projection error andwhere denotes mathematical expectation, N is the number of simulation and el denotes projection error in lth simulation. If u(t) is any deterministic function then Eq. (5.1) becomesSimilarly for bivariate function define be the projection error and
Numerical experiments
To illustrate the proposed method discussed in Section 4, we consider the following four examples. Let unum(ti, l) denotes the approximate solution of lth simulation at ti and uexact(ti, l) denotes the exact solution of lth simulation at ti. If the exact solution of SFIDE (1.1) is known then maximum absolute error and l2,Δt error are defined asrespectively, where 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 formor,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.
References (30)
- et al.
Fourth order compact scheme for space fractional advection-diffusion reaction equations with variable coefficients
J. Comput. Appl. Math.
(2020) - et al.
A difference scheme for the time-fractional diffusion equation on a metric star graph
Appl. Numer. Math.
(2020) - et al.
Existence and uniqueness results for a nonlinear caputo fractional boundary value problem on a star graph
J. Math. Anal. Appl.
(2019) - et al.
A bessel collocation method for solving fractional optimal control problems
Appl. Math. Modell.
(2015) - et al.
Multistep schemes for one and two dimensional electromagnetic wave models based on fractional derivative approximation
J. Comput. Appl. Math.
(2020) - et al.
Anomalous diffusion modeling by fractal and fractional derivatives
Comput. Math. Appl.
(2010) - et al.
Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions
Math. Comput. Modell.
(2012) - et al.
Application of orthonormal bernstein polynomials to construct an efficient scheme for solving fractional stochastic integro-differential equation
Optik
(2017) - et al.
Numerical solution of stochastic fractional integro-differential equation by the spectral collocation method
J. Comput. Appl. Math.
(2017) - et al.
On the numerical solution of fractional stochastic integro-differential equations via meshless discrete collocation method based on radial basis functions
Eng. Anal. Boundary Elem.
(2019)
Wavelets collocation methods for the numerical solution of elliptic bv problems
Appl. Math. Modell.
Collocation methods based on Gegenbauer and Bernoulli wavelets for solving neutral delay differential equations
Math. Comput. Simul.
Fast wavelet-Taylor Galerkin method for linear and non-linear wave problems
Appl. Math. Comput.
The legendre wavelet method for solving initial value problems of Bratu-type
Comput. Math. Appl.
Convergence and condition number of multi-projection operators by Legendre wavelets
Comput. Math. Appl.
Cited by (32)
Application of modified Fourier's law in a fuzzy environment to explore the tangent hyperbolic fluid flow over a non-flat stretched sheet using the LWCM approach
2024, International Communications in Heat and Mass TransferSolving nonlinear stochastic differential equations via fourth-degree hat functions
2023, Results in Control and OptimizationA wavelet collocation method based on Gegenbauer scaling function for solving fourth-order time-fractional integro-differential equations with a weakly singular kernel
2023, Applied Numerical MathematicsCitation Excerpt :Recently, polynomial-based wavelet methods have been widely used to investigate differential equations numerically. In particular, Gegenabuer and Bernoulli wavelet [6], Jacobi wavelet [5], Hermite wavelet [15,17], Legendre wavelet [39], Chebyshev wavelet [7], and so on. We have shown the procedure of finding the approximate solution of Eq. (1.1) in Algorithm 1.
Operational matrix method for solving fractional weakly singular 2D partial Volterra integral equations
2023, Journal of Computational and Applied MathematicsCitation Excerpt :Moreover, in [12] the authors have developed two-dimensional BPFs to find the solution of the nonlinear mixed Volterra–Fredholm IDEs. The application of the Wavelet collocation approach to solve stochastic fractional integration differential equations has been investigated by authors of [13]. The authors of [14] used Legendre polynomials to solve 2D Volterra integral equations.
Data-driven fault-tolerant control with fuzzy-rules equivalent model for a class of unknown discrete-time MIMO systems and complex coupling
2022, Journal of Computational ScienceCitation Excerpt :Unfortunately, it is complicated to verify this condition in practice, especially for a class of unknown discrete-time systems [17]. For systems with high complexity, fractional-order dynamics have been applied to model many phenomena of controlled plants according to the combination of integration and differentiation of arbitrary order [18,19]. Furthermore, fractional-order systems have been successfully utilized for control engineering applications such that optimal control [20], fuzzy robust controller [21,22], PID control [23], sliding mode control [24] and so on when controlled plants have been considered as a class of single input and single output systems (SISO) with partial known dynamics.
A Tikhonov regularization method for solving a backward time–space fractional diffusion problem
2022, Journal of Computational and Applied MathematicsCitation Excerpt :A tutorial on inverse problems for anomalous diffusion processes can be found in [14]. Recently, some numerical methods are proposed to deal with fractional ordinary differential equations, such as [15–19]. A pair of the direct and the inverse problems are as follows.
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.
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.