Abstract
The main purpose of this paper is to develop a new method based on operational matrices of the linear cardinal B-spline (LCB-S) functions to numerically solve of the fractional stochastic integro-differential (FSI-D) equations. To reach this aim, LCB-S functions are introduced and their properties are considered, briefly. Then, the operational matrices based on LCB-S functions are constructed, for the first time, including the fractional Riemann-Liouville integral operational matrix, the stochastic integral operational matrix, and the integer integral operational matrix. The main characteristic of the new scheme is to convert the FSI-D equation into a linear system of algebraic equations which can be easily solved by applying a suitable method. Also, the convergence analysis and error estimate of the proposed method are studied and an upper bound of error is obtained. Numerical experiments are provided to show the potential and efficiency of the new method. Finally, some numerical results, for various values of perturbation in the parameters of the main problem are presented which can indicate the stability of the suggested method.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The most physical phenomena can be modeled as different stochastic equations including stochastic integral equations or stochastic integro-differential equations.
An integral equation is called a stochastic integral equation if it contains at least one term as a stochastic process, accordingly the solution of this equation is a stochastic process. In most cases, such equations can not be solved explicitly. So, the different numerical methods and different basis functions have been presented for solving of them such as cubic B-spline and bicubic B-spline collocation method [19], hybrid of block-pulse and parabolic functions [22], Bernoulli polynomials and Bernoulli wavelet metrhods [23, 36], delta basis functions [29], quintic B-spline collocation method [21], Euler polynomial [27], wavelet-based computational method [30] and the other methods [5,6,7, 12, 13, 16, 18, 26, 31, 34].
Fractional differential equations (FDEs) arise in the various sciences including finance, physics, signal processing and control theory [3, 8,9,10,11, 32, 33]. Sometimes, these equations are combined with the stochastic integral equations. A stochastic integro–differential equation is called a FSI-D equation if the order of derivative is non–integer. Due to non–existing of the analytical solution for these type equations, it is clear that obtaining the numerical solutions of them can be interesting issue for the researchers.
The basis functions are one of the most common methods used to solve such equations [1, 20, 24, 25, 28]. When the basis functions are applied, the solution of the considering problem is approximated as the linear combination of the basis functions with unknown coefficients. The main characteristic of such methods is to convert the differential or integral equation to a algebraic system of equations. It not only simplified the problem but also speeds up the computation.
In this work, a new scheme based on the LCB-S basis functions is introduced for solving FSI-D equation as
wherein \(^{C}_{0}D_{\tau }^{\upsilon }\) is the Caputo fractional derivative of order \(0<\upsilon <1\). Also, \(U(t)\in C^2([0,1])\),\( \ \lambda _i(\tau ,s)\in C^2([0,1]\times [0,1])\) for \(i=1, 2\) are stochastic processes defined on the probability space \((\Omega , F, P)\) and \({\mathcal {B}}(\tau )\) is Brownian motion process. In Eq. (1.1), \(Y(\tau )\) is an unknown function that must be determined. Lakestani et al. [15] have applied the LCB-S functions for solving the fractional differential equations. They have constructed the operational matrix of derivative and fractional derivative. Here, the LCB-S functions are developed for solving the FSI-D Eq. (1.1). For this purpose, the operational matrices based on LCB-S functions are constructed, for the first time, including the fractional Riemann-Liouville integral operational matrix, the stochastic integral operational matrix, and the integer integral operational matrix. The mentioned matrices are all upper triangular which can be calculated simply. It is worth noting that the LCB-S functions have the interpolation properties. Therefore, the coefficients of each known function can be easily calculated without using any integration and this is a major advantage of the LCB-S functions respect to the other basis functions [4]. Moreover, these functions have cardinality properties and using this characteristic, the cost of calculation is decreased(see cpu times of proposed method in the tables of numerical verifications section).
The organization of this work is as follows: Some definitions, notations, the formulation of LCB-S functions on [0, 1] and the derivative operational matrix are expressed in Sect. 2. The operational matrices of fractional Riemann-Liouville, stochastic and integer integrals are constructed in Sect. 3. The proposed method is used to the approximate solution of the FSI-D equation in Sect. 4. Convergence analysis and error estimate of the proposed method are studied in Sect. 5. In Sect. 6, numerical experiments are reported. Also in this section, some numerical results, for various values of perturbation in the parameters of the main problem are presented which can indicate the stability of the suggested method. A brief conclusion is presented in Sect. 7.
2 Some definitions and notations
Definition 2.1
[14] For the given function \(Z(\tau )\)
are called the Riemann-Liouville integral, the Riemann-Liouville derivative and the Caputo derivative of order \(\upsilon \), respectively.
Definition 2.2
[14] If \(Z\in C^{q}[\tau _0,\tau ]\) then
wherein \(\Phi (\tau )=\displaystyle {\sum _{j=0}^{q-1}}\frac{Z^{(j)}(\tau _0)}{\Gamma (j+1)}(\tau -\tau _0)^j\).
2.1 LCB-S functions on [0, 1]
In this part, a brief definition of the cardinal B-spline (CB-S) functions is expressed. In the sequel, the operational matrix of derivative based on the LCB-S functions is presented.
Definition 2.3
[17] Let the CB-S function of first order, represented by \({\mathcal {N}}_1(\tau )={\mathcal {X}}_{[0,1)}(\tau )\), wherein \({\mathcal {X}}_{[0,1)}(\tau )\) is a characteristic function on the interval [0, 1). Also, The CB-S function of order r, defined as
recursively, and supp\([{\mathcal {N}}_{r}(\tau )]=[0,r]\).
In explicit form, the LCB-S function of 2 order is presented as
Let \({\mathcal {N}}_{r,q}(\tau )={\mathcal {N}}_{2}(2^r\tau -q),\ \ r,q\in {\mathbb {Z}}\). It can easily be shown that
Define \({\mathcal {S}}_{r}=\{q: [2^{-r}q,2^{-r}(2+q)]\cap (0, 1)\ne \emptyset \}, \,\,\,\, r\in {\mathbb {Z}}.\) It can be derived that \(min\{{\mathcal {S}}_{r}\}=-1 \) and \(max \{{\mathcal {S}}_{r}\}=2^r-1, r\in {\mathbb {Z}}.\)
Since it is required that the support of \({\mathcal {N}}_{r,q}(\tau )\) is restricted on [0, 1], so put
As a result, one can write
2.2 Vector form and function approximation
Suppose
where \(r\in {\mathbb {Z}} \) is a fixed number. A function \(U(\tau )\in L^2[0,1]\) is approximated by the LCB-S functions as
where
Let \(e_q\) be the \({(q+2)}\)-th column of unit matrix of order \(2^r+1\), so it is easy to verify that
Also, a function \(\lambda (\tau ,s)\in L^2([0,1]\times [0,1]) \) may be approximated by LCB-S functions as
where \(\varvec{\Lambda }\) is the \((2^r+1)\times (2^r+1)\) coefficient matrix with entries \(\varvec{\Lambda }_{q,p}\) as follows
2.3 Operational matrix of the derivative
Suppose
The differentiation of \(\varvec{\Psi }(\tau )\) in (2.7) is calculated as
where \({\mathbf {D}}\) recalls \((2^r+1)\times (2^r+1)\) operational matrix of derivative for the LB-S functions on [0, 1] (for more details see [15]).
3 Operational matrices of the fractional integral, stochastic integral and integer integral
In this section, the fractional integral, stochastic integral and integer integral of the LCB-S functions on [0, 1] are constructed.
3.1 Operational matrix of the fractional integral
Let \(\upsilon \in (0,1)\) then applying Definition 2.1, one can write
To calculate the above integral, the following three cases are considered:
Case 1: \(q=-1\).
In this case, if \(\tau \in [0,2^{-r}]\) then Eq. (3.1) can be calculated as follows
where above integral can be solved using integration by parts. Therefore, one can write
If \(\tau >2^{-r}\) then Eq. (3.1) can calculated as
Case 2: \(q=0,\ldots ,2^r-2\).
In this case, if \(\tau \in [0, 2^{-r}q]\) then it can be proven \(_{0}{\mathcal {J}}_{\tau }^{\upsilon }\psi _{q}(\tau )=0\).
If \(\tau \in [2^{-r}q,2^{-r}(q+1)]\) then
If \(\tau \in [2^{-r}(q+1),2^{-r}(q+2)]\) then
If \(\tau >2^{-r}(q+2)\) then
Case 3: \(q=2^r-1\).
In this case, if \(\tau \in [0, 2^{-r}q]\) then it can be derived \(_{0}{\mathcal {J}}_{\tau }^{\upsilon }\psi _{2^r-1}(\tau )=0\).
If \(\tau \in [1-2^{-r},1]\) then
If \(\tau >1\) then
Using Eqs. (3.2)–(3.8), one can write
Applying Eqs. (2.7)–(2.8) on (3.9), it can be obtained
wherein
It can be proven that \({ \mathbf{J}}_{\upsilon }\) is a \((2^r+1)\times (2^r+1)\) matrix as follows
where
and \(\lambda =\frac{1}{2^{r\upsilon }\Gamma (\upsilon +2)}\).
3.2 Operational matrix of the stochastic integral
Suppose
To calculate the above integral, the following three cases are considered:
Case 1: \(q=-1\).
In this case, if \(\tau \in [0,2^{-r}]\) then Eq. (3.14) is calculated as
where the above integral can be solved using integration by parts. Therefore, it can be written
In Eq. (3.16), the integral can be approximated by Simpson’s rule. So it can be obtained that
If \(\tau >2^{-r}\) then Eq. (3.17) can be written as
Applying Simpson’s rule on the above integral, one can derive that
Case 2: \(q=0,\ldots ,2^r-2\).
In this case, if \(\tau \in [0, 2^{-r}q]\) then it can be proven \({\mathcal {I}}^{s}\psi _{q}(\tau )=0\).
If \(\tau \in [2^{-r}q,2^{-r}(q+1)]\) then using integral by parts and Simpson’s rule, it is obtained that
If \(\tau \in [2^{-r}(q+1),2^{-r}(q+2)]\) then
If \(\tau >2^{-r}(q+2)\) then
Case 3: \(q=2^r-1\).
In this case, if \(\tau \in [0, 2^{-r}q]\) then it can be derived \({\mathcal {I}}^{s}\psi _{2^r-1}(\tau )=0\).
If \(\tau \in [1-2^{-r},1]\) then
If \(\tau >1\) then
Using Eqs. (3.15)–(3.24), one can write
Applying Eqs. (2.7)–(2.8) on (3.25), it can be achieved as
wherein
It can be demonstrated that \({ \mathbf{I}}_{s}\) is a \((2^r+1)\times (2^r+1)\) matrix as follows
where
3.3 Operational matrix of the integer integral
Define
To calculate the above integral, the following three cases are investigated:
Case 1: \(q=-1\).
In this case, if \(\tau \in [0,2^{-r}]\) then Eq. (3.1) is calculated as follows
If \(\tau >2^{-r}\) then Eq. (3.32) can be obtained as
Case 2: \(q=0,\ldots ,2^r-2\).
In this case, if \(\tau \in [0, 2^{-r}q]\) then it can be proven \({\mathcal {I}}\psi _{q}(\tau )=0\).
If \(\tau \in [2^{-r}q,2^{-r}(q+1)]\) then
If \(\tau \in [2^{-r}(q+1),2^{-r}(q+2)]\) then
If \(\tau >2^{-r}(q+2)\) then
Case 3: \(q=2^r-1\).
In this case, if \(\tau \in [0, 2^{-r}q]\) then it can be derived \({\mathcal {I}}\psi _{2^r-1}(\tau )=0\).
If \(\tau \in [1-2^{-r},1]\) then
If \(\tau >2^{-r}(q+1)\) then
Using Eqs. (3.31)–(3.37), one can write
Using Eqs. (2.7)–(2.8) on (3.38), it can be obtained
wherein
It can be proven that \({\mathbf {I}}\) is a \((2^r+1)\times (2^r+1)\) matrix as follows
where \(\displaystyle \varrho =2^{-r-1}\).
4 Numerical solution of FSI-D equation
In this section, a numerical scheme is presented for solving FSI-D Eq. (1.1).
From Definition 2.1, one can derive
Therefore Eq. (1.1) is written as follows
Now functions \(Y(\tau ),\ U(\tau ),\ \lambda _1(\tau ,s)\) and \(\lambda _2(\tau ,s)\) can be approximated by using LCB-S functions as the following form
wherein vector u from Eq. (2.9) and matrices \(\lambda _1, \lambda _2\) from Eq. (2.12) can be obtained. Moreover, y is an unknown vector that must be found.
Considering Eq. (4.2) and using the presented operational matrices in the previous section, one can write Eq. (4.2) as follows
Also, using the initial condition of problem (1.1), one can write
Collocating Eq. (4.4) at the point \(\xi _j=(j+1)2^{-r}, \ \ j=0,\ldots ,2^r-1\) and using Eq. (2.10) it can be obtained as
Moreover from Eq. (4.5) it can be derived
Now, from Eq. (4.6) and Eq. (4.7), a linear system of equations is obtained which is calculated to find the unknown function \(Y(\tau )\) in Eq. (4.3).
5 Convergence and error analysis
In this section, the error analysis of the proposed method is studied.
Theorem 5.1
Suppose \(U(t)\in C^2[0,1]\) and \({\tilde{U}}(t)=\displaystyle \sum _{q=-1}^{2^r-1}U(2^{-r}(q +1))\psi _{q}(t)\) be the LCB-S approximation of U(t), then the upper bound of error for the LCB-S method is obtained as
eventually, it is derived that
Proof
Let
and define
Then it can be written that
It is clear that when \(\frac{1}{2^r}\rightarrow 0\), one can write
By replacing the Taylor expansion of U(t) in terms of \((t-\frac{q}{2^r})\) in Eq. (5.4), one can derive that
wherein \(\zeta _q\in (\frac{q}{2^r},\frac{q+1}{2^r})\). Since \(|t-\frac{q}{2^r}|<\frac{1}{2^r}\), then it is concluded that
Thus
Also from (5.5), it can be derived
\(\square \)
Theorem 5.2
Suppose \(\lambda (t,s)\in C^2([0,1]\times [0,1])\) and
be the LCB-S aproximation of \(\lambda (t,s)\), then
finally, it is derived that
Proof
Let
It can be defined
Then one can write that
In Eq. (5.9) after using two-dimensional Taylor expansion of \(\lambda (t,s)\) in terms of \((t-\frac{q}{2^r})\) and \((s-\frac{p}{2^r})\), when \(\frac{1}{2^r}\rightarrow 0\), as a result, it can be written that
wherein \(\zeta _q\in (\frac{q}{2^r},\frac{q+1}{2^r})\) and \(\eta _p\in (\frac{p}{2^r},\frac{p+1}{2^r})\). Since \(|t-\frac{q}{2^r}|, |t-\frac{p}{2^r}|<\frac{1}{2^r}\), then it can be concluded that
Therefore
As a result
Also from Eq. (5.10), it can be derived
\(\square \)
Theorem 5.3
Suppose \(Y(\tau )\) and \({\tilde{Y}}(\tau )\) are the analytical solution and the numerical solution of Eq. (1.1), respectively. Also let
-
(a)
\(||Y(t)||\le \vartheta ,\)
-
(b)
\(||\lambda _j(\tau ,s)||\le H_j,\ \ j=1,2,\)
-
(c)
\(H_1+S_1+\vartheta S_2||B(t)|| <1,\)
then
wherein
\(\Upsilon =\displaystyle \frac{1}{2\times 2^{2r}}||U''(\tau )||\)
\(S_j=\displaystyle \frac{1}{2\times 2^{2r}}\Big (\Big |\Big |\frac{\partial ^2\lambda _j(\tau ,s)}{\partial \tau ^2}\Big |\Big |+2\Big |\Big |\frac{\partial ^2\lambda _j(\tau ,s)}{\partial \tau \partial s} \Big |\Big |+\Big |\Big |\frac{\partial ^2\lambda _j(\tau ,s)}{\partial s^2}\Big |\Big |\Big ),\ \ j=1,2.\)
Proof
Consider Eq. (1.1) in special case \(\upsilon =0\). It is clear that
consequently
Now, using Theorems (5.1)–(5.2) and assumptions (a) and (b), one can get
Replacing Eq. (5.12) in Eq. (5.11), one can obtain
Using assumption (c) in Eq. (5.13), it can be derived as
\(\square \)
6 Numerical verifications
In this section first, two numerical examples are reported in order to indicate the effectiveness of the proposed method. It is worth noting that Example 1 has the exact solution in a particular case, so the main focus is on this example to show the potential and efficiency of the proposed method. Next, numerical simulations and results of perturbation on the main problem are reported. In the presented tables the values of maximum absolute error and RMS-error are computed as
wherein \(Y(\tau )\) and \(Y_N(\tau )\) represent the exact solution and the numerical solution, respectively. Also, in these tables N indicates the number of the interpolation points used to approximate the solution function and it is as \(N=2^r+1\) for proposed method. The numerical experiments are performed on a computer Intel(R) Pentium(R) CPU G2030 @ 3.00GHz 3.00GHz 4.00GB RAM by running some codes written in Maple 18 software.
6.1 Numerical example
Example 1. Consider the FSI-D equation as
with the initial condition \(Y(0)=0\). Eq. (6.1) does not have an analytical solution. However, in the case \(\sigma =0\) the function \(Y(t)=t^3\) is the analytical solution for problem (6.1). This example has been used in [20, 25].
In order to compare the numerical solution with the exact solution of Eq. (6.1), suppose \(\sigma =0\). In Table 1, the results of this experiment are listed for \(r=5,\ \upsilon =0.35, 0.65, 0.95\). According to Table 1, it is clear that the proposed method has the considerable accuracy. In Table 2, the new scheme compared with the two different methods based on radial basis functions in [28]. The results show that the run times for the proposed method are lower than the presented methods in [28]. Furthermore, in last column of Table 2, the values of \(\frac{1}{2^{2r}}\) are listed for different values of r. Comparing this column with the column \(||e||_\infty \), one can result that \(||e||_\infty <\frac{1}{2^{2r}}\). So, the numerical results in this table confirm the presented theoretical in the previous section. Therefore, the new scheme can be used as the practical and efficient method for solving the fractional stochastic integro-differential equation. In Table 3 the LCB-S method is compared with the Bernstein [25] and the Cubic B-spline methods [20]. As a result of this table, it can be derived that the Bernstein and the Cubic B-spline methods have only good accuracy near the initial condition \(Y(0)=0\). In other words, the accuracy of these methods is greatly reduced by moving away from \(\tau =0\) and approaching to \(\tau =1\). Unlike these two methods, the proposed method has good accuracy over the whole interval [0, 1]. In Table 4, the numerical solutions are reported for case \(\sigma =1\) with \(r=4\) and various values of \(\upsilon \). In Fig. 1, the exact solution (in case \(\sigma =0)\) is compared with the numerical solutions for case \(r=4, \upsilon =0.85\) and \(\sigma =0,0.35,0.70,1\).
Example 2. Consider FSI-D equation as follows [20, 25, 28]
with the initial condition \(Y(0)=0\). Equation (6.2) does not have any analytical solution. Therefore, a numerical solution as an efficient approximation would be useful and interesting. Here, the numerical solutions for case \(r=3\) and various values of \(\upsilon \) together with the numerical solutions of [20, 25] are listed in Table 5. Also, in Table 6, the numerical solutions of proposed method are compared with the presented methods in [28] at \(\tau =0.9\). Furthermore, in this table run times for the mention methods are listed. According to the Table 6, it is clear that the computational speed of the proposed method is higher than the compared methods.
6.2 Perturbation on the main problem
In this part, firstly, the initial condition is prturbed as following form.
Consider the FSI-D equation of Example 1 as
Some numerical results, for different values of perturbation in the initial condition are presented in Table 7. This table shows that the new scheme preserves the stability of the initial value problem, with respect to small perturbations in the initial condition.
Secondly, the function U(t) of the main problem is perturbed as following form.
Consider the FSI-D equation of Example 1 as
The numerical solutions, for different values of perturbation of the function U(t) are listed in Table 8.
Finally, the function \(\lambda _2(\tau ,s)\) of the main problem is perturbed as following form.
Consider the FSI-D equation of Example 1 as
The numerical results, for different values of perturbation of the function \(\lambda _2(\tau ,s)\) are reported in Table 9. From Tables 7, 8 and 9, it is clear that a small error in the input data leads to a small error in the solution of the problem, and this can confirm the stability of the proposed method.
7 Conclusion
In this study, a numerical scheme for solving the stochastic integro-differential equation of fractional order was developed. For this purpose, linear cardinal B-spline functions were applied. Also, the convergence and error analysis of the proposed method is studied. The reported results indicated that the problem can be solved effectively by the proposed approach.
References
Asgari, M.: Block pulse approximation of fractional Stochastic integro-differential equation. Commun. Numer. Anal. 2014, 1–7 (2014)
Alipour, S., Mirzaee, F.: An iterative algorithm for solving two dimensional nonlinear stochastic integral equations: a combined successive approximations method with bilinear spline interpolation. Appl. Math. Comput. 371, 124947 (2020)
Atanackovic, T.M., Stankovic, B.: On a system of differential equations with fractional derivatives arising in rod theory. J. Phys. A-Math. Gen. 37(4), 1241–1250 (2004)
De Boor, C.A.: Practical Guide to Spline. Springer, New York (1978)
Cioica, P.A., Dahlke, S.: Spatial Besov regularity for semilinear stochastic partial differential equations on bounded Lipschitz domains. Int. J. Comput. Math. 89(18), 2443–2459 (2012)
Dimov, I., Venelin, T.: Error Analysis of Biased stochastic algorithms for the second kind Fredholm integral equation. In: Innovative Approaches and Solutions in Advanced Intelligent Systems, pp. 3–16. Springer, Berlin (2016)
Diop, M., Caraballo, T.: Asymptotic stability of neutral stochastic functional integro-dierential equations with impulses. Electron. Commun. Probab. 20, 1 (2015)
Evans, R.M., Katugampola, U.N., Edwards, D.A.: Applications of fractional calculus in solving Abel-type integral equations: Surface-volume reaction problem. Comput. Math. Appl. 73, 1346 (2017)
He, J.H.: Nonlinear oscillation with fractional derivative and its applications. Int. Conf. Vibrat. Eng. 98, 288–291 (1998)
He, J.H.: Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 15(2), 86–90 (1999)
Hilfer, R.: Applications of fractional calculus in physics. World Scientific, Singapore (2000)
Ichiba, T., Karatzas, I., Prokaj, V., Yan, M.: Stochastic integral equations for Walsh semi martingales. arXiv:1505.02504 (2015)
Jankovic, S., Ilic, D.: One linear analytic approximation for stochastic integro-differential equations. Acta Mathematica Scientia 308(4), 1073–1085 (2010)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, San Diego (2006)
Lakestani, M., Dehghan, M., Irandoust-pakchin, S.: The construction of operational matrix of fractional derivatives using B-spline functions. Commun. Nonlinear Sci. Numer. Simulat. 17, 1149–1162 (2012)
Ma, X., Huang, Ch.: Numerical solution of fractional integro-differential equations by a hybrid collocation method. Appl. Math. Comput. 219, 6750–6760 (2013)
Milovanovic, G.V., Udovicic, Z.: Calculation of coefficients of a cardinal B-spline. Appl. Math. Lett. 23, 1346–1350 (2010)
Mirzaee, F., Hamzeh, A.: A computational method for solving nonlinear stochastic Volterra integral equations. J. Comput. Appl. Math. 306, 166–178 (2016)
Mirzaee, F., Alipour, S.: An efficient cubic B-spline and bicubic B-spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations. Math. Methods Appl. Sci. 43(1), 384–397 (2019)
Mirzaee, F., Alipour, S.: Cubic B-spline approximation for linear stochastic integro-differential equation of fractional order. J. Comput. Appl. Math. 366, 112440 (2020)
Mirzaee, F., Alipour, S.: Quintic B-spline collocation method to solve n-dimensional stochastic Ito-Volterra integral equations. J. Comput. Appl. Math. 384, 113153 (2021)
Mirzaee, F., Alipour, S., Samadyar, N.: Numerical solution based on hybrid of block-pulse and parabolic func- tions for solving a system of nonlinear stochastic It-Volterra integral equations of fractional order. J. Comput. Appl. Math. 349, 157–171 (2019)
Mirzaee, F., Samadyar, N.: Application of Bernoulli wavelet method for estimating a solution of linear stochastic It-Volterra integral equations. Multidiscip. Model. Mater. Struct. 15(3), 575–598 (2019)
Mirzaee, F., Samadyar, N.: Application of hat basis functions for solving two-dimensional stochastic fractional integral equations. Comput. Appl. Math. 37(4), 4899–4916 (2018)
Mirzaee, F., Samadyar, N.: Application of orthonormal Bernstein polynomials to construct a efficient scheme for solving fractional stochastic integro-differential equation. Opt. Int. J. Light Electron. Opt. 132, 262–273 (2017)
Mirzaee, F., Samadyar, N.: Implicit meshless method to solve 2D fractional stochastic Tricomi- type equation defined on irregular domain occurring in fractal transonic flow. Numer. Methods Part. Differ. Equ. (2020)
Mirzaee, F., Samadyar, N.: Euler polynomial solutions of nonlinear stochastic It-Volterra integral equations. J. Comput. Appl. Math. 330, 574–585 (2018)
Mirzaee, F., Samadyar, N.: On the numerical solution of fractional stochastic integro-differential equations via meshless discrete collocation method based on radial basis functions. Eng. Anal. Bound. Elem. 100, 246–255 (2019)
Mirzaee, F., Samadyar, N.: On the numerical solution of stochastic quadratic integral equations via operational matrix method. Math. Methods Appl. Sci. 41(12), 4465–4479 (2018)
Mohammadi, F.: A wavelet-based computational method for solving stochastic It\({\hat{o}}\)-Volterra integral equations. J. Comput. Phys. 298, 254–265 (2015)
Mohammadi, F.: A wavelet Galerkin method for solving stochastic fractional differential equations. J. Fract. Calc. Appl. 7(1), 73–86 (2016)
Nadeem, M., Dabas, J.: Existence of solution for fractional integro-differential equation with impulsive effect. In: Mathematical Analysis and its Applications, pp. 373–380. Springer India (2015)
Panda, R., Dash, M.: Fractional generalized splines and signal processing. Signal Process 86, 2340–2350 (2006)
Picchini, U., Julie, F.: Stochastic differential equation mixed effects models for tumor growth and response to treatment (2016)
Saha, S., Singh, S.: Numerical solution of nonlinear stochastic It-Volterra integral equations driven by fractional Brownian motion. Math. Methods Appl. Sci. 41(4), 1410–1423 (2020)
Samadyar, N., Mirzaee, F.: Orthonormal Bernoulli polynomials collocation approach for solving stochastic It-Volterra integral equations of Abel type. Int. J. Numer. Model. Electron. Networks Devices Fields 33(1), e2688 (2020)
Acknowledgements
The work of first author was supported by the University of Tabriz, Iran under Grant No. 6898.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Abdi-Mazraeh, S., Kheiri, H. & Irandoust-Pakchin, S. Construction of operational matrices based on linear cardinal B-spline functions for solving fractional stochastic integro-differential equation. J. Appl. Math. Comput. 68, 151–175 (2022). https://doi.org/10.1007/s12190-021-01519-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-021-01519-8