An efficient approach based on radial basis functions for solving stochastic fractional differential equations

Ahmadi, Nafiseh; Vahidi, Alireza; Allahviranloo, Tofigh

doi:10.1007/s40096-017-0211-7

Cited by 19 publications

(10 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where and are stochastic MHFs coefficient vectors and and j , j = 1, 2, … , n are stochastic MHFs coefficient matrices. Substituting (9)- (12) in relation (8), we obtain Using the 6-th property in relation (13), we get Utilizing operational matrices defined in relations (5) and (6) in (14), we have…”

Section: Solving Stochastic Itô-volterra Integral Equation With Multimentioning

confidence: 99%

“…Nowadays, modelling different problems in different issues of science leads to stochastic equations [1]. These equations arise in many fields of science such as mathematics and statistics [2][3][4][5][6][7], finance [8][9][10], physics [11][12][13], mechanics [14,15], biology [16][17][18], and medicine [19,20]. Whereas most of them do not have an exact solution, the role of numerical methods and finding a reliable and accurate numerical approximation have become highlighted [21].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A computational method for solving stochastic Itô–Volterra integral equation with multi-stochastic terms

2018

Self Cite

View full text Add to dashboard Cite

In this paper, a linear combination of quadratic modified hat functions is proposed to solve stochastic Itô-Volterra integral equation with multi-stochastic terms. All known and unknown functions are expanded in terms of modified hat functions and replaced in the original equation. The operational matrices are calculated and embedded in the equation to achieve a linear system of equations which gives the expansion coefficients of the solution. Also, under some conditions the error of the method is O(h 3). The accuracy and reliability of the method are studied and compared with those of block pulse functions and generalized hat functions in some examples.

show abstract

Section: Solving Stochastic Itô-volterra Integral Equation With Multimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A computational method for solving stochastic Itô–Volterra integral equation with multi-stochastic terms

2018

Self Cite

View full text Add to dashboard Cite

show abstract

“…Numerical methods for solving a variety of stochastic integral equations, stochastic differential equations, and stochastic integro‐differential equations are presented in several papers. Heydari et al used hat basis functions for solving stochastic Itô‐Volterra integral equations, Ahmadi et al presented an efficient approach on the basis of radial basis functions for solving stochastic fractional differential equations, and Mirzaee et al applied combinatorial functions to solve a system of stochastic integral equations of fractional order . Also, Taheri et al provide the numerical solution of stochastic integro‐differential equation of fractional order via the spectral collocation method .…”

Section: Introductionmentioning

confidence: 99%

An efficient cubic B‐spline and bicubic B‐spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

Mirzaee

Alipour

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper aims to develop a novel numerical approach on the basis of B-spline collocation method to approximate the solution of one-dimensional and two-dimensional nonlinear stochastic quadratic integral equations. The proposed approach is based on the hybrid of collocation method, cubic B-spline, and bi-cubic B-spline interpolation and Itô approximation. Using this method, the problem solving turns into a nonlinear system solution of equations that is solved by a suitable numerical method. Also, the convergence analysis of this numerical approach has been discussed. In the end, examples are given to test the accuracy and the implementation of the method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient. KEYWORDS collocation method, cubic and bicubic B-spline functions, nonlinear quadratic integral equations, stochastic integral equations MSC CLASSIFICATION 60H20; 97N50; 65D30; 41A15; 65D05 wileyonlinelibrary.com/journal/mma

show abstract

“…In more situations, exact solutions of these equations are not available and therefore providing an efficient algorithm to get their approximate solutions is an essential requirement. In recent decade, some numerical methods such as operational matrix method, meshless method, and wavelet method are applied to solve stochastic integral equations. The development of an efficient and accurate numerical method to solve stochastic integral equations is still an update topic.…”

Section: Introductionmentioning

confidence: 99%

On the numerical solution of stochastic quadratic integral equations via operational matrix method

Mirzaee

Samadyar

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, stochastic operational matrix of integration based on delta functions is applied to obtain the numerical solution of linear and nonlinear stochastic quadratic integral equations (SQIEs) that appear in modelling of many real problems. An important advantage of this method is that it dose not need any integration to compute the constant coefficients. Also, this method can be utilized to solve both linear and nonlinear problems. By using stochastic operational matrix of integration together collocation points, solving linear and nonlinear SQIEs converts to solve a nonlinear system of algebraic equations, which can be solved by using Newton's numerical method. Moreover, the error analysis is established by using some theorems. Also, it is proved that the rate of convergence of the suggested method is O(h 2 ). Finally, this method is applied to solve some illustrative examples including linear and nonlinear SQIEs. Numerical experiments confirm the good accuracy and efficiency of the proposed method. KEYWORDSBrownian motion process, error analysis, operational matrix, quadratic integral equations, stochastic integral equations | INTRODUCTIONThe various kinds of integral equations are important mathematical tools for describing knowledge models that appear in different areas of applied science. Because of extensive application of integral equations and not having the exact solutions in many cases, numerical solution of integral equations has attracted researcher's attention to develop numerical method for approximating solution of these equations. Among these methods, we refer to wavelet method, 1-4 homotopy perturbation method, 5-8 collocation method, 9,10 and meshless method. 11-13 A novel algorithm to get approximate solution of these equations is to express the solution as linear combination of orthogonal or nonorthogonal basis functions and polynomials such as block-pulse functions, 14,15 hat functions, 16,17 Bernoulli polynomials, 18 Legendre polynomials, 19 Bessel polynomials, 20 Chebyshev polynomials, 21 Fibonacci polynomials, 22 and orthonormal Bernstein polynomials. 23 Mathematical modeling of various phenomena in real world is leaded to a special kind of integral equations named quadratic integral equations. Quadratic integral equations always arise in many problems of mathematical physics and chemical such as theory of radiative transfer, the kinetic theory of gases, the theory of neutron transport, the queuing theory, and the traffic theory and many other applications. Existence solution and numerical method to solve these type of integral equations have been studied in previous papers. [24][25][26][27][28] Recently, with increasing computational power, it becomes feasible to utilize more accurate equations to model some problems that are often dependent on a noise source. So modeling these problems naturally require the use of

show abstract

An efficient approach based on radial basis functions for solving stochastic fractional differential equations

Cited by 19 publications

References 31 publications

A computational method for solving stochastic Itô–Volterra integral equation with multi-stochastic terms

A computational method for solving stochastic Itô–Volterra integral equation with multi-stochastic terms

An efficient cubic B‐spline and bicubic B‐spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

On the numerical solution of stochastic quadratic integral equations via operational matrix method

Contact Info

Product

Resources

About