A new scheme for solving nonlinear Stratonovich Volterra integral equations via Bernoulli’s approximation

Int J Numerical Modelling

2019

Self Cite

In this paper, an efficient matrix method based on 2D orthonormal Bernoulli polynomials are developed to obtain numerical solution of weakly singular fractional partial integro‐differential equations (FPIDEs). Operational matrix of integration, almost operational matrix of integration, and product operational matrix are employed to transform the solution of Volterra singular FPIDEs to the system of linear or nonlinear algebraic equations. The obtained algebraic system can be easily solved by using an appropriate numerical method. Also, some useful theorems concerning the convergence and error analysis associated to the mentioned approach are proven in the current paper. Finally, some numerical examples are included to illustrate the accuracy, efficiency, and applicability of the proposed approach.

“…Definition The Bernoulli basis polynomials of order n are defined as follows:

true \sum_{i = 0}^{n} ((), \binom{n + 1}{i}) B_{i} false(x false) = false(n + 1 false) x^{n}, n = 0, 1, 2, ⋯,

where the binomial coefficients are given by

((), \binom{n + 1}{i}) = \frac{false(n + 1 false)!}{i! false(n + 1 - i false)!} .

…”

Section: Preliminaries and Initial Definitionsmentioning

confidence: 99%

Numerical scheme for solving singular fractional partial integro‐differential equation via orthonormal Bernoulli polynomials

Int J Numerical Modelling

2019

Self Cite

“…Heydari et al utilized wavelets Galerkin method for solving stochastic heat equation [13]. Mirzaee and Samadyar applied operational matrix method based on different bases to obtain the numerical solution of Itô stochastic integral equations [14][15][16], or Stratonovich integral equations [17][18][19]. Dehghan and Shirzadi utilized meshless simulation based on radial basis functions (RBFs) to get the approximate solution of stochastic advection-diffusion equations [20].…”

Section: Introductionmentioning

confidence: 99%

Implicit meshless method to solve 2D fractional stochastic Tricomi‐type equation defined on irregular domain occurring in fractal transonic flow

Numerical Methods Partial

2020

Self Cite

In the current paper, we develop a new method based on the finite difference formulation and meshless method to solve 2D time fractional stochastic Tricomi‐type equation with Caputo derivative in temporal direction and Neumann boundary condition defined on irregular domain, numerically. In this method, first the fractional order derivative is discretized by a finite difference scheme and then meshless method based on radial basis functions is employed to approximate the functions in the spatial directions. So, the solution of considered problem is transformed to the solution of a linear system of algebraic equations which will be solved at each time step. The most advantage of this method is that it has no triangular, quadrangular, or other type of meshes and therefore is very applicable to solve various high dimensional problems. Finally, some test problems with different domains have been solved via present method to verify the accuracy and efficiency of the newly developed meshless formulation. Numerical results confirm the present technique is very effective to approximate the solution of 2D fractional stochastic Tricomi‐type equation with Neumann boundary condition.

“…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

Math Methods in App Sciences

2018

Self 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