An iterative algorithm for solving two dimensional nonlinear stochastic integral equations: A combined successive approximations method with bilinear spline interpolation

Alipour, Sahar; Mirzaee, Farshid

doi:10.1016/j.amc.2019.124947

Cited by 23 publications

(5 citation statements)

References 16 publications

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“…The hierarchical identification principle is an effective tool [8][9][10] to solve the parameter identification issue of large-scale systems with the heavy computational burden and can be applied to multivariable systems, 11,12 nonlinear systems, [13][14][15] and bilinear systems. 16 The hierarchical identification principle is divided into three steps: the identification model decomposition, the submodel identification, and the coordination of correlation terms among sub-algorithms. For nonlinear controlled autoregressive systems, Chaudhary et al studied the standard hierarchical gradient descent algorithm and the fractional hierarchical gradient descent algorithm by using the hierarchical identification principle.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, the key of identifying MIMO systems is to improve the calculation efficiency. The hierarchical identification principle is an effective tool 8–10 to solve the parameter identification issue of large‐scale systems with the heavy computational burden and can be applied to multivariable systems, 11,12 nonlinear systems, 13–15 and bilinear systems 16 . The hierarchical identification principle is divided into three steps: the identification model decomposition, the submodel identification, and the coordination of correlation terms among sub‐algorithms.…”

Section: Introductionmentioning

confidence: 99%

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Hierarchical recursive least squares parameter estimation methods for multiple‐input multiple‐output systems by using the auxiliary models

Xing,

Ding,

Pan

et al. 2023

Adaptive Control & Signal

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SummaryMultiple‐input multiple‐output (MIMO) models are widely used in practical engineering. This article derives a new identification model of the MIMO system by decomposing the MIMO system into several multiple‐input single‐output subsystems. By means of the auxiliary model identification idea, an auxiliary model‐based recursive least squares (AM‐RLS) algorithm is derived for identifying the MIMO systems. In order to reduce the computational burden for identifying MIMO systems, this article presents a hierarchical identification model for the MIMO systems. By applying the hierarchical identification principle, an auxiliary model‐based hierarchical least squares (AM‐HLS) algorithm is proposed for improving the computational efficiency. The computational efficiency analysis indicates that the AM‐HLS algorithm is effective in reducing the calculation amount compared with the AM‐RLS algorithm. Moreover, this article analyzes the convergence of the AM‐HLS algorithm. The simulation example shows that the AM‐RLS and AM‐HLS algorithms studied in this article are effective.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hierarchical recursive least squares parameter estimation methods for multiple‐input multiple‐output systems by using the auxiliary models

Xing,

Ding,

Pan

et al. 2023

Adaptive Control & Signal

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show abstract

“…In the quest for such solutions, numerous numerical methods have been developed, among which polynomial and non-polynomial splines have surfaced as promising techniques for approximating solutions to nonlinear time-fractional differential equations [8,9]. Various methods have been employed in the field, including the approaches based on finite difference and spline approximation [10], Haar-Wavelet and optimal homotopy asymptotic methods [11], the B-spline collocation technique [12], cubic and quintic B-spline techniques [13][14][15], utilization of the Hermite-Galerkin method [16], the application of bilinear spline interpolation [17], utilization of bicubic B-spline functions [18], and the implementation of the quadratic spline-based with integral scheme [19], among several other methodologies. The difficulty in obtaining analytical solutions for nonlinear time-fractional differential equations leads to the adoption of numerical methods as valuable assets for approximating solutions.…”

Section: Introductionmentioning

confidence: 99%

A Hybrid Non-Polynomial Spline Method and Conformable Fractional Continuity Equation

Yousif,

Hamasalh

2023

Mathematics

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This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach’s applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool for solving nonlinear time-fractional differential equations. We further demonstrate its broad applicability by testing it on the Burgers–Fisher equations and comparing it with existing approaches, highlighting its superiority in biology, ecology, physics, and other fields. Moreover, meticulous evaluations of accuracy and efficiency using (L2 and L∞) norm errors reinforce its robustness and suitability for real-world applications. In conclusion, this paper presents a novel numerical technique for nonlinear time fractional differential equations, with the CCE and NPS methods’ unique combination driving its effectiveness and broad applicability in computational mathematics, scientific research, and engineering endeavors.

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“…We only mentioned the referenced such as [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and other relevant literatures. On the other hand, some authors obtained the numerical solution of stochastic Volterra integral equation by Euler-Maruyama approximation or iterative algorithm, for example [22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Multidimensional Stochastic Itô-Volterra Integral Equation Based on the Least Squares Method and Block Pulse Function

Jiang

Deng

2021

Mathematical Problems in Engineering

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In this paper, a method based on the least squares method and block pulse function is proposed to solve the multidimensional stochastic Itô-Volterra integral equation. The Itô-Volterra integral equation is transformed into a linear algebraic equation. Furthermore, the error analysis is given by the isometry property and Doob’s inequality. Numerical examples verify the effectiveness and precision of this method.

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An iterative algorithm for solving two dimensional nonlinear stochastic integral equations: A combined successive approximations method with bilinear spline interpolation

Cited by 23 publications

References 16 publications

Hierarchical recursive least squares parameter estimation methods for multiple‐input multiple‐output systems by using the auxiliary models

Hierarchical recursive least squares parameter estimation methods for multiple‐input multiple‐output systems by using the auxiliary models

A Hybrid Non-Polynomial Spline Method and Conformable Fractional Continuity Equation

Numerical Solution of Multidimensional Stochastic Itô-Volterra Integral Equation Based on the Least Squares Method and Block Pulse Function

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