Stability and error analysis of the reproducing kernel Hilbert space method for the solution of weakly singular Volterra integral equation on graded mesh

Beyrami, Hossein; Lotfi, Taher; Mahdiani, Katayoun

doi:10.1016/j.apnum.2017.05.010

Cited by 14 publications

(12 citation statements)

References 25 publications

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“…In response to the above-unsolved problems and considering our previous research results, this paper proposes a novel identification method for crack fault evolutionary characteristics of planetary gearboxes based on the Volterra highorder kernel function theory [24,25]. The main academic ideas of this method can be summarized as follows: First, a nonlinear Volterra system model of the system is established by providing vibration signals of planetary gearboxes as the input and output; second, through the rational construction of the Volterra series Hilbert reproducing kernels [26][27][28], the high-order kernels of the Volterra model are calculated. Avoiding the "dimensional disaster" problem that may occur [29], third, through the generalized frequency response characteristics of high-order kernel of the system Volterra model, the crack faults and degree characteristics of planetary gearbox gears are identified efficiently.…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Research on Fault Evolution Feature Extraction and Identification of Sun Gear Cracks in Planetary Gearbox Based on Volterra High-Order Kernel Generalized Frequency Response Graphic Analysis

Wang

Kang

Shi

et al. 2018

Mathematical Problems in Engineering

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In this paper, relying on the Volterra series nonlinear system model and the high-order kernel Hilbert’s reconstructed kernel fast solved algorithm, a fault feature frequency domain identification method based on Volterra high-order kernel generalized frequency response graph analysis is proposed. Firstly, the method uses the system input and output vibration signals to determine the Volterra model. Then, the Volterra high-order kernel function is solved quickly by reproducing kernel Hilbert space method, and the generalized frequency response function is used to identify the model. Finally, multidimensional high-order spectral pattern analysis is used to separate and extract the fault and degree characteristic information implied by frequency and phase coupling in the third-order kernel function. Following the theoretical approach, in the experimental part, this paper uses the planetary gearbox fault loading test rig to complete the data collection and establishes the Volterra experimental model through the measured data. The generalized frequency responses of each order kernel function are compared and analyzed and the capability of distinguishing and the adaptability of different order kernel functions for the degree of crack failure are discussed. The effects of changing the memory length of the Volterra model and the order of the kernel function on the recognition result are verified. The final experimental results show that the use of reproducing kernel Hilbert space can effectively avoid the dimension disaster problem that occurs in the high-order kernel solution process. Moreover, the third-order kernel can describe more intuitively the nonlinear system model under multifactor coupling than the second-order kernel. Finally, Volterra series model the third-order kernel’s generalized frequency response can effectively distinguish between nondefective and faulty gears, and its resolution is enough to distinguish the degree of failure of gear cracks.

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Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Research on Fault Evolution Feature Extraction and Identification of Sun Gear Cracks in Planetary Gearbox Based on Volterra High-Order Kernel Generalized Frequency Response Graphic Analysis

Wang

Kang

Shi

et al. 2018

Mathematical Problems in Engineering

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“…Sahihi et al Sahihi et al, Sahihi et al, (2020) have studied on solving system of second-order BVPs using a new algorithm based on reproducing kernel Hilbert space. For interesting results and more details about this method, we refer the reader to (Bergman, 1950;Beyrami et al, 2017;Foroutan et al, 2018;Zaremba, 1907Zaremba, , 1908 and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Reproducing kernel method for the solutions of non-linear partial differential equations

Yildirim

Akgül

2021

Arab Journal of Basic and Applied Sciences

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In modeling of a lots of complex physical problems and engineering process, the non-linear partial differential equations have a very important role. Development of dependable and effective methods to solve such types equations are constructed. In the suggested technique, reproducing kernel method is examined to approximate the solutions together with reproducing kernel functions. In order to demonstrate accuracy, the performance and reliability of the proposed method, the results of the experiments and the available results are compared. There is high stability for a higher degree of accuracy between the solutions.

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“…This theory has been successfully applied to integral equations [14,15], partial differential equations [16], boundary value problems [17][18][19][20][21][22], fractional differential equations [23], and so on [24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

A numerical approach for solving the high-order nonlinear singular Emden–Fowler type equations

Lotfi

Mahdiani

2018

Adv Differ Equ

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Reproducing kernel Hilbert space method (RKHSM) is an analytical technique, which can overcome the difficulty at the singular point of non-homogeneous, linear singular initial value problems; especially when the singularity appears on the right-hand side of this type of equations, so it can solve powerfully linear singular initial value problems. Therefore, using advantages of this method, in this paper the high-order nonlinear singular Emden-Fowler type equations are investigated by RKHSM. Then we present five numerical examples to show that the proposed scheme is accurate and reliable.

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Stability and error analysis of the reproducing kernel Hilbert space method for the solution of weakly singular Volterra integral equation on graded mesh

Cited by 14 publications

References 25 publications

Research on Fault Evolution Feature Extraction and Identification of Sun Gear Cracks in Planetary Gearbox Based on Volterra High-Order Kernel Generalized Frequency Response Graphic Analysis

Research on Fault Evolution Feature Extraction and Identification of Sun Gear Cracks in Planetary Gearbox Based on Volterra High-Order Kernel Generalized Frequency Response Graphic Analysis

Reproducing kernel method for the solutions of non-linear partial differential equations

A numerical approach for solving the high-order nonlinear singular Emden–Fowler type equations

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