Numerical Solution of Variable-Order Time Fractional Weakly Singular Partial Integro-Differential Equations With Error Estimation

Dehestani, Haniye; Ordokhani, Yadollah; Razzaghi, Mohsen

doi:10.3846/mma.2020.11692

Cited by 13 publications

(3 citation statements)

References 36 publications

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“…Numerical methods have been widely used in various sciences [19,28,29,42]. These methods have high accuracy and acceptable convergence rate [8,18,34]. Thus, numerical methods have been presented to find an accelerate and possible solution in this case.…”

Section: Introductionmentioning

confidence: 99%

Bézier Base Extended Isogeometric Numerical Method for Thermo Elastic-Plastic Analysis of Crack Propagation in Cracked Plate Under Welding Residual Stress and Thermal Load

Shoheib

Shahrooi

Shishesaz

et al. 2022

Mathematical Modelling and Analysis

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A new procedure in the field of Bézier base extended isogeometric method (XIGA) has been introduced to analyze the effect of welding residual stress and thermal load on crack propagation rate and fatigue life. This new procedure is based on the constitutive thermoelastic plastic equation. The main parts of this procedure are using the B´ezier base XIGA method to calculate the redistribution of welding residual stress due to crack growth and to compute the value of stress intensity factor (SIF) in the welding residual stress field. For this purpose, the grid points of Bézier elements (with C0-continuity) around the crack line and the crack tip are identified by the level set representation. Then, discontinuous enrichment functions are added to the isogeometric analysis approximation. Thus, this method does not require the re-meshing process. The results show that there is a good agreement between the results of proposed numerical method and the Hole-Drilling Strain-Gage method. The interaction integral method has been used to extract SIF. The effects of welding residual stress and thermal load on the SIF are considered using the superposition method. Also, the Walker equation has been modified to calculate the fatigue life caused by thermal loading and welding residual stress. The results display a good agreement between the proposed method and the finite element method. Due to the advantages of the Bézier based XIGA method, which eliminates parametric space and allows the precise addition of enrichment functions to the basis functions of cracked elements (crack line or crack tip), the obtained results are highly accurate that shows this method is effective for analyzing discontinuous problems.

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Section: Introductionmentioning

confidence: 99%

Bézier Base Extended Isogeometric Numerical Method for Thermo Elastic-Plastic Analysis of Crack Propagation in Cracked Plate Under Welding Residual Stress and Thermal Load

Shoheib

Shahrooi

Shishesaz

et al. 2022

Mathematical Modelling and Analysis

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show abstract

“…In recent years, due to the many applications of fractional calculus in modeling natural phenomena, much attentions have been paid to the numerical solution of the fractional equations (see, e.g., previous studies 4–24 ).…”

Section: Introductionmentioning

confidence: 99%

A numerical study on the non‐smooth solutions of the nonlinear weakly singular fractional Volterra integro‐differential equations

Sajjadi

Najafi

Aminikhah

2022

Math Methods in App Sciences

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The solutions of weakly singular fractional Volterra integro-differential equations involving the Caputo derivative typically have solutions whose derivatives are unbounded at the left end-point of the interval of integration. In this paper, we design an algorithm to prevail on this non-smooth behavior of solutions of the nonlinear fractional Volterra integro-differential equations with a weakly singular kernel. The convergence of the proposed method is investigated.The proposed scheme is employed to solve four numerical examples in order to test its efficiency and accuracy.

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“…Avazzadeh et al [29] established a hybrid technique by blending the Legendre wavelets, and operational matrix of fractional integration. A numerical technique based on Legendre-Laguerre and the collocation method has been considered by Dehestani et al [30].…”

Section: Introductionmentioning

confidence: 99%

A Numerical Study of Nonlinear Fractional Order Partial Integro-Differential Equation with a Weakly Singular Kernel

2021

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Fractional differential equations can present the physical pathways with the storage and inherited properties due to the memory factor of fractional order. The purpose of this work is to interpret the collocation approach for tackling the fractional partial integro-differential equation (FPIDE) by employing the extended cubic B-spline (ECBS). To determine the time approximation, we utilize the Caputo approach. The stability and convergence analysis have also been analyzed. The efficiency and reliability of the suggested technique are demonstrated by two numerical applications, which support the theoretical results and the effectiveness of the implemented algorithm.

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Numerical Solution of Variable-Order Time Fractional Weakly Singular Partial Integro-Differential Equations With Error Estimation

Cited by 13 publications

References 36 publications

Bézier Base Extended Isogeometric Numerical Method for Thermo Elastic-Plastic Analysis of Crack Propagation in Cracked Plate Under Welding Residual Stress and Thermal Load

Bézier Base Extended Isogeometric Numerical Method for Thermo Elastic-Plastic Analysis of Crack Propagation in Cracked Plate Under Welding Residual Stress and Thermal Load

A numerical study on the non‐smooth solutions of the nonlinear weakly singular fractional Volterra integro‐differential equations

A Numerical Study of Nonlinear Fractional Order Partial Integro-Differential Equation with a Weakly Singular Kernel

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