Jacobi spectral solution for integral algebraic equations of index-2

Hadizadeh, M.; Ghoreishi, F.; Pishbin, S.

doi:10.1016/j.apnum.2010.08.009

Cited by 48 publications

(20 citation statements)

References 17 publications

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“…First, by setting h = 1, we compare these methods with the Jacobi spectral method of degree m + 1 (if the criteria of the comparison are the size of linear system it may be m). In Table 4, we compare these results with the corresponding results of Hadizadeh et al (2011), which shows the efficiency of the introduced methods. Now, since the convergence results were only guaranteed for sufficiently small h, let us fix m = 7 and compare those with the results of h = 1/2 N , N = 1, .…”

Section: Example 3 Letmentioning

confidence: 94%

“…In this section, we illustrate the efficiency of the introduced methods by applying them to some linear and nonlinear problems, in comparison with the existing methods (e.g., Kauthen, 2001;Hadizadeh et al, 2011). In the designed package for these methods, we have provided techniques for solving linear and nonlinear systems.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…For this system, Hadizadeh et al (2011) to make a comparison between the presented methods and Jacobi spectral methods of Hadizadeh et al (2011).…”

Section: Example 3 Letmentioning

confidence: 99%

“…For the 'degree' of ill-posedness, Lamm (2005) as well as Lamm and Scofield (2000) introduced 'v-smoothing' for Volterra integral equations of the first kind, which is equivalent with a differential index. The tractable index 1 and 2 problems are defined respectively by Brunner (2004) and Hadizadeh et al (2011).…”

Section: B Shiri Et Almentioning

confidence: 99%

See 3 more Smart Citations

Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of the Hessenberg type

Shiri¹,

Shahmorad²,

Hojjati³

2013

International Journal of Applied Mathematics and Computer Science

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In this paper, we deal with a system of integral algebraic equations of the Hessenberg type. Using a new index definition, the existence and uniqueness of a solution to this system are studied. The well-known piecewise continuous collocation methods are used to solve this system numerically, and the convergence properties of the perturbed piecewise continuous collocation methods are investigated to obtain the order of convergence for the given numerical methods. Finally, some numerical experiments are provided to support the theoretical results.

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Section: Example 3 Letmentioning

confidence: 94%

Section: Numerical Experimentsmentioning

confidence: 99%

“…For this system, Hadizadeh et al (2011) to make a comparison between the presented methods and Jacobi spectral methods of Hadizadeh et al (2011).…”

Section: Example 3 Letmentioning

confidence: 99%

Section: B Shiri Et Almentioning

confidence: 99%

See 2 more Smart Citations

Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of the Hessenberg type

Shiri¹,

Shahmorad²,

Hojjati³

2013

International Journal of Applied Mathematics and Computer Science

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“…When they do not include integral or differential equations, they are called differential-algebraic equations (DAE) or integralalgebraic equations (IAE), respectively. While the qualitative theory of DAE is widely available (see [2,14,[16][17][18]), there exist fewer results for IAE and DIDE (see, [3,5,8,10,15] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Existence and uniqueness of solutions to weakly singular integral-algebraic and integro-differential equations

Bulatov¹,

Lima²,

Weinmüller

2013

Open Mathematics

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We consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix ( ) vanishes when = . Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples. MSC:65R20, 45F15

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Adaptive iterative regularization schemes for two‐dimensional integral‐algebraic systems

Farahani

Hadizadeh

Bulatov

et al. 2019

Math Methods in App Sciences

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The main idea of this paper is to utilize the adaptive iterative schemes based on regularization techniques for moderately ill-posed problems that are obtained by a system of linear two-dimensional Volterra integral equations with a singular matrix in the leading part. These problems may arise in the modeling of certain heat conduction processes as well as in the dynamic simulation packages such as compressible flow through a plant piping network. Owing to the ill-posed nature of the first kind Volterra equation that appears in the system, we will focus on the two families of regularization algorithms, ie, the Landweber and Lavrentiev type methods, where we treat both the exact and perturbed data. Our aim is to work directly with the original Volterra equations without any kind of reduction. Two fast iterative algorithms with reasonable computational complexity are developed. Numerical experiments on a few test problems are used to illustrate the validity and efficiency of the proposed iterative methods in comparison with the classical regularization methods.

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Jacobi spectral solution for integral algebraic equations of index-2

Cited by 48 publications

References 17 publications

Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of the Hessenberg type

Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of the Hessenberg type

Existence and uniqueness of solutions to weakly singular integral-algebraic and integro-differential equations

Adaptive iterative regularization schemes for two‐dimensional integral‐algebraic systems

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