A local meshless method for Cauchy problem of elliptic PDEs in annulus domains

Shirzadi, Ahmad; Takhtabnoos, Fariba

doi:10.1080/17415977.2015.1061521

Cited by 17 publications

(3 citation statements)

References 22 publications

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“…According to Jiao et al [21], "although the series can be rapidly convergent in a very small region, it has very slow convergence rate in the wider region and the truncated series solution is an inaccurate solution in that region, which will greatly restrict the application area of the method". Moreover, the BVP (1), (2) has been analyzed by meshless method through radial basis functions, though they are often used for high-dimensional problems [22][23][24][25][26][27][28][29][30][31][32][33][34][35]. The interested readers are also referred to [36][37][38][39] for other useful techniques, which can be applied for the same problems.…”

Section: Preliminaries and Mathematical Formulationmentioning

confidence: 99%

Buckling of Doubly Clamped Nano-Actuators in General Form through Optimized Chebyshev Polynomials with Interior Point Algorithm

Shivanian

Ansari

2019

Acta Phys. Pol. A

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In current work, a novel intelligent computational technique is adopted for searching the behaviour of the problem of buckling of nano-actuators in the presence of various nonlinear forces. In order to accomplish this aim, the governing integro-differential equation in general form is taken into account for nano-actuators, which contains various nonlinear forces as well. This generalized form for the nano-actuators is in fact a non-linear fourth-order Fredholm integro-differential boundary value problem. The boundary value problem is transformed into an equivalent problem whose boundary conditions are such that it is convenient to apply reformed version of the Chebyshev polynomials of the first kind. These Chebyshev polynomials based functions construct approximate series solution with unknown weights. The mathematical formulation of optimization problem consists of an uninspected error which is minimized through adjusting weights via interior point method. The trial approximate solution is validated by imposing tolerance constrained into optimization problem. This numerical technique enables us to overcome all kind of nonlinearities in the mentioned boundary value problem, and then to obtain an accurate solution. Thus, it can expedite the design of nano-actuators.

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Section: Preliminaries and Mathematical Formulationmentioning

confidence: 99%

Buckling of Doubly Clamped Nano-Actuators in General Form through Optimized Chebyshev Polynomials with Interior Point Algorithm

Shivanian

Ansari

2019

Acta Phys. Pol. A

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“…The local meshless technique, based on the finite collocation method for solving Cauchy problems of elliptic PDEs in annulus domains, was developed in ref. [35]. A novel meshless numerical solution method for the inverse Cauchy problem for a semilinear elliptic-type PDE in an arbitrary doubly connected plane domain was developed in [36].…”

Section: Introductionmentioning

confidence: 99%

Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation

Koleva,

Vulkov

2024

Axioms

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In this work, we consider Cauchy-type problems for Laplace’s equation with a dynamical boundary condition on a part of the domain boundary. We construct a discrete-in-time, meshless method for solving two inverse problems for recovering the space–time-dependent source and boundary functions in dynamical and Dirichlet boundary conditions. The approach is based on Green’s second identity and the forward-in-time discretization of the non-stationary problem. We derive a global connection that relates the source of the dynamical boundary condition and Dirichlet and Neumann boundary conditions in an integral equation. First, we perform time semi-discretization for the dynamical boundary condition into the integral equation. Then, on each time layer, we use Trefftz-type test functions to find the unknown source and Dirichlet boundary functions. The accuracy of the developed method for determining dynamical and Dirichlet boundary conditions for given over-determined data is first-order in time. We illustrate its efficiency for a high level of noise, namely, when the deviation of the input data is above 10% on some part of the over-specified boundary data. The proposed method achieves optimal accuracy for the identified boundary functions for a moderate number of iterations.

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“…Related contributions are found in several places, for example, an spectral method for a Cauchy problem associated with the Laplace equation can be found in Bernston and Eldén [8]. Mesh-less radial point interpolation methods are employed in [40,41]. An inverse boundary element method (BEM) for determining the heat transfer coefficients on solid surfaces of arbitrary shape is described by Martin and Dulikravich [28]; BEM-based methods are also employed in [30,43].…”

Section: Introductionmentioning

confidence: 99%

Identification of heat transfer coefficient through linearization: explicit solution and approximation

Bazán

Bedin

2017

Inverse Problems

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We derive an explicit reconstruction formula for the heat flux on the internal surface of a duct based on external temperature data. With the heat flux at hand we determine the internal convective heat transfer coefficient. Technically, by determining explicitly the singular system of a compact operator and assuming noise-free data, the heat flux is expressed as an infinite singular value expansion, whereas for noisy data, the truncated expansion is used as regularized solution. An error bound is derived for the case where the truncation parameter is determined by the Morozov's discrepancy principle. The method has high flexibility in that it is readily adapted to the finite data case without any substantial modification. Moreover, it does not require any linear system solver and it is therefore easy to implement and extremely inexpensive when compared to existing heat transfer coefficient estimation techniques. The method is illustrated by means of numerical results using both synthetic and experimental data.

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A local meshless method for Cauchy problem of elliptic PDEs in annulus domains

Cited by 17 publications

References 22 publications

Buckling of Doubly Clamped Nano-Actuators in General Form through Optimized Chebyshev Polynomials with Interior Point Algorithm

Buckling of Doubly Clamped Nano-Actuators in General Form through Optimized Chebyshev Polynomials with Interior Point Algorithm

Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation

Identification of heat transfer coefficient through linearization: explicit solution and approximation

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