Ritz–Galerkin method for solving an inverse heat conduction problem with a nonlinear source term via Bernstein multi-scaling functions and cubic B-spline functions

Dehghan, Mehdi; Yousefi, S. A.; Rashedi, Kamal

doi:10.1080/17415977.2012.701627

Cited by 30 publications

(22 citation statements)

References 39 publications

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“…Equation (2) and the other by imposing the Neuman condition, i.e. Equation (3). So, the matrix (l) and the vector d (l) are either …”

Section: The Proposed Local Meshless Approach For Cauchy Problemmentioning

confidence: 99%

“…Now, let us divide this set of nodal points into four groups 1 , 2 , 3 and 4 where 1 is the set of nodes located on the outer boundary out , 2 is the set of nodes located next to the outer boundary out (i.e. the dashed line in Figure 1), 3 is the set of nodes located on the inner boundary in and 4 represents the remaining nodes. Let us describe the method by considering these set of nodes separately as follows:…”

Section: The Proposed Local Meshless Approach For Cauchy Problemmentioning

confidence: 99%

“…Many engineering problems such as identification of material parameters, identification of unknown boundaries, identification of initial boundary conditions, identification of inaccessible boundary conditions can be considered as inverse problems, see for example [1][2][3]. Unfortunately, inverse problems are generally ill-posed problems in the Hadamard sense, [4] since the existence or uniqueness or the continuous dependence on the data of their solutions may not be ensured.…”

Section: Introductionmentioning

confidence: 99%

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

Shirzadi

Takhtabnoos

2015

Inverse Problems in Science and Engineering

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This paper is concerned with the development of a meshless local approach based on the finite collocation method for solving Cauchy problems of 2-D elliptic PDEs in annulus domains. In the proposed approach, besides the collocation of unknown solution, the governing equation is also enforced in the local domains. Moreover, to improve the accuracy, the method considers auxiliary points in local subdomains and imposes the governing PDE operator at these points, without changing the global system size. Localization property of the method reduces the ill-conditioning of the problem and makes it efficient for Cauchy problem.To show the efficiency of the method, four test problems containing Laplace, Poisson, Helmholtz and modified Helmholtz equations are given. A numerical comparison with traditional local RBF method is given in the first test problem.

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“…Equation (2) and the other by imposing the Neuman condition, i.e. Equation (3). So, the matrix (l) and the vector d (l) are either …”

Section: The Proposed Local Meshless Approach For Cauchy Problemmentioning

confidence: 99%

Section: The Proposed Local Meshless Approach For Cauchy Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Shirzadi

Takhtabnoos

2015

Inverse Problems in Science and Engineering

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“…In [16], Dehghan and Shakeri proposed a numerical scheme to solve two dimensions parabolic equations by the method of lines. A Ritz-Galerkin method was proposed to solve the one-dimensional parabolic equation in [18].…”

Section: Introductionmentioning

confidence: 99%

Galerkin-Chebyshev spectral method and block boundary value methods for two-dimensional semilinear parabolic equations

Liu

Sun

2015

Numer Algor

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In this paper, we present a high-order accurate method for twodimensional semilinear parabolic equations. The method is based on a GalerkinChebyshev spectral method for discretizing spatial derivatives and a block boundary value methods of fourth-order for temporal discretization. Our formulation has high-order accurate in both space and time. Optimal a priori error bound is derived in the weighted L 2 ω -norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence properties of the method.

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“…Therefore, it is also significant mathematically.The Ritz-Galerkin method in the Bernstein polynomials basis is the method to convert a continuous operator problem to a discrete problem, which essentially converts the equation to a weak formulation, and then applies some constraints on the function space to characterize the space with a finite set of basis functions. It has been widely used in many areas of mathematics, especially in the field of numerical analysis [22][23][24][25][26].The presence of time-dependent and non-local boundary conditions can greatly complicate the application of standard numerical techniques such as finite-difference procedures [27,28], finite-element methods, spectral techniques, and boundary integral equation schemes [29,30]. Therefore, it is most important to convert time-dependent and non-local boundary value problems to a more desirable equivalent form, which is often a tough task.In present paper, approximation solution of this problem is implemented by the Ritz-Galerkin method for the first time.…”

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confidence: 99%

Ritz–Galerkin method for solving a parabolic equation with non‐local and time‐dependent boundary conditions

Zhou

2015

Math Methods in App Sciences

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Communicated by W. SprößigThe paper is devoted to the investigation of a parabolic partial differential equation with non-local and time-dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz-Galerkin method, which is a first attempt at tackling parabolic equation with such non-classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non-local boundary condition, we use a trick of introducing the transition function G.x, t/ to convert non-local boundary to another non-classical boundary, which can be handled with the Ritz-Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper.

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Ritz–Galerkin method for solving an inverse heat conduction problem with a nonlinear source term via Bernstein multi-scaling functions and cubic B-spline functions

Cited by 30 publications

References 39 publications

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

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

Galerkin-Chebyshev spectral method and block boundary value methods for two-dimensional semilinear parabolic equations

Ritz–Galerkin method for solving a parabolic equation with non‐local and time‐dependent boundary conditions

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