Heat source identification of some parabolic equations based on the method of fundamental solutions

Rashedi, Kamal; Sarraf, Aydin

doi:10.1140/epjp/i2018-12193-8

Cited by 5 publications

(8 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where the constant τ > 2 (large enough), h(δ) = (log 1 δ ) − σβ α , σ ∈ (0, 1), which satisfies h(δ) ≥ δ when δ is sufficiently small. Before we proceed to the actual estimation, we need to guarantee the existence of the parameter µ for (18). Because lim µ→0 Pµ (ξ) = 1, then lim µ→0 K f δ µ (ξ) − ĝδ (ξ) = ĝδ (ξ) − ĝδ (ξ) = 0 < τh(δ).…”

Section: The New Discrepancy Principlementioning

confidence: 99%

“…Lemma 4. Assume µ D is chosen by the generalized discrepancy principle (18), and the a priori condition (12) holds. Therefore, we can obtain…”

Section: The New Discrepancy Principlementioning

confidence: 99%

“…A few theoretical articles concern conditional stability and the data compatibility of the solution notably in [1,3,8,[12][13][14][15]. As yet, there are many published works in this field based in several numerical methods, such as the Lie-group shooting method (LGSM) [5], the coupled method [16], the fundamental solutions method [6,17], the meshless method [7,18], the genetic algorithm [4], and the conjugate gradient method (CGM) [19][20][21]. In [22,23], these numerical methods were considered to solve two and three-dimensional inverse transient heat source problems.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Identifying the Unknown Source in Linear Parabolic Equation by a Convoluting Equation Method

Xiong

Cheng

2022

Mathematics

View full text Add to dashboard Cite

This article is devoted to identifying a space-dependent source term in linear parabolic equations. Such a problem is ill posed, i.e., a small perturbation in the input data may cause a dramatically large error in the solution (if it exists). The conditional stability of the solution is analyzed. Based on a convoluting equation method, we can deal with the problem under the a priori parameter choice rule. Meanwhile, a modified version of Morozov’s discrepancy principle is provided to decide on an a posteriori regularization parameter choice strategy and a log-type error estimate is obtained. Two numerical results show that our proposed method works well.

show abstract

Section: The New Discrepancy Principlementioning

confidence: 99%

“…Lemma 4. Assume µ D is chosen by the generalized discrepancy principle (18), and the a priori condition (12) holds. Therefore, we can obtain…”

Section: The New Discrepancy Principlementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Identifying the Unknown Source in Linear Parabolic Equation by a Convoluting Equation Method

Xiong

Cheng

2022

Mathematics

View full text Add to dashboard Cite

show abstract

“…31 The study of differential equations with integral conditions has grown significantly over the last 30 years because they have played a significant role in modeling many important physical phenomena related to thermoelasticity and control problems. 32 A novel discussion on the derivation of mathematical models and the analysis of such nonlocal boundary problems can be found in Cushman and Ginn. 33 The interested reader can refer to previous studies [34][35][36][37] and many other references therein for a summary of numerical solutions to parabolic problems with nonlocal boundary conditions.…”

Section: Literature Reviewmentioning

confidence: 99%

“…The application of the parabolic partial differential equations with nonlocal boundary conditions including integral conditions was introduced in Cannon 31 . The study of differential equations with integral conditions has grown significantly over the last 30 years because they have played a significant role in modeling many important physical phenomena related to thermoelasticity and control problems 32 . A novel discussion on the derivation of mathematical models and the analysis of such nonlocal boundary problems can be found in Cushman and Ginn 33 .…”

Section: Introductionmentioning

confidence: 99%

A numerical solution of an inverse diffusion problem based on operational matrices of orthonormal polynomials

Rashedi

2021

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

The inverse problem of identifying the diffusion coefficient in the one‐dimensional parabolic heat equation is studied. We assume that the information of Dirichlet boundary conditions along with an integral overdetermination condition is available. By applying the given assumptions, the problem is reformulated as a nonclassical parabolic equation along with the initial and boundary conditions. Then we employ the direct technique based on the operational matrices for integration, differentiation, and the product of the orthonormal polynomials together with the Ritz–Galerkin technique to reduce the main problem to the solution of a system of nonlinear algebraic equations. Numerical simulations are presented to show the applicability of the proposed method.

show abstract

Reconstruction of a time‐dependent coefficient in nonlinear Klein–Gordon equation using Bernstein spectral method

Rashedi

2022

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

In this work, we present a spectral method for recovering an unknown time-dependent lower-order coefficient and unknown wave displacement in a nonlinear Klein-Gordon equation with overdetermination at a boundary condition. We apply the initial and boundary conditions to construct the satisfier function and use this function in a transformation to convert the main problem to a nonclassical hyperbolic equation with homogeneous initial and boundary conditions. Then, we utilize the orthonormal Bernstein basis functions to approximate the solution of the reformulated problem and use a direct technique based on the operational matrices of integration and differentiation of these basis functions together with the collocation technique to reduce the problem to a system of nonlinear algebraic equations. Regarding the perturbed measurements, the method takes advantage of the mollification method in order to derive stable numerical derivatives. Numerical simulations for solving several test examples are presented to show the applicability of the proposed method for obtaining accurate and stable results.

show abstract

Heat source identification of some parabolic equations based on the method of fundamental solutions

Cited by 5 publications

References 49 publications

Identifying the Unknown Source in Linear Parabolic Equation by a Convoluting Equation Method

Identifying the Unknown Source in Linear Parabolic Equation by a Convoluting Equation Method

A numerical solution of an inverse diffusion problem based on operational matrices of orthonormal polynomials

Reconstruction of a time‐dependent coefficient in nonlinear Klein–Gordon equation using Bernstein spectral method

Contact Info

Product

Resources

About