A gradient reproducing kernel collocation method for high order differential equations

Mahdavi, Ashkan; Chi, Sheng‐Wei; Zhu, Hongping

doi:10.1007/s00466-019-01724-0

Cited by 27 publications

(7 citation statements)

References 40 publications

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“…The mathematical proof of reduced complexity in comparison with reproducing kernel approximation is described in a previous study [13]. For direct problems solved by using high-order implicit gradient reproducing kernel approximation, the details are described in a recent study [11]. In the present study, the fourth-order gradient reproducing kernel approximation is adopted, and the corresponding implicit RK shape functions to the first-, second-, third-, and fourth-order are expressed explicitly as follows:…”

Section: Review Of High-order Gradient Reproducing Kernel Approximationmentioning

confidence: 99%

“…Other advanced versions of smooth or gradient reproducing kernel collocation methods have shown the efficacy of solving both the second-order and fourth-order partial differential equations with superconvergent rates recently [8][9][10]. With the aid of high-order gradient approximation [11], the direct differentiation of reproducing kernel (RK) shape functions is avoided. It should be noted that the weighted collocation method is established on the basis of the least-squares minimization [12,13].…”

Section: Introductionmentioning

confidence: 99%

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Recovering Heat Source from Fourth-Order Inverse Problems by Weighted Gradient Collocation

Yang

2022

Mathematics

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The weighted gradient reproducing kernel collocation method is introduced to recover the heat source described by Poisson’s equation. As it is commonly known that there is no unique solution to the inverse heat source problem, the weak solution based on a priori assumptions is considered herein. In view of the fourth-order partial differential equation (PDE) in the mathematical model, the high-order gradient reproducing kernel approximation is introduced to efficiently untangle the problem without calculating the high-order derivatives of reproducing kernel shape functions. The weights of the weighted collocation method for high-order inverse analysis are first determined. In the benchmark analysis, the unclear illustration in the literature is clarified, and the correct interpretation of numerical results is given particularly. Two mathematical formulations with four examples are provided to demonstrate the viability of the method, including the extreme cases of the limited accessible boundary.

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Section: Review Of High-order Gradient Reproducing Kernel Approximationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Recovering Heat Source from Fourth-Order Inverse Problems by Weighted Gradient Collocation

Yang

2022

Mathematics

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“…As an extension of the gradient reproducing kernel approximation [15], the second derivatives of RK shape functions are obtained by using the high-order gradient reproducing kernel (HG-RK) approximation [17]. Unlike the direct derivatives of RK shape functions, the implicit derivatives of RK shape functions in two dimensions are constructed as follows:…”

Section: High-order Gradient Reproducing Kernel Approximationmentioning

confidence: 99%

“…Still, the gradient approximation of RK shape functions requires computing the derivatives once in solving the second-order PDEs. As pointed out in the literature [17], by using the gradient reproducing conditions, the higher-order gradient RK shape functions can be constructed implicitly without taking direct derivatives of RK shape functions, which benefits the second-and fourth-order PDEs. To overcome discretization restriction, the harmonic-enriched reproducing kernel approximation is proposed to solve highly oscillatory PDEs, in which the idea of implicit derivatives of RK shape functions is also adopted [18].…”

Section: Introductionmentioning

confidence: 99%

Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method

Yang

Lam

2020

Mathematics

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The weighted reproducing kernel collocation method exhibits high accuracy and efficiency in solving inverse problems as compared with traditional mesh-based methods. Nevertheless, it is known that computing higher order reproducing kernel (RK) shape functions is generally an expensive process. Computational cost may dramatically increase, especially when dealing with strong-from equations where high-order derivative operators are required as compared to weak-form approaches for obtaining results with promising levels of accuracy. Under the framework of gradient approximation, the derivatives of reproducing kernel shape functions can be constructed synchronically, thereby alleviating the complexity in computation. In view of this, the present work first introduces the weighted high-order gradient reproducing kernel collocation method in the inverse analysis. The convergence of the method is examined through the weights imposed on the boundary conditions. Then, several configurations of multiply connected domains are provided to numerically investigate the stability and efficiency of the method. To reach the desired accuracy in detecting the outer boundary for two special cases, special treatments including allocation of points and use of ghost points are adopted as the solution strategy. From four benchmark examples, the efficacy of the method in detecting the unknown boundary is demonstrated.

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“…The meshfree methods recognized as powerful tools to solve practical problems governed by partial differential equations (PDEs) have attracted considerable attention [1][2][3]. These methods have the advantage that it does not require mesh construction [4]. Engineering problems involving irregular geometry are usually intractable.…”

Section: Introductionmentioning

confidence: 99%

Multiquadrics without the Shape Parameter for Solving Partial Differential Equations

Liu

Xiao

et al. 2020

Symmetry

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In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF at each center point coinciding with an interior point, we separated the center points from the interior points, in which the center points were regarded as the fictitious sources collocated outside the domain. The interior, boundary, and source points were therefore collocated within, on, and outside the domain, respectively. Since the radial distance between the interior point and the source point was always greater than zero, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Accordingly, the shape parameter was no longer required in the MQ and IMQ RBFs. Numerical examples with the domain in symmetry and asymmetry are presented to verify the accuracy and robustness of the proposed method. The results demonstrated that the proposed method using MQ RBFs without the shape parameter acquires more accurate results than the conventional RBF collocation method with the optimum shape parameter. Additionally, it was found that the locations of the fictitious sources were not sensitive to the accuracy.

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A gradient reproducing kernel collocation method for high order differential equations

Cited by 27 publications

References 40 publications

Recovering Heat Source from Fourth-Order Inverse Problems by Weighted Gradient Collocation

Recovering Heat Source from Fourth-Order Inverse Problems by Weighted Gradient Collocation

Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method

Multiquadrics without the Shape Parameter for Solving Partial Differential Equations

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