A finite difference scheme with arbitrary mesh systems for solving high-order partial differential equations
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Cited by (16)
A new approach for solving heat conduction under zero and non-zero initial conditions
2022, Engineering Analysis with Boundary ElementsCitation Excerpt :Several mesh-based approaches are alternatives if the domain is discretized and allows the simulation of non-homogeneous media. For instance, Zhang et al. [16] and Mitsoulis et al. [17] used the finite element method (FEM), Cai et al. [18] applied the finite volume method (FVM) and Ozisik [19], Tseng et al. [20] used the finite difference method (FDM) in large models with more complex shapes but finite dimensions. Cascio et al. [21] presented a hybrid formulation coupling the virtual element method (VEM) [22–24] with the boundary element method formulation for heterogeneous materials.
The generalized finite difference method with third- and fourth-order approximations and treatment of ill-conditioned stars
2021, Engineering Analysis with Boundary ElementsCitation Excerpt :However, higher-order approximations have been rarely used in the GFDM, due to two main reasons: the need to increase the minimum number of points per star, which often causes problems of ill-conditioning, and the high computational cost. To avoid the use of the fourth-order approximation when the differential equation is of fourth-order, Orkisz [4] and Tseng and Gu [24] suggested decomposing the fourth-order operators by successive approximations using second-order operators. Nowadays, the power of computers allows using the fourth-order approximation with many more points per star than the minimum necessary, thus avoiding ill-conditioned stars.
Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: A survey
2011, Computers and Mathematics with ApplicationsCitation Excerpt :A series of problems in many scientific fields can be modeled with the use of differential equations such as problems in physics [1–3], chemistry [4–6], biology [7,8], economics [9], etc. Due to the importance of differential equations many methods have been proposed in the relevant literature for their solution such as Runge–Kutta methods [10,11], Predictor–Corrector methods [12,13], Finite Difference methods [14–16], Finite Element methods [17–19], Splines [20–23] and other methods [24–41] etc. These methods require the discretization of domain into the number of finite elements where the functions are approximated locally.
A generalized finite difference method using Coatmèlec lattices
2009, Computer Physics CommunicationsCitation Excerpt :The list of methods referred to as meshless is very large and it is continuously growing. For example, smooth particle hydrodynamics [2], generalized finite differences [3,4], moving least squares techniques [5], diffuse elements [6], element-free Galerkin [7], to name only a few (a more comprehensive list can be found in [1]). Meshless methods are useless without an evaluation of the nodal connectivity bounded in time and a computational cost which grows linearly with the total number of nodes in the domain [1].
Numerical solution of quenching problems using mesh-dependent variable temporal steps
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Visiting Scholar, originally with the Department of Mathematics and Mechanics, Lanzhou University, People's Republic of China.