A localized MAPS using polynomial basis functions for the fourth-order complex-shape plate bending problems

Localized method of approximate particular solutions with polynomial basis functions

Local method of approximate particular solutions for two-dimensional unsteady Burgers’ equations

A novel upwind technique for the localized method of approximate particular solutions (LMAPS) is proposed to solve the convection-diffusion equations. An upwind approximation to the convective terms is implemented by choosing upwind interpolation stencils while the central interpolation stencils are used for the diffusive terms. The proposed upwind LMAPS scheme is also been compared with conventional LMAPS without upwind technique, to demonstrate its superiority in generating high accurate solutions than the latter. Numerical results show that the proposed upwind LMAPS has high accuracy and efficiency for a variety of convection-diffusion equations.

In-silico study about the influence of electroporation parameters on the chemotherapeutic drug transport in cancer tissues using the meshless method of approximate particular solutions

Localized method of approximate particular solutions with Cole–Hopf transformation for multi-dimensional Burgers equations

This paper attempts to obtain bending solutions to plates under uniformly distributed and hydrostatic load distributions using Ritz variational methods and basis functions that are found by superposing trigonometric series and third degree polynomials. Two cases of boundary conditions were considered. In one case, three edges were simply supported and the fourth edge was clamped (SSCS thin plate). In the second case, the adjacent edges were clamped and the other edges were simply supported (SCCS thin plate). This work presents first principles, rigorous derivation of the governing Ritz variational functional and the displacement basis functions for the boundary conditions investigated. The solution is presented in analytical form. The obtained results are compared with previous results obtained using Levy series and Ritz methods and found to be in close agreement . The disadvantage of the method is the associated computational rigour, but the benefit is the accuracy of the results. Comparisons of the present results for center deflections and center bending moments with results in the literature show that there is negligible difference. Double series expressions were found for deflections and bending moments for the plate bending problems solved. Evaluation of the double series expressions at the plate center gave center deflection results that differed from the exact solutions by for to for for uniformly loaded thin plates with three simply supported edges and one clamped edge (SSSC). The differences in the center bending moments Mxx were found to vary from for to for In general, the present results yielded reasonably accurate solutions for the plate bending problems studied.

Integral equation–based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.

1. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards, Applied Mathematics Series 55, 3rd printing, with corrections, March, 1965. 2. P. F. Hsieh and Y. Sibuya, On the asymptotic integration of second order linear ordinary differential equations with polynomial coefficients, J. Math. Anal. Appl. 16 (1966), 84-103. 3. M. A. Neumark, Lineare Differential-Operatoren, Akademie-Verlag, Berlin, 1960. 4. H. L. Turrittin, Solvable related equations pertaining to turning point problems, Proc. Sympos. Asymptotic Solutions of Differential Equations and Their Applications, pp. 27-52, Wiley, New York, 1964.

This paper introduces an innovative method for numerically integrating fourth-order initial value problems by utilizing Chebyshev polynomials as the fundamental basis function. The block integrator base on Chebyshev polynomial demonstrates significant improvements in accuracy and stability, rendering it a valuable tool across various scientific and engineering fields. By leveraging the characteristics of Chebyshev polynomials, this approach accurately estimates solutions for fourth-order differential equations without reducing it to a system of first order Ordinary Differential Equations while at the same time effectively managing error accumulation within a block integration framework and thereby enhancing its accuracy over extended intervals. Through rigorous numerical experiments, the effectiveness and reliability of the new integrator are demonstrated and compared with existing methods. The new method is consistent, zero stable and convergent. The method also shows an appreciable error constants. The new method performed better in terms of accuracy than the existing methods in the literature in both linear and nonlinear problems.

A meshless local collocation method for time fractional diffusion wave equation

Based on the improved complex variable moving least-squares (ICVMLS) approximation, the improved complex variable element-free Galerkin (ICVEFG) method for the bending problem of Kirchhoff plate is presented. Compared with the moving least-squares (MLS) approximation, in the ICVMLS approximation, the approximation function of two-dimensional problems can be obtained with one-dimensional basis function, then the computational efficiency of the shape functions is higher. Compared with the meshless methods based on the MLS approximation, under the same node distributions, the ones using the ICVMLS approximation can obtain the solutions with higher computational accuracy; and under the similar computational accuracy, the ones using the ICVMLS approximation have higher computational efficiency. The ICVMLS approximation is used to form the approximation function of the deflection of a Kirchhoff plate, the Galerkin weak form of the bending problem of Kirchhoff plates is adopted to obtain the discretized system equations, and the penalty method is employed to enforce the essential boundary conditions, then the corresponding formulae of the ICVEFG method for the bending problem of Kirchhoff plates are presented. By computing and analyzing four typical examples, it is shown that the ICVEFG method of Kirchhoff plates in this paper is efficient. And the computational precision of the numerical solutions is analyzed to select the basis function, weight function, scaling factor, node distribution and penalty factor in the ICVEFG method. Numerical examples show that the method in this paper has better convergence and higher accuracy. When the quadratic polynomial basis function and the cubic or quartic spline weight function are used, and d m a x = 4.2−4.4 , the ICVEFG method of Kirchhoff plates in this paper can obtain the solutions with high computational accuracy. And more nodes are distributed in the problem domain, higher computational accuracy of the solution will obtained, which shows that the method in this paper has better convergence.

Point interpolation collocation method for the solution of partial differential equations

High dynamic range (HDR) image, when displayed on ordinary dynamic range devices, its visual effect will be weakened. To relieve this problem, this paper proposes a new method which combines global and local tone mapping together, to produce HDR image of better visual effect The first step is to convert a HDR image to ordinary dynamic range using either global mapping or local mapping, depending on the mean intensity of the image. Then, Curvelet transform is applied to the mapped result to obtain a set of transform coefficients which are fused. Finally, taking advantages of both global and local mapping methods, a high-quality image is obtained. Experimental results show that the processed method provides images with more details and better visual effect, compared with other mapping methods such as Reinhard tonal mapping.

A review of the meshless formulations based on local boundary integral equation (LBIE) methods is presented. Physical quantities are approximated by the moving least-squares method. A summary of recent developments in the application of the LBIE method to potential problems, elastostatics, elastodynamics, thermoelasticity, and plate bending problems is given. The efficiency and generality of the present formulation in a wide class of engineering problems are confirmed.

The local meshless collocation method for numerical simulation of shallow water waves based on generalized equal width (GEW) equation