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• Different types of RBFs are utilized for localized meshless methods in the context of interface PDEs. • Both sharp edged and smooth edged interfaces are taken into account. • Fully scattered and uniform nodal distributions are considered. • Numerical order of convergence are calculated. • Shape parameter sensitivity analysis of the meshless methods is performed. In the present paper, we develop a local meshless procedure for solving a steady state two-dimensional interface problem having discontinuous coefficients and curved interfaces with sharp corners. The proposed local meshless methods are based on three types of radial basis functions (RBFs): a local meshless method based on multiquadric RBF (LMM1P), a local meshless method based on integrated multiquadric RBF (LMM2P) and a local meshless method based on hybrid Gaussian-Cubic RBF (LMM3P). Stencils are designed at the interface and interior regions to cope with discontinuities and sharp corners. Due to the localized nature of the procedure and a sparse matrix representation, the local meshless methods become computationally less expensive than global meshless methods. The methods are augmented with linear polynomial to improve accuracy and ensure stable computation. Comparison with some existing versions of finite element methods is also performed to show better accuracy of the proposed meshless methods. Accuracies of the proposed local meshless methods are also compared among themselves. Flexibility of the meshless methods with respect to complex geometries, adapting to different shapes of the interfaces and selection of the shape parameter is also considered. [ABSTRACT FROM AUTHOR]

• Novel computing method is designed for nonlinear Thomas-Fermi system. • Neural networks, active set algorithm, interior-point and sequential quadratic programming are exploited for the system. • Comparison from exact and numerical results verify the correctness of the scheme. • Accuracy and convergence of the solver is validated from the statistical analysis. • Proposed scheme is an alternate, accurate, and reliable methods for such systems. We propose a computational intelligence technique to solve the nonlinear Thomas-Fermi equation for a finite interval. This technique uses an optimized approximation to reduce the error between the reference solution and optimized numerical solution with trigonometric and hyperbolic functions. Computational intelligence technique is based on optimization tools like Active Set Technique (AST), Interior Point Technique (IPT) with Sequential Quadratic Programming (SQP) through artificial neural networks (ANN). Furthermore, the proposed model results can be applied in atomic models phenomena. Numerical results showed that the optimizers provided a high level of accuracy, as well as the efficiency with marginal computational requirements infinite domain. Statistical analysis is presented, particularly fitting of normal distribution based on absolute errors (AEs) of proposed solvers and Chi-square distribution data curve fitting. It provides evidence that the present solution is highly accurate in the context of confidence bands. [ABSTRACT FROM AUTHOR]

A modified form of Levin's method based on multi-quadric radial basis function is presented for numerical evaluation of integrals having squared oscillatory Bessel function of the first kind of order μ and oscillatory Airy function. The method converts the oscillatory integrals into a system of coupled ordinary differential equations (ODEs) and subsequently, numerical solution of the coupled ODEs is obtained by meshless collocation procedure. A multi-resolution quadrature is used to tackle singularity of the proposed method. Theoretical error analysis of the proposed methods is performed in asymptotic sense. Numerical test problems are included to verify accuracy and efficiency of the new methods in the context of oscillatory Airy and Bessel integrals. [ABSTRACT FROM AUTHOR]

Differential equations with nonlocal boundary conditions are used to model a number of physical phenomena encountered in situations where data on the boundary cannot be measured directly. This study explores numerical solutions to elliptic, parabolic and hyperbolic equations with two different types of nonlocal integral boundary conditions. The numerical solutions are obtained using the Haar wavelet collocation method with the aid of Finite Differences for time derivatives. The method is applicable to both linear and nonlinear problems. To obtain the numerical solutions, Gauss elimination method is used for linear and Newton’s method for nonlinear differential equations. The validity of the proposed method is demonstrated by solving several benchmark test problems from the literature: two elliptic linear and two nonlinear samples covering both types of nonlocal integral boundary conditions; one nonlinear and two linear test problems for parabolic partial differential equations; two linear samples for hyperbolic partial differential equations. The accuracy of the method is verified by comparing the numerical results with the analytical solutions. The numerical results confirm that the method is simple and effective. [ABSTRACT FROM AUTHOR]

Topology optimization is a renowned structural optimization approach used to compute the optimal topology for the enhancement of structural performance. It has been in common practice in different engineering fields such as automobile and aerospace. However, there still exist gaps between topology optimization and its engineering operations, which considerably impedes topology optimization's applications. One of these gaps is how to make (especially finite element-based) topology optimization results machine-readable, which means, how to transform these into computer-aided design (CAD) models, which are ready-to-use models for manufacturing. In the proposed work the authors adopted a unique methodology that plugs the gap between topology optimization and manufacturing. This methodology seamlessly integrates the structural analysis tool, boundary description tool and standard CAD geometry representation; equipped with the Hausdorff distance approach for 3D printing/additive manufacturing of optimal structures. Initially, we extract the skeleton of the optimal design (in the form of a points cloud) using the level set method, followed by an interpolation technique with shape preservation property, the extracted skeleton is made dense. Incorporating the shape and curvature information of the data set, an adaptive B-spline approximation is devised for fitting a smooth curve through the points cloud. This approach interprets topology optimization results as a parametric CAD model with minimum possible control points and minimum approximation error. The CAD-based optimal designs are then used for additive manufacturing. The numerical results exhibit in detail the validity and accuracy of the proposed method. • High accuracy structural response evaluation through Element free Galerkin method for structural evolution. • Conversion of topology optimization results into parametric CAD model through a new Hausdorff distance and standard deviation measures. • Adaptive B-spline approximation using the local curvature maximum points based relative curvature and relative arc-length. • Interpretation of topology optimization results as parametric CAD model with minimum possible control points and minimum approximation error. • Additive Manufacturing of the CAD based optimal designs. [ABSTRACT FROM AUTHOR]

Differential equations with nonlocal boundary conditions are used to model a number of physical phenomena encountered in situations where data on the boundary cannot be measured directly. This study explores numerical solutions to elliptic, parabolic and hyperbolic equations with two different types of nonlocal integral boundary conditions. The numerical solutions are obtained using the Haar wavelet collocation method with the aid of Finite Differences for time derivatives. The method is applicable to both linear and nonlinear problems. To obtain the numerical solutions, Gauss elimination method is used for linear and Newtonâs method for nonlinear differential equations. The validity of the proposed method is demonstrated by solving several benchmark test problems from the literature: two elliptic linear and two nonlinear samples covering both types of nonlocal integral boundary conditions; one nonlinear and two linear test problems for parabolic partial differential equations; two linear samples for hyperbolic partial differential equations. The accuracy of the method is verified by comparing the numerical results with the analytical solutions. The numerical results confirm that the method is simple and effective.

Abstract: A mesh-free method based on radial basis functions (RBFs) is proposed for the solution of complex modified Korteweg–de Vries (CMKdV) equation. The mesh-free algorithm is based on scattered data interpolation along with approximating functions known as radial basis functions (RBFs). A set of scattered nodes provided by initial data is used for solution of the problem. Computational experiments examine the accuracy of the method in terms of error norms , and the three invariants , and . The results obtained by the method are in good agreement with exact solutions and the earlier work. [Copyright &y& Elsevier]

Abstract: In this paper, radial basis function (RBFs) based mesh-free method is implemented to find numerical solution of the Kuramoto–Sivashinsky equations. This approach has an edge over traditional methods such as finite-difference and finite element methods because it does not require a mesh to discretize the problem domain, and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method. The accuracy of the method is assessed in terms of the error norms , number of nodes in the domain of influence, free parameter, dependent parameter RBFs and time step length. Numerical experiments demonstrate accuracy and robustness of the method for solving a class of nonlinear partial differential equations. [Copyright &y& Elsevier]

Abstract: A generalized scheme based on quartic non-polynomial spline functions is proposed. This scheme is designed for numerical solution of singularly perturbed two-point boundary-value problems arising in the study of science and engineering. The scheme leads to a five-diagonal linear system of equations. Convergence analysis of the method is briefly discussed. Two numerical examples each of constant and variable coefficients are given to show practical usefulness of the method. [Copyright &y& Elsevier]

In this research paper, an innovative bio-inspired algorithm based on evolutionary cubic splines method (CSM) has been utilized to estimate the numerical results of nonlinear ordinary differential equation Painlevé-I. The computational mechanism is used to support the proposed technique CSM and optimize the obtained results with global search technique genetic algorithms (GAs) hybridized with sequential quadratic programming (SQP) for quick refinement. Painlevé-I is solved by the proposed technique CSM-GASQP. In this process, variation of splines is implemented for various scenarios. The CSM-GASQP produces an interpolated function that is continuous upto its second derivative. Also, splines proved to be stable than a single polynomial fitted to all points, and reduce wiggles between the tabulated points. This method provides a reliable and excellent procedure for adaptation of unknown coefficients of splines by searching globally exploiting the performance of GA-SQP algorithms. The convergence, exactness and accuracy of the proposed scheme are examined through the statistical analysis for the several independent runs. [ABSTRACT FROM AUTHOR]

In the present paper, a reaction-diffusion epidemic mathematical model is proposed for analysis of the transmission mechanism of the novel coronavirus disease 2019 (COVID-19). The mathematical model contains six-time and space-dependent classes, namely; Susceptible, Exposed, Asymptomatically infected, Symptomatic infected, Quarantine, and Recovered or Removed (SEQI a I s R). The threshold number R 0 is calculated by utilizing the next-generation matrix approach. In addition to the simple explicit procedure, the mathematical epidemiological model with diffusion is simulated through the operator splitting approach based on finite difference and meshless methods. Stability analysis of the disease free and endemic equilibrium points of the model is investigated. Simulation results of the model with and without diffusion are presented in detail. A comparison of the obtained numerical results of both the models is performed in the absence of an exact solution. The correctness of the solution is verified through mutual comparison and partly, via theoretical analysis as well., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2022 Elsevier Ltd. All rights reserved.)