In this article, we will explore the numerical simulation of the Allen–Cahn equation and provide effective combination methods to efficiently solve it. The Allen–Cahn equation, an equation of mathematical physics, represents a singularly perturbed reaction–diffusion phenomenon that elucidates the phase separation mechanism occurring in multi-component alloy systems. Finding a numerical solution for the Allen–Cahn equation presents a difficult challenge for computational science and engineering researchers. To solve the Allen–Cahn equation, we will use Lie–Trotter’s and Strang’s splitting techniques combined with the radial basis function partition of unity method. We will discretize the problem’s spatial domain using the classical and direct RBF-PU methods. We will also provide suitable theorems for analytical support of the presented numerical methods. We will provide several numerical examples which include examples with exact solutions to check the accuracy and efficiency of the method and examples in the field of phase transition to demonstrate the adaptability, performance, and efficiency of the proposed method.

We present an effective practical approach for solving multi-order nonlinear fractional differential equations. Our method uses integrated Bernoulli polynomials and comes with a comprehensive convergence analysis. The integrated Bernoulli polynomials are combined with the collocation and simple iteration methods to approximate the solutions. We have provided several numerical examples to demonstrate the effectiveness, strength, and flexibility of our method. The results obtained from implementing the method have been compared with exact solutions and results obtained from other methods mentioned in the articles.

Heat transfer can be enhanced in various equipment by utilizing fins that have pores in their media. Efforts have been made to solve the porous fins problem by numerical, semianalytical, and analytical methods. In this paper, after the extraction of equations and the expression of boundary conditions, the Sinc collocation method is employed for problem solving the first time. By using the properties of the Sinc collocation method, the nonlinear problem is discritized into a nonlinear system of equations. The results support the accuracy and speed of the solution. The effects of effective physical parameters such as fins porosity, thermal conductivity of fin, Raley's number, and Darcy parameter have been considered in solving the problem. Also, the variations in temperature distribution, efficiency, and performance of fins are investigated. Therefore, changes in the parameters of the solving method indicate the convergence of the problem in the domain of many issues. In general, the results show that the proposed method can be considered as a powerful method for solving different types of fines.

An iterative shooting-like method based on the shifted Chebyshev polynomials is proposed for solving the nonlinear fractional boundary value problems with the multi-point boundary conditions. The proposed method can be applied easily to various nonlocal linear boundary conditions. Here, we investigate the convergence of the proposed method for the nonlinear problems with the multi-point boundary condition. We investigate the convergence of the proposed method for the nonlinear problems with the multi-point boundary conditions. The obtained numerical results confirm the theoretical results and show the efficiency and accuracy of the proposed method.

An iterative kernel-based method is proposed for solving the nonlinear generalized Benjamin-Bona-Mahony-Burgers equation in regular and irregular domains. The method is based on the positive definite kernels pseudospectral method. We have used an iterative linearization scheme to overcome the nonlinearity of the problem. The convergence of the linearization scheme is proved. The space-time kernels are used for the full discretization of the problems in the temporal and spatial domains. Here, we construct the multidimensional kernels using the product of one-dimensional kernels. Numerical results and comparisons are presented for the several two and three-dimensional problems with the various domains. The obtained results confirm the efficiency and accuracy of the proposed method.

In this paper, we study the uniqueness and multiplicity of the solutions of a strongly nonlinear mathematical model arising from chemical reactor theory. The analysis is based on the reproducing kernel Hilbert space method. The main aim of this work is to find how much information can be predicted using numerical computations. The dependence of the number of solutions on the parameters of the model is also studied. Furthermore, the analytical approximations of all branches of solutions can be calculated by the proposed method. The convergence of the proposed method is proved. Some numerical simulations are presented.

In this work, we have studied a numerical scheme based on Sinc collocation method to solve a class of nonlinear singularly perturbed boundary value problems. The solution of the problems exhibit a boundary layer on the both sides or one side of the domain due to the presence of perturbation parameter ?. The Sinc method can control the oscillations in computed solutions at boundary layer regions naturally because the distribution of Sinc points is denser at near the boundaries. The convergence analysis is discussed and the method is shown to be an exponential convergent. The numerical results support the theoretical results and illustrate the efficiency and accuracy of the method compared with the results in the existing methods.

In this study, TiO 2 /ZnO nanostructures were synthesized by sol-gel solid-state dispersion method and hydrothermal methods for degradation of Rhodamine B (RhB).The ZnO weight percentage, calcination temperature and irradiation time were investigated for synthesizing the different catalyst samples.Then, the X-ray diffraction, Brunauer-Emmett-Teller, Fourier-transform infrared spectroscopy and field emission scanning electron microscopy analysis was performed to characterize the as-prepared catalysts.The experimental design was utilized for photocatalytic degradation of RhB.The effects of operating parameters such as pH, irradiation time, initial RhB concentration and catalyst concentration were investigated through this study.The degradation tests were performed by UV of 24 W. The best degradation performance of 99.28% was obtained.The results of characterization and degradation via two methods of preparation including the; sol-gel and hydrothermal were analyzed and compared.Besides, the optimum conditions for reaching the highest RhB degradation were reported.

In this study, sinc‐Galerkin and Sinc collocation methods coupled with double exponential transformation have been developed to obtain a numerical solution of nonlinear two‐point boundary value problem arising in steady state nonlinear reaction diffusion equation containing a nonlinear term related to Michaelis‐Menten of the enzymatic reaction. Application of these methods reduce the solution of the Michaelis‐Menten equation to the solution of nonlinear system of algebraic equations. The upper bound of the error was calculated. Two problems of this model, chosen from the literature, were considered and for several parameters, numerical solution was shown. Obtained numerical results and error of the methods show that these methods are very fast with high efficiency in convergence.

In this study, improved Sinc-Galerkin and Sinc-collocation methods are developed based on double exponential transformation to solve a one-dimensional Bratu-type equation. The properties of these methods are used to reduce the solution of the nonlinear problem to the solution of nonlinear algebraic equations. For simplicity in solving the nonlinear system, a matrix vector form of the nonlinear system is found. The upper bound of the error for the Sinc-Galerkin is determined. Also the numerical approximations are compared with the best results reported in the literature. The results confirm that both the Sinc-Galerkin and the Sinc-collocation methods have the same accuracy, but they are significantly more accurate than the other existing methods.

This work is motivated by the frequent occurrence of boundary value problems with various boundary conditions in the modeling of some problems in engineering and physical science. Here we propose a new technique to force the positive definite kernels such as some radial basis functions to satisfy the boundary conditions exactly. It can improve the applications of existing methods based on positive definite kernels and radial basis functions especially the kernel based pseudospectral method for handling the differential equations with more complicated boundary conditions. In the proposed technique some new kernels are constructed using the positive definite kernels in a manner that they satisfy the required conditions. In addition, we prove the positive definiteness of the newly constructed kernel, and also the non-singularity of the collocation matrix is proved under some conditions. The proposed method is verified through the numerical solution of some benchmark problems such as a singularly perturbed steady-state convection–diffusion problem, two and three dimensional Poissons equations with various boundary conditions.

In this paper, we combine the theory of the reproducing kernel Hilbert spaces with the field of collocation methods to solve boundary value problems with a special emphasis on the reproducing property of kernels. Using the reproducing property of the kernels, a new efficient algorithm is proposed to obtain the cardinal functions of a reproducing kernel Hilbert space, which can be applied conveniently for multi-dimensional domains. The differentiation matrices are constructed and also a pointwise error estimate of applying them is derived. In addition, we prove the non-singularity of the collocation matrix. The proposed method is truly meshless, and can be applied conveniently and accurately for high order and also multi-dimensional problems. Numerical results are presented for the several problems such as second- and fifth-order two-point boundary value problems, one- and two-dimensional unsteady Burgers’ equations, and a three-dimensional parabolic partial differential equation. In addition, we compare the numerical results with the best-reported results in the literature to show the high accuracy and efficiency of the proposed method.

This paper presents a computational method to solve nonlinear boundary value problems with multi-point boundary conditions. These problems have important applications in the theoretical physics and engineering problems. The method is based on reproducing kernel Hilbert spaces operational matrices and an iterative technique is used to overcome the nonlinearity of the problem. Furthermore, a rigorous convergence analysis is provided and some numerical tests reveal the high efficiency and versatility of the proposed method. The results of numerical experiments are compared with analytical solutions and the best results reported in the literature to confirm the good accuracy of the presented method.

In this investigation, the Sinc collocation method based on double exponential transformation is developed to solve the Troesche's problem. Properties of this method are utilized to reduce the system of strongly nonlinear two point boundary value problem to same nonlinear algebraic equations. Combining double exponential transformation through Sinc collocation method causes the remarkable results. To illustrate the high accuracy of the method, the obtained solutions are compared with results of other methods in open literature. The demonstrated results show the simplicity and considerably accuracy of this method in comparison with other methods.

Sinc-Galerkin method based upon double exponential transformation for solving Troesch's problem is given in this study. Properties of the Sinc-Galerkin approach are utilized to reduce the solution of nonlinear two-point boundary value problem to same nonlinear algebraic equations, also, the matrix form of the nonlinear algebraic equations is obtained.The error bound of the method is found. Moreover, in order to illustrate the accuracy of presented method, the obtained results compare with numerical results in the open literature. The demonstrated results confirm that proposed method is considerably efficient and accurate.

A study of Sinc-Galerkin method based on double exponential transformation for solving a class of nonlinear weakly singular two point boundary value problems with non-homogeneous boundary conditions is given. The properties of the Sinc-Galerkin approach are utilized to reduce the computation of nonlinear problem to nonlinear system of equations with unknown coefficients. This method tested on several test examples. We compare our numerical results with several numerical results of existing methods. The demonstrated results confirm that proposed method is considerably efficient, accurate nature and rapidly converge.