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This study aims to apply a numerical meshless method, namely, the boundary knot method (BKM) combined with the meshless analog equation method (MAEM) in space and use a semi-implicit scheme in time for finding a new numerical solution of the advection–reaction–diffusion and reaction–diffusion systems in two-dimensional spaces, which arise in biology.

First, the BKM is applied to approximate the spatial variables of the studied mathematical models. Then, this study derives fully discrete scheme of the studied models using a semi-implicit scheme based on Crank–Nicolson idea, which gives a linear system of algebraic equations with a non-square matrix per time step that is solved by the singular value decomposition. The proposed approach approximates the solution of a given partial differential equation using particular and homogeneous solutions and without considering the fundamental solutions of the proposed equations.

This study reports some numerical simulations for showing the ability of the presented technique in solving the studied mathematical models arising in biology. The obtained results by the developed numerical scheme are in good agreement with the results reported in the literature. Besides, a simulation of the proposed model is done on buttery shape domain in two-dimensional space.

This study develops the BKM combined with MAEM for solving the coupled systems of (advection) reaction–diffusion equations in two-dimensional spaces. Besides, it does not need the fundamental solution of the mathematical models studied here, which omits any difficulties.

This study aims to use the polynomial approximation method based on the Pascal polynomial basis for obtaining the numerical solutions of partial differential equations. Moreover, this method does not require establishing grids in the computational domain.

In this study, the authors present a meshfree method based on Pascal polynomial expansion for the numerical solution of the Sobolev equation. In general, Sobolev-type equations have several applications in physics and mechanical engineering.

The authors use the Crank-Nicolson scheme to discrete the time variable and the Pascal polynomial-based (PPB) method for discretizing the spatial variables. But it is clear that increasing the value of the final time or number of time steps, will bear a lot of costs during numerical simulations. An important purpose of this paper is to reduce the execution time for applying the PPB method. To reach this aim, the proper orthogonal decomposition technique has been combined with the PPB method.

The developed procedure is tested on various examples of one-dimensional, two-dimensional and three-dimensional versions of the governed equation on the rectangular and irregular domains to check its accuracy and validity.

The purpose of this study is to develop a new numerical algorithm to simulate the phase-field model.

First, the derivative of the temporal direction is discretized by a second-order linearized finite difference scheme where it conserves the energy stability of the mathematical model. Then, the isogeometric collocation (IGC) method is used to approximate the derivative of spacial direction. The IGC procedure can be applied on irregular physical domains. The IGC method is constructed based upon the nonuniform rational B-splines (NURBS). Each curve and surface can be approximated by the NURBS. Also, a map will be defined to project the physical domain to a simple computational domain. In this procedure, the partial derivatives will be transformed to the new domain by the Jacobian and Hessian matrices. According to the mentioned procedure, the first- and second-order differential matrices are built. Furthermore, the pseudo-spectral algorithm is used to derive the first- and second-order nodal differential matrices. In the end, the Greville Abscissae points are used to the collocation method.

In the numerical experiments, the efficiency and accuracy of the proposed method are assessed through two examples, demonstrating its performance on both rectangular and nonrectangular domains.

This research work introduces the IGC method as a simulation technique for the phase-field crystal model.

The main aim of the current paper is to find a numerical plan for hydraulic fracturing problem with application in extracting natural gases and oil.

First, time discretization is accomplished via Crank-Nicolson and semi-implicit techniques. At the second step, a high-order finite element method using quadratic triangular elements is proposed to derive the spatial discretization. The efficiency and time consuming of both obtained schemes will be investigated. In addition to the popular uniform mesh refinement strategy, an adaptive mesh refinement strategy will be employed to reduce computational costs.

Numerical results show a good agreement between the two schemes as well as the efficiency of the employed techniques to capture acceptable patterns of the model. In central single-crack mode, the experimental results demonstrate that maximal values of displacements in x- and y- directions are 0.1 and 0.08, respectively. They occur around both ends of the line and sides directly next to the line where pressure takes impact. Moreover, the pressure of injected fluid almost gained its initial value, i.e. 3,000 inside and close to the notch. Further, the results for non-central single-crack mode and bifurcated crack mode are depicted. In central single-crack mode and square computational area with a uniform mesh, computational times corresponding to the numerical schemes based on the high order finite element method for spatial discretization and Crank-Nicolson as well as semi-implicit techniques for temporal discretizations are 207.19s and 97.47s, respectively, with 2,048 elements, final time T = 0.2 and time step size τ = 0.01. Also, the simulations effectively illustrate a further decrease in computational time when the method is equipped with an adaptive mesh refinement strategy. The computational cost is reduced to 4.23s when the governed model is solved with the numerical scheme based on the adaptive high order finite element method and semi-implicit technique for spatial and temporal discretizations, respectively. Similarly, in other samples, the reduction of computational cost has been shown.

This is the first time that the high-order finite element method is employed to solve the model investigated in the current paper.

The current paper aims to develop a reduced order discontinuous Galerkin method for solving the generalized Swift–Hohenberg equation with application in biological science and mechanical engineering. The generalized Swift–Hohenberg equation is a fourth-order PDE; thus, this paper uses the local discontinuous Galerkin (LDG) method for it.

At first, the spatial direction has been discretized by the LDG technique, as this process results in a nonlinear system of equations based on the time variable. Thus, to achieve more accurate outcomes, this paper uses an exponential time differencing scheme for solving the obtained system of ordinary differential equations. Finally, to decrease the used CPU time, this study combines the proper orthogonal decomposition approach with the LDG method and obtains a reduced order LDG method. The circular and rectangular computational domains have been selected to solve the generalized Swift–Hohenberg equation. Furthermore, the energy stability for the semi-discrete LDG scheme has been discussed.

The results show that the new numerical procedure has not only suitable and acceptable accuracy but also less computational cost compared to the local DG without the proper orthogonal decomposition (POD) approach.

The local DG technique is an efficient numerical procedure for solving models in the fluid flow. The current paper combines the POD approach and the local LDG technique to solve the generalized Swift–Hohenberg equation with application in the fluid mechanics. In the new technique, the computational cost and the used CPU time of the local DG have been reduced.

This paper aims to propose an adaptive method for the numerical solution of the shallow water equations (SWEs). The authors provide an arbitrary high-order method using high-order spline wavelets. Furthermore, they use a non-linear shock capturing (SC) diffusion which removes the necessity of post-processing.

The authors use a space-time weak formulation of SWEs which exploits continuous Galerkin (cG) in space and discontinuous Galerkin (dG) in time allowing time stepping, also known as cGdG. Such formulations along with SC term have recently been proved to ensure the stability of fully discrete schemes without scarifying the accuracy. However, the resulting scheme is expensive in terms of number of degrees of freedom (DoFs). By using natural adaptivity of wavelet expansions, the authors devise an adaptive algorithm to reduce the number of DoFs.

The proposed algorithm uses DoFs in a dynamic way to capture the shocks in all time steps while keeping the representation of approximate solution sparse. The performance of the proposed scheme is shown through some numerical examples.

An incorporation of wavelets for adaptivity in space-time weak formulations applied for SWEs is proposed.

This paper aims to show that the variational iteration method (VIM) and the homotopy perturbation method (HPM) are powerful and suitable methods to solve the Fornberg‐Whitham equation.

Using HPM the explicit exact solution is calculated in the form of a quickly convergent series with easily computable components. Also, by using VIM the analytical results of this equation have been obtained in terms of convergent series with easily computable components.

Numerical solutions obtained by these methods are compared with the exact solutions, revealing that the obtained solutions are of high accuracy.

Also the results show that the introduced methods are efficient tools for solving the nonlinear partial differential equations.

Rosenau‐Hyman equation was discovered as a simplified model to study the role of nonlinear dispersion on pattern formation in liquid drops. Also, this equation has important roles in the modelling of various problems in physics and engineering. The purpose of this paper is to present the solution of Rosenau‐Hyman equation.

This paper aims to present the solution of the Rosenau‐Hyman equation by means of semi‐analytical approaches which are based on the homotopy perturbation method (HPM), variational iteration method (VIM) and Adomian decomposition method (ADM).

These techniques reduce the volume of calculations by not requiring discretization of the variables, linearization or small perturbations. Numerical solutions obtained by these methods are compared with the exact solutions, revealing that the obtained solutions are of high accuracy. These results reveal that the proposed methods are very effective and simple to perform.

Efficient techniques are developed to find the solution of an important equation.

The purpose of this paper is to show that the meshless method based on radial basis functions (RBFs) collocation method is powerful, suitable and simple for solving one and two dimensional time fractional telegraph equation.

In this method the authors first approximate the time fractional derivatives of mentioned equation by two schemes of orders O(τ3−α) and O(τ2−α), 1/2<α<1, then the authors will use the Kansa approach to approximate the spatial derivatives.

The results of numerical experiments are compared with analytical solution, revealing that the obtained numerical solutions have acceptance accuracy.

The results show that the meshless method based on the RBFs and collocation approach is also suitable for the treatment of the time fractional telegraph equation.

The purpose of this paper is to implement the Ritz‐Galerkin method in Bernstein polynomial basis to give approximation solution of a parabolic partial differential equation with non‐local boundary conditions.

The properties of Bernstein polynomial and Ritz‐Galerkin method are first presented, then the Ritz‐Galerkin method is utilized to reduce the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

The authors applied the method presented in this paper and solved three test problems.

This is the first time that the Ritz‐Galerkin method in Bernstein polynomial basis is employed to solve the model investigated in the current paper.