This article presents an alternative explicit solution to the one-dimensional Bratu problem obtained through an adaptation of the Kudryashov expansion method. The analysis revisits the classical bifurcation structure of the problem, demonstrating that the set of solutions contains zero, one, or two branches depending on the value of the critical parameter λ _c . The explicit formulation developed here provides additional analytical insight into the Bratu equation and offers a useful tool for further studies of nonlinear boundary value problems with similar exponential nonlinearities.

This paper introduces an enhanced Iterative Finite Difference (IFD) method for efficiently solving strongly nonlinear, time-dependent problems. Extending the original IFD framework for nonlinear ordinary differential equations, we generalize the approach to address nonlinear partial differential equations with time dependence. An improved strategy is developed to achieve high-order accuracy in space and time. A finite difference discretization is applied at each iteration, yielding a flexible and robust iterative scheme suitable for complex nonlinear equations, including the Sine-Gordon, Klein–Gordon, and generalized Sinh-Gordon equations. Numerical experiments confirm the method’s rapid convergence, high accuracy, and low computational cost.

This paper introduces a novel approach for solving systems of boundary value problems (BVPs) by employing the recently developed Discontinuous Galerkin (DG) method, which removes the necessity for auxiliary variables.This marks the initial installment in a sequence of publications dedicated to exploring DG methods for solving partial differential equations (PDEs).In fact, through a systematic application of the DG method to each spatial variable within the PDE, employing the method of lines, we convert the initial problem into a system of ordinary differential equations (ODEs).In the current study, we developed a global error analysis of the DG method applied to systems of ODEs.Our analysis shows that using p-degree piecewise polynomials and h-mesh step size, the DG solutions achieve optimal O(h p+1 ) convergence rates in the L 2 -norm.

In this paper, we develop a novel discontinuous Galerkin (DG) finite element method for solving the Poisson's equation uxx+uyy=f(x,y) on Cartesian grids. The proposed method consists of first applying the standard DG method in the x-spatial variable leading to a system of ordinary differential equations (ODEs) in the y-variable. Then, using the method of line, the DG method is directly applied to discretize the resulting system of ODEs. In fact, we propose a fully DG scheme that uses p-th and q-th degree DG methods in the x and y variables, respectively. We show that, under proper choices of numerical fluxes, the method achieves optimal convergence rate in the L2-norm of O(hp+1)+O(kq+1) for the DG solution, where h and k denote, respectively, the mesh step sizes for the x and y variables. Our theoretical results are validated through several numerical experiments.

In this paper, we present an innovative approach to solve a system of boundary value problems (BVPs), using the newly developed discontinuous Galerkin (DG) method, which eliminates the need for auxiliary variables. This work is the first in a series of papers on DG methods applied to partial differential equations (PDEs). By consecutively applying the DG method to each space variable of the PDE using the method of lines, we transform the problem into a system of ordinary differential equations (ODEs). We investigate the convergence criteria of the DG method on systems of ODEs and generalize the error analysis to PDEs. Our analysis demonstrates that the DG error’s leading term is determined by a combination of specific Jacobi polynomials in each element. Thus, we prove that DG solutions are superconvergent at the roots of these polynomials, with an order of convergence of O(hp+2).

In this paper, we develop, for the first time, a closed-form solution of Bratu’s problem u”+λ e^u=0 in terms of elementary functions. We use several changes of variables to reduce the nonlinear equation to a linear first-order differential equation. We then deduce the exact solution of Bratu’s problem. From the closed-form solution, we prove that Bratu’s problem has zero solution, a unique solution and two bifurcated solutions, respectively, when λ >λ _c , λ =λ _c and 0<λ <λ _c , where λ _c=3.5138307191251608 is the critical value. A numerical evaluation of the closed-form solution is presented to show the bifurcation behavior.

We consider a novel approach to providing highly accurate analytic solutions to non-classical, non-linear problems. This approach is then implemented to the highly sensitive Troesch-problem, which possesses a boundary layer at the right-end: we determine an upper and lower envelope solution, and compare the average to existing numerical results. Computer simulations show that the proposed analytic solution is highly accurate when compared to the existing benchmark. This confirms that the proposed approach to deal with such non-linear problems can be used to treat similar non-classical boundary-value problems.

In this paper, we present and analyze a new space–time ultra-weak discontinuous Galerkin (UWDG) finite element method for the second-order wave equation in one space dimension. The UWDG finite element approximations are used in space variable and also for the temporal approximation. The space–time UWDG discretization is presented in detail, including the definition of the numerical fluxes, which are necessary to obtain optimal error estimates. The proposed scheme can be made arbitrarily high-order accurate in both space and time. The error estimates of the presented semi-discrete and fully-discrete schemes are both analyzed. Several numerical examples are provided to confirm the theoretical results.

In this paper, we propose a discontinuous Galerkin (DG) method for systems of stochastic differential equations (SDEs) driven by m-dimensional Brownian motion. We first construct a new approximate system of SDEs on each element using whose converges to the solution of the original system. The new system is then discretized using the standard DG method for deterministic ordinary differential equations (ODEs). For the case of additive noise, we prove that the proposed scheme is convergent in the mean-square sense. Our numerical experiments suggest that our results hold true for the case of multiplicative noise as well. Several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method. In particular, the proposed scheme is illustrated by considering different examples arising in population biology, physics, and mathematical finance.

In this paper, we present a discontinuous Galerkin (DG) finite element method for stochastic ordinary differential equations (SDEs) driven by additive noises. First, we construct a new approximate SDE whose solution converges to the solution of the original SDE in the mean-square sense. The new approximate SDE is obtained from the original SDE by approximating the Wiener process with a piecewise constant random process. The new approximate SDE is shown to have better regularity which facilitates the convergence proof for the proposed scheme. We then apply the DG method for deterministic ordinary differential equations (ODEs) to approximate the solution of the new SDE. Convergence analysis is presented for the numerical solution based on the standard DG method for ODEs. The orders of convergence are proved to be one in the mean-square sense, when p-degree piecewise polynomials are used. Finally, we present several numerical examples to validate the theoretical results. Unlike the Monte Carlo method, the proposed scheme requires fewer sample paths to reach a desired accuracy level.

In this paper, a deterministic model is formulated in the aim of performing a thorough investigation of the transmission dynamics of influenza. The main advantage of our model compared to existing models is that it takes into account the effects of hospitalization as well as the diffusion. The proposed model consisting of a dynamical system of partial differential equations with diffusion terms is numerically solved using fast and accurate numerical techniques for partial differential equations. Furthermore, the basic reproduction number that guarantees the local stability of disease-free steady state without diffusion term is calculated. Various numerical simulation for different values of the model input parameters are finally presented in order to show the effect of the effective contact rate on the steady state of the different population compartments.

In this paper, we propose an iterative finite difference (IFD) scheme to simultaneously approximate both branches of a two-branched solution to the one-dimensional Bratu's problem. We first introduce a transformation to convert Bratu's problem into a simpler one. The transformed nonlinear ordinary differential equation is discretized using the Newton–Raphson–Kantorovich approximation in function space. The convergence of the sequence of approximations is proved to be quadratic. Then, we apply the classical finite difference method to approximate the sequence of approximations. The proposed new scheme has two main advantages. First, it produces accurate numerical solutions with low computational cost. Second, it is able to compute the two branches of the solution of Bratu's problem, even for small values of the transition parameter λ, where the numerical computation of the upper branch of the solution becomes challenging. Numerical examples are provided to show the efficiency and accuracy of the proposed scheme.

In this paper, we propose fast iterative methods based on the Newton–Raphson–Kantorovich approximation in function space [Bellman and Kalaba, (1965)] to solve three kinds of the Lane–Emden type problems. First, a reformulation of the problem is performed using a quasilinearization technique which leads to an iterative scheme. Such scheme consists in an ordinary differential equation that uses the approximate solution from the previous iteration to yield the unknown solution of the current iteration. At every iteration, a further discretization of the problem is achieved which provides the numerical solution with low computational cost. Numerical simulation shows the accuracy as well as the efficiency of the method.

In this paper, we study the nonlinear boundary-layer equation of Falkner-Skan defined on a semi-infinite domain. An iterative finite difference (IFD) scheme is proposed to numerically solve such nonlinear ordinary differential equation. A computational iterative scheme is developed based on Newton-Kantorovich quasilinearization. At every iteration, the obtained linearized differential equation is numerically solved using the standard finite difference method. Numerical experiments show the accuracy and efficiency of the method compared to existing solvers. The computation is performed for different parameter values, including the special case of Blasius problem.

Finding numerical solutions to the Troesch’s problem is known to be challenging, especially when the sensitivity parameter \(\lambda \) is large. In this manuscript, we propose a numerical method for solving the Troesch’s problem which combines efficiency and accuracy, even for large sensitivity parameter. Our method can be summarized as a finite difference method formulated on a Shishkin mesh, which is piecewise uniform with smaller step size in the boundary layer in the neighborhood of the point \(x=1\). In fact, we treat the Troesch’s problem as a singularly perturbed problem with a special transition point on the mesh. The transition point for the Shishkin mesh is first defined and computed. The application of the finite difference method on the piecewise uniform mesh leads to a nonlinear matrix system, which is solved numerically using the generalized Newton’s method. While, existing numerical solvers suffer significantly when sensitivity parameter \(\lambda \) becomes large and fail to provide accurate numerical solutions for \(\lambda >20\) (Chang, Appl Math Comput 216(11):3303–3306, 2010; Khuri and Sayfy, Math Comput Model 54(9):1907–1918, 2011; Raja, Inf Sci 279:860–873, 2014; Temimi, Appl Math Comput 219(2):521–529, 2012), our method provides accurate numerical solution for large values of this sensitivity parameter, up to \(\lambda =100\). Moreover, the implementation of the method is straight forward and the computation cost is low. Numerical experiments show that the method provides accurate solutions for different values of the sensitivity parameter \(\lambda \).

In this paper, we study the convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a linear convection-diffusion problem in one-dimensional setting. We prove that the DG solution and its derivative exhibit optimal O(h) and O(h) convergence rates in the L-norm, respectively, when p-degree piecewise polynomials with p ≥ 1 are used. We further prove that the p-degree DG solution and its derivative are O(h) superconvergent at the downwind and upwind points, respectively. Numerical experiments demonstrate that the theoretical rates are optimal and that the DG method does not produce any oscillation. We observed optimal rates of convergence and superconvergence even in the presence of boundary layers when Shishkin meshes are used.

The main purpose of this paper is to examine the influence of time delay associated with a semi-active variable viscous (SAVV) damper on the response of seismically excited linear and nonlinear structures. The maximum time delay is estimated on the basis of stability criteria, which consist of analyses of structural modal properties. Numerical computation of the critical time delay is performed by using dichotomic approach, which is based on multiple solving of the eigenvalue problem. Simulation results indicate that variable dampers can be effective in reducing the seismic response of structures, and that time-delay effects are important factors in control design of seismically excited structures. Furthermore, simulation results show degradation of performance whenever the actual delay exceeds the calculated critical time delay, which shows the accuracy and reliability of the proposed approach.

In this paper, we investigate the superconvergence criteria of the discontinuous Galerkin (DG) method applied to one-dimensional nonlinear differential equations. We show numerically that the p-degree finite element (DG) solution is \(O(\Delta x^{p+2})\) superconvergent at the roots of specific combined Jacobi polynomials. Moreover, we used these results to construct efficient and asymptotically exact a posteriori error estimates.

 Abstract—In this paper, we propose a new discontinuous Galerkin finite element (DG) method to solve Troesch’s problem, which is highly sensitive for large values of the parameter. This twopoint boundary value problem has been heavily studied since 1960, however, only a few papers have provided a reliable solution for high sensitivity. Therefore, we developed the DG method which has been proved its efficiency for many decades to be a new numerical solver. We demonstrate through computational results compared with those computed by other methods, that the discontinuous Galerkin method provides a quite efficient, accurate and reliable solution. Thus, the DG method is an attractive and competitive alternative to other numerical and semi-analytical techniques to solve highly sensitive nonlinear problems.

This paper aims to investigate the differences of national culture between countries belonging to similar cluster and sharing some attributes such as language, religion or geographic location based on two of Hofstede’s cultural dimensions (Hofstede, 1980; Hofstede and Hofstede, 1991), power distance and uncertainty avoidance, within the context of Takaful industry (Islamic Insurance). Hofstede classified all Arabic countries within same cluster and gave them same scores for his cultural dimensions. In this study only power distance and uncertainty avoidance were chosen as the dimensions of this study for many reasons of which the ability of power distance to eliminate the information gap in all directions such as informal, horizontal, top and bottom information sharing and gathering and also to strengthen the quality of decisions it makes Khatri (2009), and uncertainty avoidance which stimulates organisations to provide customer activities Todeva (1999). This research was conducted on two Arabic countries Kuwait and Egypt.