Applied Numerical Mathematics最新文献

We introduce a novel least squares functional specifically formulated to solve linear partial differential equations with operators that have a nonempty null space. Our method involves projecting the solution onto the orthogonal complement of the operator's null space to overcome challenges encountered by conventional numerical methods when nonzero null components are present. We describe the theoretical framework of the proposed method and validate it through numerical examples that show improved accuracy and usability in cases where traditional methods are less effective due to significant null space components. Overall, this approach provides a practical and reliable solution for partial differential equations with substantial null space components.

This paper numerically solves Fredholm integro-differential equations with small parameters and integral boundary conditions. The solution of these equations has a boundary layer at the right boundary. A central difference scheme approximates the second-order derivative, a backward difference (upwind scheme) approximates the first-order derivative, and the trapezoidal rule is used for the integral term with a Shishkin mesh. It is shown that theoretically, the proposed scheme is uniformly convergent with almost first-order convergence. Further to improve the order of convergence from first order to second order, we use the post-processing and the hybrid scheme. Two numerical examples are computed to support the theoretical results.

We propose a nonstandard finite difference scheme for the Susceptible–Infected–Removed (SIR) continuous model. We prove that our discretized system is dynamically consistent with its continuous counterpart and we derive its exact solution. We end with the analysis of the long-term behavior of susceptible, infected and removed individuals, illustrating our results with examples. In contrast with the SIR discrete-time model available in the literature, our new model is simultaneously mathematically and biologically sound.

In this article, we propose a numerical method for a general form of nonlinear cordial Volterra integral equations. We discuss conditions that under them the problem has solutions. Since the existence of solutions for the problem depends on the solvability of a scalar equation and also a linear form of the problem, then we employ quasilinearization technique in which solving a nonlinear problem is reduced to solve a sequence of linear equations. The existence of solutions of linear equations and their quadratically convergence to the solutions of the nonlinear problem is considered. For the numerical solution of the produced linear equations we apply Legendre-collocation method along with a regularization technique for the quadrature formulas. We discuss the error analysis of the collocation method considering that the cordial Volterra integral operators are noncompact. To test the efficiency and accuracy of the proposed method, the solution of different cases of numerical examples are reported.

In this paper, we study a numerical scheme to solve the nonlinear Maxwell's equations. The discrete scheme is based on the triangular spectral element method (TSEM) in space and the exponential time differencing fourth-order Runge-Kutta (ETDRK4) method in time. TSEM has the advantages of spectral accuracy and geometric flexibility. The ETD method involves exact integration of the linear part of the governing equation followed by an approximation of an integral involving the nonlinear terms. The RK4 scheme is introduced for the time integration of the nonlinear terms. The stability region of the ETDRK4 method is depicted. Moreover, the contour integral in the complex plan is utilized and improved to compute the matrix function required by the implementation of ETDRK4. The numerical results demonstrate that our proposed method is of exponential convergence with the order of basis function in space and fourth order accuracy in time.

In this study, a numerical scheme to a system of second-kind linear Fredholm integral equations featuring a Green's kernel function is proposed. This involves introducing Galerkin and iterated Galerkin (IG) methods based on piecewise polynomials to tackle the integral model. A thorough analysis of convergence and error for these proposed methods is carried out. Firstly, the existence and uniqueness of solutions for the Galerkin and iterated Galerkin methods are established. Later, the order of convergence is derived using tools from functional analysis and the boundedness property of Green's kernel. The Galerkin scheme has $O\left(h^{\alpha }\right)$ order of convergence. Next, the superconvergence of the iterated Galerkin (IG) method is established. The IG method exhibits $O\left(h^{\alpha +{\alpha }^{⁎}}\right)$ order of convergence. Theoretical findings are validated through extensive numerical experiments.