Search Results (39)

Singularly perturbed integro-partial differential equations with reaction–diffusion behavior present significant challenges due to boundary layers arising from small perturbation parameters, which complicate the development of accurate and efficient numerical schemes for physical and engineering models. In this study, a uniformly convergent higher-order method is proposed to address these challenges. The approach applies the implicit Euler method for temporal discretization on a uniform mesh and central differences on a Shishkin mesh for spatial approximation, and utilizes the trapezoidal rule for evaluating integral terms; further, extrapolation techniques are incorporated in both time and space to increase accuracy. Numerical analysis demonstrates that the base scheme achieves first-order convergence, while extrapolation enhances convergence rates to second-order in time and fourth-order in space. Theoretical results confirm uniform convergence with respect to the perturbation parameter, and comprehensive numerical experiments validate these analytical claims. Findings indicate that the proposed scheme is reliable, efficient, and particularly effective in attaining fourth-order spatial accuracy when solving singularly perturbed integro-partial differential equations of reaction–diffusion type, thus providing a robust numerical tool for complex applications in science and engineering. Full article

This paper considers impulsive systems with singularities. The main novelty of this study is that the impulses (impulsive functions) and the initial value are singular. The asymptotic convergence of the solution to a singularly perturbed initial problem with an infinitely large initial value, as $\epsilon \to 0,$ to the solution to a corresponding modified degenerate initial problem is proved. It is established that the solution to the initial problem at point $t=0$ has an initial jump phenomenon, and the value of this initial jump is determined. The theoretical results are supported by illustrative examples with simulations. Singularly perturbed problems are characterized by the presence of a small parameter multiplying the highest derivatives in the differential equations. This leads to rapid changes in the solution near the boundary or at certain points inside the domain. In our problem, symmetry is violated due to the emergence of a boundary layer at the initial point and at the moments of discontinuity. As a result, the problem as a whole is asymmetric. Such asymmetry in the behavior of the solution is a main feature of singularly perturbed problems, setting them apart from regularly perturbed problems in which the solutions usually exhibit smoother changes. Full article

This paper presents a uniformly convergent cubic spline numerical method for a singularly perturbed two-perturbation parameter ordinary differential equation. The considered differential equation is discretized using the cubic spline numerical method on a Bakhvalov-type mesh. The uniform convergence via the error analysis is established very well. The numerical findings indicate that the proposed method achieves second-order uniform convergence. Four test examples have been considered to perform numerical experimentations. Full article

This paper introduces a stable non-standard finite difference (NSFD) method to solve time-fractional singularly perturbed convection–diffusion problems. The fractional derivative in time is defined in the Caputo sense. The proposed method shows high efficiency when applied using a uniform mesh and can be easily extended to a Shishkin mesh in the spatial domain. We discuss error estimates to demonstrate the convergence of the numerical scheme. Additionally, various numerical examples are presented to illustrate the behavior of the solution for different values of the perturbation parameter $ϵ$ and the order of the fractional derivative. Full article

This article describes an effective computing method for singularly perturbed parabolic problems with small negative shifts in convection and reaction terms. To handle the small negative shifts, the Taylor series expansion is used. The asymptotically equivalent singularly perturbed parabolic convection–diffusion–reaction problem is then discretized with the Crank–Nicolson method on a uniform mesh for the time derivative and a hybrid method on Shishkin-type meshes for the space derivative. The method’s stability and parameter-uniform convergence are established. To substantiate the theoretical findings, the numerical results are presented in tables and graphs are plotted. The present results improve the existing methods in the literature. Due to the effect of the small negative shifts in Examples 1 and 2, the numerical results using Shishkin and Bakhvalov–Shishkin meshes are almost the same. Since there are no small shifts in Examples 3 and 4, the numerical results using the Bakhvalov–Shishkin mesh are more efficient than using the Shishkin mesh. We conclude that the present method using the Bakhvalov–Shishkin mesh performs well for singularly perturbed problems without small negative shifts. Full article

A set of singularly perturbed systems comprising Fredholm integro-differential equations associated with reaction–diffusion problems is considered. To approximate solutions to these systems, a second-order scheme for the derivatives and the trapezoidal rule for the integral terms are utilized. The discretization is performed on non-standard grids known as Shishkin-type meshes. The numerical method demonstrates a second-order rate of convergence with respect to small parameters in the equations, and error estimates are derived in the discrete maximum norm. Numerical experiments are conducted to verify the theoretical results. Full article

This paper presents a robust fitted mesh finite difference method for solving a system of n singularly perturbed two parameter convection–reaction–diffusion delay differential equations defined on the interval $[0,2]$. Leveraging a piecewise uniform Shishkin mesh, the method adeptly captures the solution’s behavior arising from delay term and small perturbation parameters. The proposed numerical scheme is rigorously analyzed and proven to be parameter-robust, exhibiting nearly first-order convergence. A numerical illustration is included to validate the method’s efficiency and to confirm the theoretical predictions. Full article

This paper proposes a robust finite difference method on a fitted Shishkin mesh to solve a system of n singularly perturbed convection–reaction–diffusion differential equations with two small parameters. Defined on the interval $[0,1]$, this system exhibits boundary layers due to the presence of small parameters, making accurate numerical approximations challenging. The method employs a piecewise uniform Shishkin mesh that adapts to layer regions and efficiently captures the solution’s behavior. The scheme is proven to be uniformly convergent with respect to the perturbation parameters, achieving nearly first-order accuracy. Comprehensive numerical experiments validate the theoretical results, illustrating the method’s robustness and efficiency in handling parameter-sensitive boundary layers. Full article

A class of third-order singularly perturbed two-parameter delay differential equations of boundary value problems is studied in this paper. Regular and singular components are used to estimate the solution’s a priori bounds and derivatives. A fitted finite-difference method is constructed to solve the problem on a Shishkin mesh. The numerical solution converges uniformly to the exact solution; it is validated via numerical test problems. The order of convergence of the numerical method is almost first-order, which is independent of the parameters $\epsilon$ and $\mu$. Full article

Piecewise analytical solutions to scalar, nonlinear, first-order, ordinary differential equations based on the second-order Taylor series expansion of their right-hand sides that result in Riccati’s equations are presented. Closed-form solutions are obtained if the dependence of the right-hand side on the independent variable is not considered; otherwise, the solution is given by convergent series. Discrete solutions also based on the second-order Taylor series expansion of the right-hand side and the discretization of the independent variable that result in algebraic quadratic equations are also reported. Both the piecewise analytical and discrete methods are applied to two singularly perturbed initial-value problems and the results are compared with the exact solution and those of linearization procedures, and implicit and explicit Taylor’s methods. It is shown that the accuracy of piecewise analytical techniques depends on the number of terms kept in the series expansion of the solution, whereas that of the discrete methods depends on the location where the coefficients are evaluated. For Riccati equations with constant coefficients, the piecewise analytical method presented here provides the exact solution; it also provides the exact solution for linear, first-order ordinary differential equations with constant coefficients. Full article

This paper presents a robust fitted mesh finite difference method for solving a dynamical system of two parameter convection–reaction–diffusion delay differential equations defined on the interval $[0,2]$. The method incorporates a piecewise uniform Shishkin mesh to accurately resolve the solution behavior caused by small perturbation parameters and delay terms. The proposed numerical scheme is proven to be parameter-robust and achieves almost first-order convergence. Numerical illustrations are provided to showcase the method’s effectiveness, highlighting its capability to address boundary and interior layers with improved accuracy. The results, supported by symmetrical considerations in the figures, enhance the precision and serve as validation for the theoretical results. Full article

This article proposes various new approximate analytical solutions of the boundary value problem for the non-stationary system of Nernst–Planck–Poisson (NPP) equations in the diffusion layer of an ideally selective ion-exchange membrane at overlimiting current densities. As is known, the diffusion layer in the general case consists of a space charge region and a region of local electroneutrality. The proposed analytical solutions of the boundary value problems for the non-stationary system of Nernst–Planck–Poisson equations are based on the derivation of a new singularly perturbed nonlinear partial differential equation for the potential in the space charge region (SCR). This equation can be reduced to a singularly perturbed inhomogeneous Burgers equation, which, by the Hopf–Cole transformation, is reduced to an inhomogeneous singularly perturbed linear equation of parabolic type. Inside the extended SCR, there is a sufficiently accurate analytical approximation to the solution of the original boundary value problem. The electroneutrality region has a curvilinear boundary with the SCR, and with an unknown boundary condition on it. The article proposes a solution to this problem. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transfer in membrane systems. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transport in membrane systems. Full article

Several types of linear and nonlinear singularly perturbed time-delay differential systems are considered. Asymptotic stability of the linear systems and asymptotic stability of the trivial solution of the nonlinear systems, valid for any sufficiently small value of the parameter of singular perturbation, are analyzed. For the stability analysis in the linear case, a partial exact slow–fast decomposition of the original system and an application of the Symmetric Matrix Riccati Equation method are proposed. Such an analysis yields parameter-free conditions, providing the asymptotic stability of the considered linear singularly perturbed time-delay differential systems for any sufficiently small value of the parameter of singular perturbation. Using the asymptotic stability results for the considered linear systems and the method of asymptotic stability in the first approximation, parameter-free conditions, guaranteeing the asymptotic stability of the trivial solution to the considered nonlinear systems for any sufficiently small value of the parameter of singular perturbation, are derived. Illustrative examples are presented. Full article

This article derives approximate formulations for Rayleigh waves on a coated orthorhombic elastic half-space with a prescribed vertical load acting as an elastic Winkler foundation. In addition, perfect continuity conditions are imposed between the coating layer and the substrate, while suitable decaying conditions are slated along the infinite depth of the half-space. The effect of the thin layer is modeled using appropriate effective boundary conditions within the long-wave limit. By applying the Radon transform and using the perturbation method, the derived model successfully captures the physical characteristics of elastic surface waves in coated half-spaces. The model consists of a pesudo-static elliptic equation decaying over the interior of the half-space and a singularly perturbed hyperbolic equation with a pseudo-differential operator. The pseudo-differential equation gives the approximate dispersion of surface waves on the coated half-space structure and is analyzed numerically at the end. Full article

We propose a Quantum Neural Network (QNN) for predicting stabilization parameter for solving Singularly Perturbed Partial Differential Equations (SPDE) using the Streamline Upwind Petrov Galerkin (SUPG) stabilization technique. SPDE-Q-Net, a QNN, is proposed for approximating an optimal value of the stabilization parameter for SUPG for 2-dimensional convection-diffusion problems. Our motivation for this work stems from the recent progress made in quantum computing and the striking similarities observed between neural networks and quantum circuits. Just like how weight parameters are adjusted in traditional neural networks, the parameters of the quantum circuit, specifically the qubits’ degrees of freedom, can be fine-tuned to learn a nonlinear function. The performance of SPDE-Q-Net is found to be at par with SPDE-Net, a traditional neural network-based technique for stabilization parameter prediction in terms of the numerical error in the solution. Also, SPDE-Q-Net is found to be faster than SPDE-Net, which projects the future benefits which can be earned from the speed-up capabilities of quantum computing. Full article