Accelerated fitted operator finite difference method for singularly perturbed delay differential equations with non-local boundary condition

We design a suitable fitted operator finite difference method (FOFDM) to solve a model of an unsteady transient magnetohydrodynamic (MHD) free convective and mass transfer flow with thermophoresis past an inclined permeable plate in the presence of chemical reaction, thermal radiation, and temperature dependent viscosity. The governing nonlinear partial differential equations in the current model are transformed by a suitable similarity transformation into a system of ordinary differential equations which are then solved numerically using the FOFDM. The superiority of the proposed FOFDM over the standard finite difference method is proved theoretically and numerically. Furthermore, we illustrate the effect of various parameters on the temperature, concentration and velocity profiles using the FOFDM. It is observed that an increase in the viscosity parameter leads to a decrease in the temperature and an increase in the velocity. Results also show that an increase in the thermophoretic parameter leads to a decrease in the concentration. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 106–120, 2016

In this paper, we extend a system of coupled first order non-linear system of delay differential equations (DDEs) arising in modeling of stoichiometry of tumour dynamics, to a system of diffusion-reaction system of partial delay differential equations (PDDEs). Since tumor cells are further modified by blood supply through the vascularization process, we determine the local uniform steady states of the homogeneous tumour growth model with respect to the vascularization process. We show that the steady states are globally stable, determine the existence of Hopf bifurcation of the homogeneous tumour growth model with respect to the vascularization process. We derive, analyse and implement a fitted operator finite difference method (FOFDM) to solve the extended model. This FOFDM is analyzed for convergence and we observe seen that it has second-order accuracy. Some numerical results confirming theoretical observations are also presented. These results are comparable with those obtained in the literature.

On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions

Fourth-order compact finite difference schemes for solving the time fractional diffusion equation with nonlocal boundary conditions are considered in this paper. The second-order derivative in one space variable is approximated by a fourth-order compact finite difference. Two different time derivatives, the Caputo derivative and Riemann–Liouville derivative are considered in the equations that we discuss. The Caputo time derivative is discretized by the L1-approximation and a second-order approximation is used for the Riemann–Liouville derivative. Stability analysis using the discrete energy method for Scheme I is given, and error estimation of this scheme is derived by conducting the local truncation error analysis, using the stability result. Numerical results are provided to show that both two fully discrete schemes have fourth-order accuracy in space, and $$2-\gamma $$ order of accuracy in time for Scheme I with $$\gamma $$ being the order of the fractional derivative, while Scheme II has second-order accuracy in time when the theoretical solutions are smooth with zero initial values.

In this article, we study an inverse problem (IP) for a fourth-order hyperbolic equation with nonlocal boundary conditions. This IP is reduced to the not self-adjoint boundary value problem (BVP) with corresponding boundary condition. Then, we use the separation of variables method, to reduce the not self-adjoint BVP to an integral equation. The existence and uniqueness of the integral equation are established by the contraction mappings principle and it is concluded that this solution is unique for a not-adjoint BVP. The existence and uniqueness of a nonlocal BVP with integral condition is proved. In addition, the fourth-order hyperbolic PDE is discretized using a collocation technique based on the quintic B-spline (QnB-spline) functions and reformed by the Tikhonov regularization function. The noise and analytical data are considered. The numerical outcome for a standard numerical example is discussed. Furthermore, the stability of the discretized system is also analyzed. The rate of convergence (ROC) of the method is also obtained.

In this paper, accelerated finite difference method for solving singularly perturbed delay reaction-diffusion equations is presented. First, the solution domain is discretized. Then, the derivatives in the given boundary value problem are replaced by finite difference approximations and the numerical scheme that provides algebraic systems of equations is obtained, which can easily be solved by Thomas algorithm. The consistency, stability and convergence of the method have been established. To increase the accuracy of our established scheme we used Richardson's extrapolation techniques. To validate the applicability of the proposed method, four model examples have been considered and solved for different values of perturbation parameters and mesh sizes. The numerical results have been presented in tables and graphs to illustrate; the present method approximates the exact solution very well. Moreover, the present method gives better accuracy than the existing numerical methods mentioned in the literature.

A composite preconditioner for the electromagnetic scattering from a large cavity

Optimal analysis for optimal design of cyclic symmetric structures subject to frequency constraints

A higher-order collocation technique based on Haar wavelets for fourth-order nonlinear differential equations having nonlocal integral boundary conditions

We consider a class of nonlocal Cahn–Hilliard equations in a bounded domain Ω ⊂ R d ( d ∈ { 2 , 3 } ) , subject to a nonlocal kinetic rate dependent dynamic boundary condition. This diffuse interface model describes phase separation processes with possible long-range interactions both within the bulk material and on its boundary. The kinetic rate 1 / L , with L ∈ [ 0 , + ∞ ] , distinguishes different types of bulk-boundary interactions. For the initial boundary value problem endowed with general singular potentials, including the physically relevant logarithmic potential, we first establish the existence and uniqueness of global weak solutions when the bulk and boundary chemical potentials are coupled through a Robin-type boundary condition, i.e. L ∈ ( 0 , + ∞ ) . The proof of existence is based on a Yosida approximation of singular potentials and a suitable Faedo–Galerkin scheme. Subsequently, we investigate asymptotic limits as the kinetic rate approaches zero or infinity, which yield weak well-posedness for the limiting cases L = + ∞ and L = 0, respectively. Under appropriate assumptions on the interaction kernels, we derive explicit convergence rates for the Yosida approximation as ɛ → 0 and for the asymptotic limits as L → 0 or L → + ∞ . Finally, we demonstrate that every global weak solution exhibits a propagation of regularity over time and when L ∈ ( 0 , + ∞ ] , we establish the instantaneous strict separation property by means of a suitable De Giorgi’s iteration scheme.

New parameter identification method for calibrating stiffness of AFM probes

Space-dependent heat source determination problem with nonlocal periodic boundary conditions

This study presents an experimental and computational fluid dynamics (CFD) analysis to evaluate the performance of a two-stroke aviation engine under constant operating conditions. The experiments were conducted at 4800 RPM and full load, where high-precision measurement devices recorded critical performance parameters, including total mass flow rate, fuel mass flow rate, torque, volumetric efficiency, and in-cylinder pressure and temperature. For CFD analysis, a three-dimensional combustion chamber model of the engine was developed, and two turbulence models (Standard k-ε and RNG k-ε) were employed using three different mesh sizes (2 mm, 3 mm, and 4 mm). The numerical results were compared with experimental data to determine the most accurate simulation configuration. The findings indicate that decreasing the mesh size improves the accuracy of the simulations, with the Standard k-ε model using a 2 mm mesh producing the most precise predictions across multiple parameters. The error rate for total mass flow rate and volumetric efficiency was reduced to 9%, demonstrating the effectiveness of fine mesh resolution. However, the RNG k-ε model yielded better accuracy for specific fuel consumption, achieving a 0.1% error rate at a 2 mm mesh size. Velocity and temperature distributions revealed that flow gradients were more accurately captured with smaller mesh sizes, particularly in the bypass and exhaust port regions. The Standard k-ε model was more effective in predicting in-cylinder pressure and heat release characteristics, while the RNG k-ε model predicted slightly higher maximum in-cylinder temperatures. Mesh refinement significantly influenced combustion behavior, with smaller mesh sizes leading to more accurate heat transfer and reaction modeling. Overall, this study highlights the impact of mesh size selection and turbulence modeling on CFD-based performance analysis of two-stroke aviation engines. The results provide valuable insights for optimizing CFD modeling strategies and can serve as a reference for future research on engine design, performance enhancement, and computational accuracy improvements.

We investigate some new inequalities by estimating the rate of convergence by means of the complete modulus of continuity and a class of Lipschitz functions for the bivariate Bernstein-type operator including Bézier basis and present an example including numerical results comparing its rate of convergence. Moreover, we introduce its GBS (Generalized Boolean Sum) operator and obtain its rate of convergence with the help of the mixed modulus of continuity and Lipschitz class of Bögel continuous functions with exemplifying numerical results. Our research will demonstrate that the GBS operator possesses at least better numerical results than the bivariate Bernstein-type operator. All mentioned results point out the novelty of this study.

Two closely-related fictitious domain methods for solving problems involving multiple interfaces are introduced. Like other fictitious domain methods, the proposed methods simplify the task of finite element mesh generation and provide access to solvers that can take advantage of uniform structured grids. The proposed methods do not involve the Lagrange multipliers, which makes them quite different from existing fictitious domain methods. This difference leads to an advantageous form of the inf–sup condition, and allows one to avoid time-consuming integration over curvilinear surfaces. In principle, the proposed methods have the same rate of convergence as existing fictitious domain methods. Nevertheless it is shown that, at the cost of introducing additional unknowns, one can improve the quality of the solution near the interfaces. The methods are presented using a two-dimensional model problem formulated in the context of linearized theory of elasticity. The model problem is sufficient for presenting method details and mathematical foundations. Although the model problem is formulated in two dimensions and involves only one interface, there are no apparent conceptual difficulties to extending the methods to three dimensions and multiple interfaces. Further, it is possible to extend the methods to nonlinear problems involving multiple interfaces.