Abstract: 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 solutions 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.

Abstract: Finite difference methods are commonly used in the pricing of discretely monitored exotic options in the Black–Scholes framework, but they tend to converge slowly due to discontinuities contained in terminal conditions. We present an effective analytical modification to existing finite difference methods that greatly enhances their performance on discretely monitored options with non-smooth terminal conditions. We apply this modification to the popular Crank–Nicolson method and obtain highly accurate option pricing results with significantly reduced CPU cost. We also introduce an adaptive mesh refinement technique that further improves the computational speed of the modified finite difference method. The proposed method is especially useful for options with high monitoring frequencies, which are difficult to price using other existing methods.

Abstract: <p>This paper presents a numerical analysis of discretization errors arising in the finite difference modeling of symmetric p–n junctions. Unlike previous comparative studies, the focus here is on the internal structure of errors introduced by discretization when solving the Poisson equation in equilibrium. The dependence of accuracy on grid step, doping concentration, and built-in potential is explored. The results are presented as heatmaps and tables, suitable for grid optimization in both educational and scientific applications.</p>

Abstract: This study presents a numerical investigation of the viscous Burgers’ equation subjected to an external sinusoidal source term, utilizing an explicit finite difference method. The model incorporates nonlinear advection, viscous diffusion, and spatially varying forcing, representing a physically relevant nonlinear partial differential equation. The numerical scheme uses forward time stepping, upwind discretization for the convective term, and central differencing for the diffusive term. Through simulation, the evolution of the solution profile is visualized using 3D surface plots, space-time contours, and cross-sectional curves. The results demonstrate a consistent amplification near the domain center due to the source term and show the smoothing influence of viscosity over time. The study also addresses the scheme's stability and convergence under suitable discretization parameters. Keywords: Viscous Burgers’ equation, nonlinear PDE, finite difference method, source term, numerical simulation, stability analysis, space-time evolution

Abstract: This paper presents a numerical framework for the low-rank approximation of the solution to three-dimensional parabolic problems. The key contribution of this work is the tensorization process based on a tensor-train reformulation of the second-order accurate finite difference method. We advance the solution in time by combining the finite difference method with an explicit and implicit Euler method and with the Crank-Nicolson method. We solve the linear system arising at each time step from the implicit and semi-implicit time-marching schemes through a matrix-free preconditioned conjugate gradient (PCG) method, appositely designed to exploit the separation of variables induced by the tensor-train format. We assess the performance of our method through extensive numerical experimentation, demonstrating that the tensor-train design offers a robust and highly efficient alternative to the traditional approach. Indeed, the usage of this type of representation leads to massive time and memory savings while guaranteeing almost identical accuracy with respect to the traditional one. These features make the method particularly suitable to tackle challenging high-dimensional problems.

Abstract: In order to investigate linear time and spatial fractional advection equations, we present a finite difference scheme (FDS) in this paper. The fractional Taylor series method for u at tj+1 and xi+1 is used to approximate the fractional derivatives. First, we construct our numerical scheme (NS) for the mathematical model. In the second part, we study the stability and convergence of our numerical scheme. Finally, the numerical simulations of the fractional advection equation, using the FDM, is plotted for several values of fractional parameters α and ν. It will be shown that the convergence is achieved properly which confirms the effectiveness of the proposed algorithm.

Abstract: The space-time generalized finite difference method was used to solve the time fractional mobile/immobile diffusion equation. Traditionally, partial differential equations are solved by separating time and space dimensions, and applying distinct numerical methods to each. Here, the time dimension was treated as an additional spatial dimension, thus transforming the $ d $-dimensional spatial problem into a new $ (d+1) $-dimensional problem in the space-time domain. In the space-time generalized finite difference method, the complexity of the numerical solution of the problem was effectively reduced by simultaneously discretizing the space and time dimensions in the space-time domain, while retaining all the advantages of the generalized finite difference method. We took the second-order Taylor series expansion as an example to show the numerical algorithm of the 2D time fractional mobile/immobile diffusion equation, and numerically simulated the one- and two-dimensional problems where the solution was a sufficiently smooth or weak singularity at the initial moment, thereby verifying the effectiveness of the algorithm.

Abstract: In this study, we proposed a quasi-transient electro-thermal analysis algorithm based on the finite difference method to comprehensively delineate the coupling phenomena within the highly integrated systems under the transient external electro-magnetic pulse (EMP). The quasi-transient coupling scheme is realized by interpolating a time factor between the two fields, thus harmonizing their time scale discrepancies efficiently. This process resolves transient physical mechanisms adeptly and saves computational cost effectively. To enhance its adaptability in diverse multi-scale targets, especially for those processed by integrated circuit’s advanced packaging techniques, including the redistribution layer (RDL), through-silicon via (TSV), and highly integrated system-in-package (SiP), we employ the subgridding discretization method and adopt the alternating direction implicit (ADI) scheme for both fields facilitating an unconditionally stable computation. On this basis, we conducted several quasi-transient electro-thermal analyses for different advanced packaging components to quantify their electrical and thermal response induced by the EMP incidence, thereby substantiating the efficacy of the method in such complex problems.

Abstract: This paper introduces a nonstandard finite difference (NSFD) approach to a reaction-diffusion SEIQR epidemiological model, which captures the spatiotemporal dynamics of infectious disease transmission. Formulated as a system of semilinear parabolic partial differential equations (PDEs), the model extends classical compartmental models by incorporating spatial diffusion to account for population movement and spatial heterogeneity. The proposed NSFD discretization is designed to preserve the continuous model's essential qualitative features, such as positivity, boundedness, and stability, which are often compromised by standard finite difference methods. We rigorously analyze the model's well-posedness, construct a structure-preserving NSFD scheme for the PDE system, and study its convergence and local truncation error. Numerical simulations validate the theoretical findings and demonstrate the scheme's effectiveness in preserving biologically consistent dynamics.

Abstract: Traffic flow in most urban areas is augmenting due to the growth in transport and continual demand for it. It is multimodal and includes use of different types vehicles, motorcycles andeven walking. The assessment of uninterrupted traffic flow is traditionally based on empirical methods. This study was based on the macroscopic model which is a mathematical model that formulates the relationships among traffic flow characteristics like density, flow, mean and speed of a traffic stream. The study considered traffic models first developed by Lighthill and whitham(1955) and later Richards (1956) shortly Called LWR traffic flow model. Simulation by use of this method enables control strategies of congestion dissipation and has suggested some recommended measures to rationalize the design of roads and implementation of regulations of road users considering some regulations and infrastructural gaps in Kisi town. This paper focus on two finite difference schemes, that is, first order Explicit Upwind Difference Scheme-EUDS (forward time. backward space) and Second order Lax-Wendroff Difference Scheme-LWDS (forward time centred space) for solving first order PDE as well the traffic density ρ(t,x) was computed by solving LWR macroscopic conservation form of traffic flow model using both schemes. The conditions of stability were numerically verified and it is shown that LWDS is superior to EUDS in terms of time step selection. The results obtained were becompared with average key data which provide initial conditions and boundary data used for numerical simulation.

Abstract: The diffusive viscous wave equation describes wave propagation in diffusive and viscous media. Examples include seismic waves traveling through the Earth's crust, taking into account of both the elastic properties of rocks and the dissipative effects due to internal friction and viscosity; acoustic waves propagating through biological tissues, where both elastic and viscous effects play a significant role. We propose a stable and high-order finite difference method for solving the governing equations. By designing the spatial discretization with the summation-by-parts property, we prove stability by deriving a discrete energy estimate. In addition, we derive error estimates for problems with constant coefficients using the normal mode analysis and for problems with variable coefficients using the energy method. Numerical examples are presented to demonstrate the stability and accuracy properties of the developed method.

Abstract: <p> In this paper, two advance numerical techniques for solving second-order boundary value problems in ordinary differential equations (ODEs) are presented. The first method, the Chebyshev Finite Difference Method (CFDM), which enhances the traditional Finite Difference Method by utilizing Chebyshev Polynomials as basis functions, resulting in improved computational <br> performance. The second method developed is the Perturbed Chebyshev Finite Difference Method (Perturbed CFDM), which incorporates perturbation techniques to further enhance the accuracy and efficiency of the method. Both methods were applied to homogeneous and non-homogeneous linear boundary value problems, with numerical results demonstrating that the Perturbed CFDM significantly outperforms both standard CFDM and the traditional finite difference method in terms of accuracy and computational efficiency. These findings establish the Perturbed CFDM as a powerful and reliable tool for solving boundary value problems. All computations were carried out using MATLAB, ensuring accurate approximation and numeri<br>cal solutions of the tested problems.</p>

Abstract: A spatial second-order conservation characteristic finite difference method is developed to solve advection-diffusion equations on the Lagrangian grid. The intermediate numerical solutions are first computed with the integration of the parabolic interpolation, which are solved by the second-order modified upwind scheme along the x-direction. The mass $ \bar {M}_{i, j}^n $ M¯i,jn on $ \bar \Omega _{i,j}(t^n) $ Ω¯i,j(tn) are computed with the integration of the parabolic interpolation along the y-direction. The stability and convergence are strictly proved in $ L^2 $ L2 norm by some auxiliary lemmas. Some numerical experiments are used to test our scheme is of second-order convergent in space and first-order convergent in time.

Abstract: Partial differential equations (PDEs) are used in mathematics to describe physical phenomena. However, analytical solutions are often unavailable, requiring approximations. The physics-informed neural network (PINN) is a recent technique that combines deep neural networks with physical knowledge leveraging automatic differentiation techniques, to provide accurate approximations. This work analyzes the convergence of the PINN method, considering the discretizations and architecture of the networks. PINN approximations were compared with Finite Difference Methods (FDM) in the case of the two-dimensional heat equation. The experiments carried out suggested that there is an optimal number of points and architectural parameters to be used, which can lead to an improvement in the generalization and estimation error. It was possible to see an advantage over FDM as it approximates the solution of all the time steps at once, without propagating the error from one step to the next.

Abstract: In this study, we introduce an 8th-order combined compact finite difference (CCD) method for solving partial dif-ferential equations (PDEs). By employing the 8th-order CCD approach to discretize spatial derivatives, we transform PDEs into a system of ordinary differential equations (ODEs). We then apply the RK(8)7 method to solve these ODEs efficiently. Our analysis demonstrates the convergence of this technique and highlights its accuracy through various numerical examples. These findings indicate the potential of the proposed high-order CCD method for delivering precise solutions to complex PDEs, making it a valuable tool for numerical analysis and a wide range of applications.

Abstract: Abstract This study presents a robust numerical framework for solving nonlinear boundary value problems (BVPs) in catalytic pores by combining the finite difference method (FDM) with MATLAB’s fsolve solver. The governing nonlinear diffusion–reaction equation is discretized using central finite differences, and the resulting system of nonlinear algebraic equations is iteratively solved with fsolve. Concentration profiles were investigated across a wide range of Thiele modulus values and reaction orders, and the predictions were compared against analytical solutions available in the literature. The results demonstrate excellent agreement with analytical benchmarks, confirming the accuracy and stability of the proposed approach. For zeroth-order kinetics, reactant depletion occurs rapidly at relatively low Thiele modulus values, whereas first-order kinetics require significantly higher Thiele modulus values for complete consumption. These findings highlight the dominant role of pore-scale transport limitations in catalyst performance. Beyond validation, the methodology provides a flexible computational platform that can be readily extended to more complex scenarios, including variable diffusivities, multidimensional geometries, and higher-order reaction mechanisms. The integration of FDM with iterative solvers such as fsolve thus offers an adaptable and efficient tool for advancing catalyst design and optimization in heterogeneous reaction engineering.

Abstract: Accurately simulating water flow movement in vadose zone is crucial for effective water resources assessment. Richards' equation, which describes the movement of water flow in the vadose zone, is highly nonlinear and challenging to solve. Existing numerical methods often face issues such as numerical dispersion, oscillation, and mass non-conservation when spatial and temporal discretization conditions are not appropriately configured. To address these problems and achieve accurate and stable numerical solutions, a finite analytic method based on water content-based Richards' equation (FAM-W) is proposed. The performance of the FAM-W is compared with analytical solutions, Finite Difference Method (FDM), and Finite Analytic Method based on the pressure Head-based Richards' equation (FAM-H). Compared to analytical solution and other numerical methods (FDM and FAM-H), FAM-W demonstrates superior accuracy and efficiency in controlling mass balance errors, regardless of spatial step sizes. This study introduces a novel approach for modelling water flow in the vadose zone, offering significant benefits for water resources management.