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In this paper, we investigate a singularly perturbed Fredholm integro-differential problem with a discontinuous source term, leading to the formation of interior layers in the solution at the point of discontinuity. We apply the exponentially fitted mesh method to solve the problem. Our analysis demonstrates that the method exhibits almost first-order convergence in the maximum norm, independent of the diffusion parameter. Numerical experiments are conducted to verify the validity of our theoretical results and they support the accuracy of our estimates. [ABSTRACT FROM AUTHOR]

The spatial–temporal spreading of a new invasive species in a habitat has interest in ecology and is modeled by a moving boundary diffusion logistic partial differential problem, where the moving boundary represents the unknown expanding front of the species. In this paper a front-fixing approach is applied in order to transform the original moving boundary problem into a fixed boundary one. A finite difference method preserving qualitative properties of the theoretical solution is proposed. Results are illustrated with numerical experiments. [ABSTRACT FROM AUTHOR]

One of the assumptions of the Black–Scholes model is that volatility is constant in the market. However, in reality, volatility cannot be constant. In this paper we examined an inverse problem of determining the time-dependent non-constant volatility from the observed market price of a European put/call option with a strike price.These unknown non-constant volatilities are from the time the option contract was signed to the time it was exercised (strike price) where the fluctuations of option price is known but the non-constant volatilities that was generated by option price within that time period is unknown. Although many studies have used the portfolio stocks to reconstruct volatility, no study have retrieved volatility from the option market. The aim is to recover unknown non-constant volatilities in vector form from one option contract period using manufacturer observed market data, that is, data from agreed contract period using Black–Scholes Partial Differential Equation (BSPDE). This would be done by using the chain and product rule to take the derivative with respect to volatility in the theoretical(Black-Schoes PDE) model to obtain the non-constant volatilities. By keeping the fluctuations data from observed option prices constant,we differentiate the volatility in the BSPDE to obtain a gradient equation and discretize the gradient equation with the finite difference method to obtain model simulated data. It is expected that non-constant volatility that would be recovered from the market using the gradient descent method would be matched with the non-constant market volatility from the model simulated data and if they are the same or approximately same, the volatility would have deem recovered. All these would be done in the least square sense. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical modeling of hyperthermia treatments by magnetic nanoparticles considering a 3D nonlinear Pennes’ bioheat transfer model with a temperature-dependent blood perfusion in order to yield more accurate results. The tissue is modeled by considering skin, fat and muscle layers in addition to the tumor. The FDM in a heterogeneous medium is employed and the resulting system of nonlinear equations in the time domain is solved by a predictor–multicorrector algorithm. Since the execution of the three-dimensional model requires a large amount of time, CUDA is used to speedup it. Experimental results showed that the parallelization with CUDA was very effective in improving performance, yielding gains up to 242 times when compared to the sequential execution time. [ABSTRACT FROM AUTHOR]

In this paper we propose a new iterative finite difference (IFD) scheme based on the Newton–Raphson–Kantorovich approximation method in function space to solve the classical one-dimensional Bratu’s problem. This new numerical method produces accurate solutions with low computational cost. The effectiveness and accuracy of the IFD method are confirmed through several numerical examples and compared to some existing numerical solvers. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a compact fourth-order finite difference scheme with low numerical dispersion to solve the 3D acoustic wave equation. Padé approximation has been used to obtain fourth-order accuracy in both temporal and spatial dimensions, while the alternating direction implicit (ADI) technique has been used to reduce the computational cost. Error analysis has been conducted to show the fourth-order accuracy, which has been confirmed by a numerical example. We have also shown that the proposed method is conditionally stable with a Courant–Friedrichs–Lewy (CFL) condition that is comparable to other existing finite difference schemes. Due to the higher-order accuracy, the new method is found effective in suppressing numerical dispersion. [Copyright &y& Elsevier]

In this paper, we propose an efficient fourth-order compact finite difference scheme with low numerical dispersion to solve the two-dimensional acoustic wave equation. Combined with the alternating direction implicit (ADI) technique and Padé approximation, the standard second-order finite difference scheme can be improved to fourth-order and solved as a sequence of one-dimensional problems with high computational efficiency. However such compact higher-order methods suffer from high numerical dispersion. To suppress numerical dispersion, the compact and non-compact stages are interlinked to produce a hybrid scheme, in which the compact stage is based on Padé approximation in both and temporal dimensions while the non-compact stage is based on Padé approximation in dimension only. Stability analysis shows that the new scheme is conditionally stable and superior to some existing methods in terms of the Courant–Friedrichs–Lewy (CFL) condition. The dispersion analysis shows that the new scheme has lower numerical dispersion in comparison to the existing compact ADI scheme and the higher-order locally one-dimensional (LOD) scheme. Three numerical examples are solved to demonstrate the accuracy and efficiency of the new method. [ABSTRACT FROM AUTHOR]

Abstract: Various finite difference methods for option pricing have been proposed. In this paper we demonstrate how a very simple approach, namely the repeated spatial extrapolation, can perform extremely better than the finite difference schemes that have been developed so far. In particular, we consider the problem of pricing vanilla and digital options under the Black–Scholes model, and show that, if the payoff functions are dealt with properly, then errors close to the machine precision are obtained in only some hundredths of a second. [Copyright &y& Elsevier]

Abstract: A higher order numerical discretization technique based on Minimum Sobolev Norm (MSN) interpolation was introduced in our previous work. In this article, the discretization technique is presented as a tool to solve two hard classes of PDEs, namely, the exterior Laplace problem and the biharmonic problem. The exterior Laplace problem is compactified and the resultant near singular PDE is solved using this technique. This finite difference type method is then used to discretize and solve biharmonic type PDEs. A simple book keeping trick of using Ghost points is used to obtain a perfectly constrained discrete system. Numerical results such as discretization error, condition number estimate, and solution error are presented. For both classes of PDEs, variable coefficient examples on complicated geometries and irregular grids are considered. The method is seen to have high order of convergence in all these cases through numerical evidence. Perhaps for the first time, such a systematic higher order procedure for irregular grids and variable coefficient cases is now available. Though not discussed in the paper, the idea seems to be easily generalizable to finite element type techniques as well. [Copyright &y& Elsevier]

Abstract: In this paper we numerically study the KdV-top equation and compare it with the Boussinesq equations over uneven bottoms. We use here a finite-difference scheme that conserves a discrete energy for the fully discrete scheme. We also compare this approach with the discontinuous Galerkin method. For the equations obtained in the case of stronger nonlinearities and related to the Camassa–Holm equation, we find several finite difference schemes that conserve a discrete energy for the fully discrete scheme. Because of its accuracy for the conservation of energy, our numerical scheme is also of interest even in the simple case of flat bottoms. We compare this approach with the discontinuous Galerkin method. [Copyright &y& Elsevier]

Abstract: The stability of three moving-mesh finite-difference schemes is studied in the norm for one-dimensional linear convection–diffusion equations. These schemes use central finite differences for spatial discretization and the method for temporal discretization, and they are based on conservative and non-conservative forms of transformed partial differential equations. The stability conditions obtained consist of the CFL condition and the mesh speed related conditions. The CFL condition is independent of the mesh speed and has the same form as that for fixed meshes. The mesh speed related conditions restrict how fast the mesh can move. The conditions of this type obtained in this paper are weaker than those in the existing literature and can be satisfied when the mesh is sufficiently fine. Illustrative numerical results are presented. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce an algorithm and a computer code for numerical differentiation of discrete functions. The algorithm presented is suitable for calculating derivatives of any degree with any arbitrary order of accuracy over all the known function sampling points. The algorithm introduced avoids the labour of preliminary differencing and is in fact more convenient than using the tabulated finite difference formulas, in particular when the derivatives are required with high approximation accuracy. Moreover, the given Matlab computer code can be implemented to solve boundary-value ordinary and partial differential equations with high numerical accuracy. The numerical technique is based on the undetermined coefficient method in conjunction with Taylor’s expansion. To avoid the difficulty of solving a system of linear equations, an explicit closed form equation for the weighting coefficients is derived in terms of the elementary symmetric functions. This is done by using an explicit closed formula for the Vandermonde matrix inverse. Moreover, the code is designed to give a unified approximation order throughout the given domain. A numerical differentiation example is used to investigate the validity and feasibility of the algorithm and the code. It is found that the method and the code work properly for any degree of derivative and any order of accuracy. [Copyright &y& Elsevier]

Abstract: In this paper, a new locally one-dimensional (LOD) scheme with error of for the two-dimensional wave equation is presented. The new scheme is four layer in time and three layer in space. One main advantage of the new method is that only tridiagonal systems of linear algebraic equations have to be solved at each time step. The stability and dispersion analysis of the new scheme are given. The computations of the initial and boundary conditions for the two intermediate time layers are explicitly constructed, which makes the scheme suitable for performing practical simulation in wave propagation modeling. Furthermore, a comparison of our new scheme and the traditional finite difference scheme is given, which shows the superiority of our new method. [Copyright &y& Elsevier]

Abstract: A nonlinear iteration method for solving a class of two-dimensional nonlinear coupled systems of parabolic and hyperbolic equations is studied. A simple iterative finite difference scheme is designed; the calculation complexity is reduced by decoupling the nonlinear system, and the precision is assured by timely evaluation updating. A strict theoretical analysis is carried out as regards the convergence and approximation properties of the iterative scheme, and the related stability and approximation properties of the nonlinear fully implicit finite difference (FIFD) scheme. The iterative algorithm has a linear constringent ratio; its solution gives a second-order spatial approximation and first-order temporal approximation to the real solution. The corresponding nonlinear FIFD scheme is stable and gives the same order of approximation. Numerical tests verify the results of the theoretical analysis. The discrete functional analysis and inductive hypothesis reasoning techniques used in this paper are helpful for overcoming difficulties arising from the nonlinearity and coupling and lead to a related theoretical analysis for nonlinear FI schemes. [Copyright &y& Elsevier]

This paper proposes an efficient localized meshless technique for approximating the viscoelastic wave model. This model is a significant methodology to explain wave propagation in solids modeled with a wide collection of viscoelastic laws. In the first method, a difference scheme with the second-order accuracy is implemented to obtain a semi-discrete scheme. Then, a localized radial basis function partition of unity scheme is adopted to get a full-discrete scheme. This localization technique consists of decomposing the initial domain into several sub-domains and constructing a local radial basis function approximation over every sub-domain. A well-conditioned resulting linear system and a low computational burden are the main merits of this technique compared to global collocation methods. Further, the stability and convergence analysis of the temporal discretization scheme are deduced using discrete energy method. Numerical results are shown to validate the accuracy and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: Partial differential equations with possibly discontinuous coefficients play an important part in engineering, physics and ecology. In this paper, we will study nonlinear partial differential equations with variable coefficients arising from population models. Generally speaking, it is difficult to analyze the behavior of nonlinear partial differential equations; therefore, we usually rely on the numerical approximation. Currently, there is an increasing interest in designing numerical schemes that preserve energy properties for differential equations. We will design the numerical schemes that preserve discrete energy property and show numerical experiments for a nonlinear partial differential equation with variable coefficients. [Copyright &y& Elsevier]

In a conventional approach to numerical computation, finite difference and finite element methods are usually implemented to determine the solution of a set of differential equations (DEs). This paper presents a novel approach to solve DEs by applying the universal approximation method through an artificial intelligence utility in a simple way. In this proposed method, neural network model (NNM) and fuzzy linguistic model (FLM) are applied as universal approximators for any nonlinear continuous functions. With this outstanding capability, the solutions of DEs can be approximated by the appropriate NNM or FLM within an arbitrary accuracy. The adjustable parameters of such NNM and FLM are determined by implementing the optimization algorithm. This systematic search yields sub-optimal adjustable parameters of NNM and FLM with the satisfactory conditions and with the minimum residual errors of the governing equations subject to the constraints of boundary conditions of DEs. The simulation results are investigated for the viability of efficiently determining the solutions of the ordinary and partial nonlinear DEs. [Copyright &y& Elsevier]