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This study presents, a numerical method for the solutions of the generalized nonlinear Benjamin-Bona-Mahony-Burgers' equation, with variable order local time fractional derivative. This derivative is expressed as a product of two functions, the usual integer order time derivative, and a function of time having a fractional exponent. Then, forward difference approximation is used for time derivative. The unknown solution of the differential problem and corresponding derivatives are estimated using Haar wavelet approximations (HWA). The collocation procedure is then implemented in HWA, to transform the given model to the system of linear algebraic equations for the determination of unknown constant coefficient of the Haar wavelet series, which update the derivatives and the numerical solutions. The sufficient condition is established for the stability of the proposed technique, and then verified computationally. To check the performance of the scheme, few illustrative examples in one and two dimensions along with l ∞ and l 2 error norms are also given. Besides this, the computational convergence rate is calculated for both type equations. Additionally, computed solutions are compared with available results in literature. Simulations and graphical data discloses, that suggested scheme works well for such complex problems. [ABSTRACT FROM AUTHOR]

We study the family of integral equations, called the Robin mean value equations (RMV), that are local averaged approximations to the Robin-Laplace boundary value problem (RL). When posed on C 1 , 1 -regular domains, we prove existence, uniqueness, equiboundedness and the comparison principle for solutions to (RMV). For the continuous right hand side of (RL), we show that solutions to (RMV) converge uniformly, in the limit of the vanishing radius of averaging, to the unique W2,p solution, which coincides with the unique viscosity solution of (RL). We further prove the lower bound on solutions to (RMV), which is consistent with the optimal lower bound for solutions to (RL). Our proofs employ martingale techniques, where (RMV) is interpreted as the dynamic programming principle along a suitable discrete stochastic process, interpolating between the reflecting and the stopped-at-exit Brownian walks. [ABSTRACT FROM AUTHOR]

We show that solutions to the Robin mean value equations (RMV), introduced in Lewicka and Peres (2022), converge uniformly in the limit of the vanishing radius of averaging, to the unique solution of the Robin-Laplace boundary value problem (RL), posed on any C 1 , 1 -regular domain and with any bounded Borel right hand side. When compared with the case of continuous right hand side, analyzed in Lewicka and Peres (2022), the present more general setting presents significant technical challenges. Along the way, we prove the asymptotic Hölder equicontinuity of solutions to (RMV): Lipschitz in the interior and C 0 , α up to the boundary, for any α ∈ (0,1). Our proofs employ martingale techniques, where (RMV) is interpreted as the dynamic programming principle for a discrete stochastic process, interpolating between the reflecting and the stopped-at-exit Brownian walks. [ABSTRACT FROM AUTHOR]

We consider a system of fully nonlinear partial differential equations that corresponds to the Hamilton–Jacobi–Bellman equations for the value functions of an optimal innovation investment problem of a monopoly firm facing bankruptcy risk. We compare several algorithms for the numerical solution of the considered problem: the collocation method, the finite difference method, WENO method and the adaptive finite element method. We discuss implementation issues for the considered schemes and perform numerical studies for different model parameters to assess their performance. [ABSTRACT FROM AUTHOR]

In this article, we propose a new computational method for second order initial value problems in ordinary differential equations. The algorithm developed is based on a local representation of theoretical solution of the second order initial value problem by a non-linear interpolating function. Numerical examples are solved to ensure the computational performance of the algorithm for both linear and non-linear initial value problems. From the results we obtained, the algorithm can be said computationally efficient and effective.

Explicit formulas for the coefficients of finite difference approximations of first and higher derivatives in any order of accuracy have been presented by Khan and Ohba. They also have provided a mathematical proof of the formulas for first derivatives. In this paper, the proof is extended for second derivatives. The proof is constructed based on Taylor series, and employs some properties of Vandermonde determinant which are not found in the proof for first derivatives. [ABSTRACT FROM AUTHOR]

Multielectronic states quantum dynamics on a grid is described in a manner motivated by on the fly classical trajectory computations. Non stationary electronic states are prepared by a few cycle laser pulse. The nuclei respond and begin moving. We solve the time dependent Schrödinger equation for the electronic and nuclear dynamics for excitation from the ground electronic state. A satisfactory accuracy is possible using a localized description on a discrete grid. This enables computing on the fly for both the nuclear and electronic dynamics including non-adiabatic couplings. Attosecond dynamics in LiH is used as an example. [ABSTRACT FROM AUTHOR]

We consider a nonstationary 3D flow of a compressible viscous heat-conducting micropolar fluid, which is in the thermodynamical sense perfect and polytropic, in the domain bounded with two concentric spheres in R 3 . In this paper we establish the existence of a global solution to the free boundary problem defined with the homogeneous boundary conditions for velocity, microrotation, heat flux on the fixed border and homogeneous boundary conditions for strain, microrotation, and heat flux on the free boundary. We suppose that the initial data are spherically symmetric, with positive initial density and temperature, having the zero density on the free boundary. Because of the spherical symmetry, the starting three-dimensional problem is transformed to the one-dimensional problem in Lagrangian coordinates in the domain that is a segment. The solution to our problem is obtained as a limit of the sequence of approximate solutions derived from suitable semi-discrete finite difference approximate systems. By using the derived bounded estimates of the approximate solutions and the results of the weak and strong compactness, we establish the convergence to the generalized solution of our problem globally in time. [ABSTRACT FROM AUTHOR]

This paper studies an effective finite difference scheme for solving two-dimensional Heston stochastic volatility option-pricing model problems. A dynamically balanced up-downwind strategy for approximating the cross-derivative is implemented and analyzed. Semi-discretized and spatially nonuniform platforms are utilized. The numerical method comprised is simple and straightforward, with reliable first order overall approximations. The spectral norm is used throughout the investigation, and numerical stability is proven. Simulation experiments are given to illustrate our results.

We present a general model for the interaction of a size-structured population with its environment. The vital rates of the individuals are assumed to depend on a number of variables including the total population and the environment. We develop a finite difference approximation for this general model and prove the convergence of the method to the unique weak solution of the nonlinear system of partial differential equations coupled with ordinary differential equations. Potential applications are presented in various fields ranging from blood cell population dynamics to the study of invasive species. [ABSTRACT FROM AUTHOR]

We consider the nonstationary 3D flow of a compressible viscous heat-conducting micropolar fluid in the domain to be a subset of R³, bounded with two concentric spheres. In the thermodynamical sense the fluid is perfect and polytropic. The homogeneous boundary conditions for velocity, microrotation, heat flux and spherical symmetry of the initial data are proposed. Due to the assumption of spherical symmetry, the problem can be considered as one--dimensional problem in Lagrangian description on the domain that is a segment. We define the approximate equations system by using the finite difference method and construct the sequence of approximate solutions for our problem. By analyzing the properties of these approximate solutions we prove their convergence to the generalized solution of our problem globally in time and establish the convergence of the defined numerical scheme, which is the main result of the paper. The practical application of the proposed numerical scheme is performed on the chosen test example. [ABSTRACT FROM AUTHOR]

Most of the fundamental theories and mathematical models of engineering and physical sciences are expressed in terms of partial differential equations (PDEs). Several studies were carried out for the numerical approximation of the second order linear Klein-Gordon equation. This study constructed a new numerical technique for the numerical approximation of second order linear Klein-Gordon equation. The new constructed scheme was based on employing non-polynomial cubic spline method (NPCSM). The second order time derivatives involved in the linear Klein-Gordon equation were decomposed into the first order derivatives. The decomposition generated a linear system of PDEs, where the first order time derivatives were approximated by the central finite differences of o(h2). Three test problems were considered for the numerical illustration of the developed scheme. For different values of spatial displacement x, step size h, and time step k, the developed numerical technique produced encouraging results which were very much close to the analytical solution. For x=0.2,h=1/5, and k=0.00001, the best observed accuracy was close to the machine precision. [ABSTRACT FROM AUTHOR]

In this article, we describe the propagation properties of the one-dimensional wave and transport equations with variable coefficients semi-discretized in space by finite difference schemes on non-uniform meshes obtained as diffeomorphic transformations of uniform ones. In particular, we introduce and give a rigorous meaning to notions like the principal symbol of the discrete wave operator and the corresponding bi-characteristic rays. The main mathematical tool we employ is the discrete Wigner transform, which, in the limit as the mesh size parameter tends to zero, yields the so-called Wigner ( semiclassical) measure. This measure provides the dynamics of the bi-characteristic rays, i.e., the solutions of the Hamiltonian system describing the propagation, in both physical and Fourier spaces, of the energy of the solution to the wave equation. We show that, due to dispersion phenomena, the high-frequency numerical dynamics does not coincide with the continuous one. Our analysis holds for the class $$C^{0,1}(\mathbb {R})$$ of globally Lipschitz coefficients and non-uniform grids obtained by means of $$C^{1,1}(\mathbb {R})$$ -diffeomorphic transformations of a uniform one. We also present several numerical simulations that confirm the predicted paths of the space-time projections of the bi-characteristic rays. Based on the theoretical analysis and simulations, we describe some of the pathological phenomena that these rays might exhibit as, for example, their reflection before touching the boundary of the space domain. This leads, in particular, to the failure of the classical properties of boundary observability of continuous waves, arising in control and inverse problems theory. [ABSTRACT FROM AUTHOR]

We consider a size-structured population model where individuals may be recruited into the population at different sizes. First- and second-order finite difference schemes are developed to approximate the solution of the model. The convergence of the approximations to a unique weak solution is proved. We then show that as the distribution of the new recruits become concentrated at the smallest size, the weak solution of the distributed states-at-birth model converges to the weak solution of the classical Gurtin–McCamy-type size-structured model in the weak* topology. Numerical simulations are provided to demonstrate the achievement of the desired accuracy of the two methods for smooth solutions as well as the superior performance of the second-order method in resolving solution-discontinuities. Finally, we provide an example where supercritical Hopf-bifurcation occurs in the limiting single state-at-birth model and we apply the second-order numerical scheme to show that such bifurcation also occurs in the distributed model. [ABSTRACT FROM AUTHOR]

Partial differential equations (PDEs) played a vital role in natural sciences and engineering. Different studies were carried out for solving fourth-order partial differential equations. These equations governed the oblique vibrations of a uniform beam. This study presented and illustrated a new technique for the numerical approximations of fourth-order partial differential equations (PDEs). The new technique was based on the fact of employing polynomial cubic spline method (PCSM) along with the Adomian decomposition method (ADM). The Adomian decomposition method was used to obtain the boundary conditions for the replaced variables, whereas continuous approximation was constructed and applied on the decomposed system of PDEs. The performance of the developed scheme was illustrated by numerical tests that involved numerical approximations with the exact solutions on a collection of test problems. [ABSTRACT FROM AUTHOR]

In this article, a numerical study for fractional reaction-subdiffusion equations is introduced using a class of finite difference methods. These methods are extensions of the weighted average methods for ordinary (non-fractional) reaction-subdiffusion equations. A stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. Simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, are given and checked numerically. Numerical test examples, figures, and comparisons have been presented for clarity. doi:10.14456/WJST.2014.50

In this paper we define a finite difference method for the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamical sense perfect and polytropic. The homogeneous boundary conditions for velocity, microrotation and heat flux are proposed. The sequence of approximate solutions for our problem is constructed by using the defined finite difference approximate equations system. We investigate the properties of these approximate solutions and establish their convergence to the strong solution of our problem globally in time, which is the main results of the paper. A numerical experiment is performed by solving the defined approximate ordinary differential equations system using strong-stability preserving (SSP) Runge-Kutta scheme for time discretization. [ABSTRACT FROM AUTHOR]

In this article, a numerical study for fractional reaction-subdiffusion equations is introduced using a class of finite difference methods. These methods are extensions of the weighted average methods for ordinary (non-fractional) reaction-subdiffusion equations. A stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. Simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, are given and checked numerically. Numerical test examples, figures, and comparisons have been presented for clarity. [ABSTRACT FROM AUTHOR]

We present a modified version of the two-player "tug-of-war" game introduced by Peres, Schramm, Sheffield, and Wilson (2009). This new tug-of- war game is identical to the original except near the boundary of the domain ??, but its associated value functions are more regular. The dynamic programming principle implies that the value functions satisfy a certain finite difference equation. By studying this difference equation directly and adapting techniques from viscosity solution theory, we prove a number of new results. We show that the finite difference equation has unique maximal and minimal solutions, which are identified as the value functions for the two tug-of-war players. We demonstrate uniqueness, and hence the existence of a value for the game, in the case that the running payoff functions is nonnegative. We also show that uniqueness holds in certain cases for sign-changing running payoff functions which are sufficiently small. In the limit ? ? 0, we obtain the convergence of the value functions to a viscosity solution of the normalized infinity Laplace equation. We also obtain several new results for the normalized infinity Laplace equation ???u = f. In particular, we demonstrate the existence of solutions to the Dirichlet problem for any bounded continuous f, and continuous boundary data, as well as the uniqueness of solutions to this problem in the generic case. We present a new elementary proof of uniqueness in the case that f > 0, f < 0, or f ? 0. The stability of the solutions with respect to f is also studied, and an explicit continuous dependence estimate from f ? 0 is obtained. [ABSTRACT FROM AUTHOR]

Abstract: Stretching computation strategies, including the hyperbolic exponential fittings along the lens interface, stretching transformations in z- and r-directions are discussed in this paper. The resulted adapted finite difference schemes are not only simple in structure, but also highly effective and efficient in the solution of paraxial Helmholtz equations for light beam propagation simulations and high performance lens designs. Cylindrically symmetric lens domains are assumed. Simulation results are given to illustrate the aforementioned numerical strategies with laboratory applications. Detailed algorithmic structures are given for immediate programming implementations and for similar lightwave propagation computation issues. [Copyright &y& Elsevier]

In this paper, we analyze finite difference discretizations for a class of control constrained elliptic optimal control problems. If the optimal control has a derivative of bounded variation, we show discrete quadratic convergence in terms of the mesh size h of the discrete optimal controls. Furthermore, based on the optimality conditions, we construct a new discrete control for which we derive continuous error estimates of order h2. [ABSTRACT FROM AUTHOR]

Abstract: A z-stretching finite difference method is developed for simulating the paraxial light beam propagation through a lens in a cylindrically symmetric domain. By introducing a domain transformation in the z-direction, we solve the corresponding complex difference equations containing an interface singularity over a computational space for great simplicity and efficiency. A specially designed matrix analysis is constructed to the study the numerical stability. Computational experiments are carried out for demonstrating our results. [Copyright &y& Elsevier]

Abstract: This paper presents a straightforward and generally applicable method for detection and elimination of errors in semi-analytical design sensitivities for any kind of FE-formulation. The basic property of the semi-analytical approach is that derivatives of the stiffness matrix and the load vector are approximated by finite differences. Obviously, truncation errors occur by this method which depend on the chosen step size and the kinematic assumptions of the mechanical model. The accuracy problems in the semi-analytical sensitivity analysis result from these approximation errors [Barthelemy B, Haftka RT. Accuracy analysis of the semi-analytical method for shape sensitivity calculation. Mech Struct Mach 1988;18:407–32]. In this contribution two beam elements (Euler–Bernoulli kinematics, Timoshenko kinematics) are utilized to emphasize the consequences of the approximation errors by an analytical computation of the error terms. These two elements show serious differences in the errors of the sensitivities. The ideas gained by these simple 1-d elements are extended further to 3-d elements with Reissner–Mindlin and Kirchhoff kinematics. The errors of the finite difference approximation of the derivatives may become serious, so it is necessary to correct them to obtain exact sensitivities. There exists a great variety of methods in the literature which try to eliminate the errors in the design sensitivities. Important contributions are published by Haftka and Adelmann, Mlejnek, Cheng and Olhoff, V. Keulen and De Boer among many others. In this paper, a method for the computation of correction factors based on product spaces of rigid body rotation vectors is presented. A straightforward derivation yields to a rigid body condition for the stiffness matrix derivative. The approximation of this derivative violates this rigid body condition due to the changed basis of the perturbed element. By the proposed method one obtains a set of correction factors related to the rigid body rotation vectors of the specific finite element. Due to the modification of the approximated stiffness matrix derivative by this set of factors one finally gets ‘exact’ sensitivities. The improved approximation of the stiffness matrix derivative satisfies the above mentioned rigid body condition. The basic advantage of the proposed method is the efficiency and the independence on the Finite Element formulation. In contrast to many other correction methods published so far, this approach is applicable to all kind of Finite Elements without major modifications. This gives rise to general shape optimization algorithms for a huge amount of finite elements without the necessity to derive each single element analytically. [Copyright &y& Elsevier]

New second- and fourth-order accurate semianalytical methods are introduced for the numerical solution of the one-dimensional nonlinear cubic Schrödinger equation. These methods are based on the combination of the Method of Lines and the Lanczos's Tau Method. The methods are self-starting and proved to be stable, accurate, and energy conservative for long time-integration periods. Approximate solutions are sought, on segmented sets of parallel lines, as finite expansions in terms of a given orthogonal polynomial basis. We have carried out numerical application concerning several cases for the propagation, collision and the bound states of N solitons, 2 ≤ N ≤ 5. Accurate results have been obtained using both shifted Chebyshev and Legendre polynomials. These results compare competitively with some other published results obtained using different methods. [ABSTRACT FROM AUTHOR]

Abstract: Parabolic partial differential equations with non-classical boundary specifications have received considerable interest in the mathematical applications in different areas of science and engineering. In this work, our goal is to obtain numerically the approximate solution of a parabolic equation with non-local conditions (on four boundaries) replacing the classical boundary specifications. Finite difference methods are given for solving this parabolic equation in two-dimensional space with non-standard boundary conditions. Several finite difference procedures are given and compared in terms of accuracy and computing time. Two fully explicit schemes, two fully implicit methods, an alternating direction implicit (ADI) formula and two explicit techniques of Saulyev are studied. The unconditional stability of the explicit procedures of Saulyev is significant. The unique advantage of the unconditionally stable implicit ADI method is that it needs only the solution of tridiagonal systems. A numerical integration procedure is employed to overcome the non-standard boundary specifications. Numerical results are given to demonstrate the efficiency and accuracy of the proposed finite difference methods. [Copyright &y& Elsevier]

A new type of Taylor series based finite difference approximations of higher-degree derivatives of a function are presented in closed forms, with their coefficients given by explicit formulas for arbitrary orders. Characteristics and accuracies of presented approximations and already presented central difference higher-degree approximations are investigated by performing example numerical differentiations. It is shown that the presented approximations are more accurate than the central difference approximations, especially for odd degrees. However, for even degrees, central difference approximations become attractive, as the coefficients of the presented approximations of even degrees do not correspond to equispaced input samples. [Copyright &y& Elsevier]

Taylor series based finite difference approximations of derivatives of a function have already been presented in closed forms, with explicit formulas for their coefficients. However, those formulas were not derived mathematically and were based on observation of numerical results. In this paper, we provide a mathematical proof of those formulas by deriving them mathematically from the Taylor series. [Copyright &y& Elsevier]

Multidimensional positive definite advection transport algorithm (MPDATA) is an iterative process for approximating the advection equation, which uses a donor cell approximation to compensate for the truncation error of the originally specified donor cell scheme. This step may be repeated an arbitrary number of times, leading to successively more accurate solutions to the advection equation. In this paper, we show how to sum the successive approximations analytically to find a single antidiffusive velocity that represents the effects of an arbitrary number of passes. The analysis is first done in one dimension to illustrate the method and then is repeated in two dimensions. The existence of cross terms in the truncation analysis of the two-dimensional equations introduces an extra complication into the calculation. We discuss the implementation of our new antidiffusive velocities and provide some examples of applications, including a third-order accurate scheme. [ABSTRACT FROM AUTHOR]

This paper studies an effective finite difference scheme for solving two-dimensional Heston stochastic volatility option-pricing model problems. A dynamically balanced up-downwind strategy for approximating the cross-derivative is implemented and analyzed. Semi-discretized and spatially nonuniform platforms are utilized. The numerical method comprised is simple and straightforward, with reliable first order overall approximations. The spectral norm is used throughout the investigation, and numerical stability is proven. Simulation experiments are given to illustrate our results. [ABSTRACT FROM AUTHOR]