Efficient numerical solution of Fisher's equation by using B-spline method

This paper seeks to develop an efficient B-spline scheme for solving Fisher's equation, which is a nonlinear reaction–diffusion equation describing the relation between the diffusion and nonlinear multiplication of a species. To find the solution, domain is partitioned into a uniform mesh and then cubic B-spline function is applied to Fisher's equation. The method yields stable and accurate solutions. The results obtained are acceptable and in good agreement with some earlier studies. An important advantage is that the method is capable of greatly reducing the size of computational work.     

A high-order finite difference scheme for a singularly perturbed fourth-order ordinary differential equation

In this paper a singularly perturbed fourth-order ordinary differential equation is considered. The differential equation is transformed into a coupled system of singularly perturbed equations. A hybrid finite difference scheme on a Vulanovi?–Shishkin mesh is used to discretize the system. This hybrid difference scheme is a combination of a non-equidistant generalization of the Numerov scheme and the central difference scheme based on the relation between the local mesh widths and the perturbation parameter. We will show that the scheme is maximum-norm stable, although the difference scheme may not satisfy the maximum principle. The scheme is proved to be almost fourth-order uniformly convergent in the discrete maximum norm. Numerical results are presented for supporting the theoretical results.     

The numerical solution of the matrix Riccati differential equation

A numerically stable and fast computational method is given for the solution of the matrix Ricatti differential equation with finite terminal time.     

Finite difference method for time-space-fractional Schrödinger equation

In this paper, an implicit finite difference scheme for the nonlinear time-space-fractional Schrödinger equation is presented. It is shown that the implicit scheme is unconditionally stable with experimental convergence order of O(τ^2?α+h²), where τ and h are time and space stepsizes, respectively, and α (0<α<1) is the fractional-order in time. In order to reduce the computational cost, the explicit–implicit scheme is proposed such that the nonlinear term is easily treated. Meanwhile, the implicit finite difference scheme for the coupled time-space-fractional Schrödinger system is also presented, which is unconditionally stable too. Numerical examples are given to support the theoretical analysis.     

The short memory principle for solving Abel differential equation of fractional order

In this paper, the short memory principle (SMP) is applied for solving the Abel differential equation with fractional order. We evaluate the approximate solution at the end of required interval, and construct a suitable iteration scheme employing this end point as initial value. Numerical experiments show that our iteration method is simple and efficient, and that a proper length of memory could maintain the validity of the short memory principle.     

广义Maxwell流体分数阶微分方程的数值解法

针对广义Maxwell粘弹性流体分数阶微分方程,建立了一种隐式差分格式,给出了数值解的求解公式,证明了隐式差分格式稳定性与收敛性。     

Non-linear feedback control of parabolic partial differential difference equation systems

This paper proposes a general method for the synthesis of non-linear output feedback controllers for single-input singleoutput quasi-linear parabolic partial differential difference equation (PDDE) systems, for which the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Initially, a non-linear model reduction scheme which is based on combination of Galerkin's method with the concept of approximate inertial manifold is employed for the derivation of differential difference equation (DDE) systems that describe the dominant dynamics of the PDDE system. Then, these DDE systems are used as the basis for the explicit construction of non-linear output feedback controllers through combination of geometric and Lyapunov techniques. The controllers guarantee stability and enforce output tracking in the closed-loop parabolic PDDE system independently of the size of the state delay, provided that the separation of the slow and fast eigenvalues of the spatial differential operator is sufficiently large and an appropriate matrix is positive definite. The methodology is successfully employed to stabilize the temperature profile of a tubular reactor with recycle at a spatially non-uniform unstable steadystate.     

用MCM方程去除椒盐噪声的数值方案

均值曲率运动（MCM）方程能有效去除脉冲噪声。给出了MCM方程的一种显示差分格式。针对ENoD-8格式在去除高密度噪声时会残留斑块的问题,提出了新的ENoD-A格式。该格式的应用减少了迭代过程中的计算量,提高了去噪效率。实验结果表明,新格式增强了方程去除椒盐噪声的能力,同时又保护了图像细节。     

Generalized trapezoidal formulas for the black–scholes equation of option pricing

For the celebrated Black–Scholes parabolic equation of option pricing, we present new time integration schemes based on the generalized trapezoidal formulas introduced by Chawla et al. [3]. The resulting GTF(α) schemes are unconditionally stable and second order in both space and time. Interestingly, since the Black–Scholes equation is linear, GTF (1/3) attains order three in time. The computational performance of the obtained schemes is compared with the Crank–Nicolson scheme for the case of European option valuation. Since the payoff is nondifferentiable having a “corner” on expiry at the exercise price, the classical trapezoidal formula used in the Crank–Nicolson scheme can experience oscillations at this corner. It is demonstrated that our present GTF (1/3) scheme can cope with this situation and performs consistently superior than the Crank–Nicolson scheme.     

Stability and convergence of F iterative scheme with an application to the fractional differential equation

In this paper, we prove that F iterative scheme is almost stable for weak contractions. Further, we prove convergence results for weak contractions as well as for generalized non-expansive mappings due to Hardy and Rogers via F iterative scheme. We also prove that F iterative scheme converges faster than the some known iterative schemes for weak contractions. An illuminative numerical example is formulated to support our assertion. Finally, utilizing our main result the solution of nonlinear fractional differential equation is approximated.

     

Almost sure stability for uncertain differential equation with jumps

Uncertain differential equation with jumps, as a crucial tool to deal with a discontinuous uncertain system, is a type of differential equation driven by both canonical Liu process and uncertain renewal process. So far, a concept of stability in measure for an uncertain differential equation with jumps has been proposed. As a supplement, this paper proposes a concept of almost sure stability for an uncertain differential equation with jumps. A sufficient condition is derived for an uncertain differential equation with jumps being stable almost surely. As a corollary, a sufficient condition is also given for a linear uncertain differential equation with jumps being stable almost surely.     

Chebyshev series solution of the two dimensional heat equation

In this paper, a computational scheme is presented for the solution of the second order parabolic partial differential equation by Chebyshev polynomials. An automated strategy of inputting the relevant coefficients of the equation and scanning a recurrence tableau gives the Chebyshev coefficients from which the solution to the equation can be obtained. Examples are given to portray the flexibility of the method.     

Finite difference procedure for solution of poisson equation over complex domains with Neumann boundary conditions

A generalized finite difference scheme for solving Poisson equation over multiply connected domain bounded by irregular boundaries at which Neumann boundary conditions are specified, is presented in this paper. The method used to treat the Neumann condition is a six-point gradient approximation method given by Greenspan[6]. The method is generalized to treat all types of grid intersections with the boundary. An efficient computational procedure is devised by eliminating the calculations at the boundary during the interations.The scheme is applied to the problem of forced convection heat transfer in a fully developed laminar flow through seven and nineteen rod-cluster assemblies. Fluid properties are assumed to be uniform. In arriving at the fast converging and efficient method from computational point of view, different iterative techniques, overrelaxation methods and boundary treatments were tried. The results of computations and the computer times are reported in the present paper.     

A new stochastic approach for solution of Riccati differential equation of fractional order

In this article, a stochastic technique has been developed for the solution of nonlinear Riccati differential equation of fractional order. Feed-forward artificial neural network is employed for accurate mathematical modeling and learning of its weights is made with heuristic computational algorithm based on swarm intelligence. In this scheme, particle swarm optimization is used as a tool for the rapid global search method, and simulating annealing for efficient local search. The scheme is equally capable of solving the integer order or fractional order Riccati differential equations. Comparison of results was made with standard approximate analytic, as well as, stochastic numerical solvers and exact solutions.     

Exponential nonlinear observer based on the differential state-dependent Riccati equation

This paper presents a novel nonlinear continuous-time observer based on the differential state-dependent Riccati equation(SDRE) filter with guaranteed exponential stability.Although impressive results have rapidly emerged from the use of SDRE designs for observers and filters,the underlying theory is yet scant and there remain many unanswered questions such as stability and convergence.In this paper,Lyapunov stability analysis is utilized in order to obtain the required conditions for exponential stability of the estimation error dynamics.We prove that under specific conditions,the proposed observer is at least locally exponentially stable.Moreover,a new definition of a detectable state-dependent factorization is introduced,and a close relation between the uniform detectability of the nonlinear system and the boundedness property of the state-dependent differential Riccati equation is established.Furthermore,through a simulation study of a second order nonlinear model,which satisfies the stability conditions,the promising performance of the proposed observer is demonstrated.Finally,in order to examine the effectiveness of the proposed method,it is applied to the highly nonlinear flux and angular velocity estimation problem for induction machines.The simulation results verify how effectively this modification can increase the region of attraction and the observer error decay rate.     

Auxiliary equation method for time-fractional differential equations with conformable derivative

In this paper, the auxiliary equation method is applied to obtain analytical solutions of (2 + 1)-dimensional time-fractional Zoomeron equation and the time-fractional third order modified KdV equation in the sense of the conformable fractional derivative. Given equations are converted to the nonlinear ordinary differential equations of integer order; and then, the resulting equations are solved using a novel analytical method called the auxiliary equation method. As a result, some exact solutions for them are successfully established. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective.     

Convergence analysis of the Hopmoc method

The Hopmoc method combines concepts of the modified method of characteristics (MMOC) and the Hopscotch method. First, Hopmoc resembles Hopscotch because it decomposes the set of grid points into two subsets. Namely, both subsets have their unknowns separately updated within one semi-step. Furthermore, each subset undergoes one explicit and one implicit update of its unknowns in order to lead to a symmetrical procedure. Such decomposition inspired the use of a convergence analysis similar to the one used in alternating direction implicit methods. Secondly, the steps are evaluated along characteristic lines in a semi-Lagrangian approach similar to the MMOC. In this work, both consistency and stability analysis are discussed for Hopmoc applied to a convection–diffusion equation. The analysis produces sufficient conditions for the consistency analysis and proves that the Hopmoc method presents unconditional stability. In addition, numerical results confirm the conducted convergence analysis.     

Numerical solution of fractional delay differential equation by shifted Jacobi polynomials

In this paper, the fractional delay differential equation (FDDE) is considered for the purpose to develop an approximate scheme for its numerical solutions. The shifted Jacobi polynomial scheme is used to solve the results by deriving operational matrix for the fractional differentiation and integration in the Caputo and Riemann–Liouville sense, respectively. In addition to it, the Jacobi delay coefficient matrix is developed to solve the linear and nonlinear FDDE numerically. The error of the approximate solution of proposed method is discussed by applying the piecewise orthogonal technique. The applicability of this technique is shown by several examples like a mathematical model of houseflies and a model based on the effect of noise on light that reflected from laser to mirror. The obtained numerical results are tabulated and displayed graphically.     

Unstructured lattice Boltzmann equation with memory

Starting from a second-order differential form of the semi-discrete Boltzmann equation, we construct a new finite-volume lattice Boltzmann equation on unstructured grids (ULBE). The new scheme (ULBE with memory) is demonstrated for the case of a Taylor-vortex flow and shown to produce stable and accurate results with time-step more than an order of magnitude above the standard LBE stability threshold.     

Stabilization of a class of nonlinear uncertain ordinary differential equation by parabolic partial differential equation controller

This article is concerned with stabilization for a class of uncertain nonlinear ordinary differential equation (ODE) with dynamic controller governed by linear 1?d heat partial differential equation (PDE). The control input acts at the one boundary of the heat's controller domain and the second boundary injects a Dirichlet term in ODE plant. The main contribution of this article is the use of the recent infinite‐dimensional backstepping design for state feedback stabilization design of coupled PDE‐ODE systems, to stabilize exponentially the nonlinear uncertain systems, under the restrictions that (a) the right‐hand side of the ODE equation has the classical particular form: linear controllable part with an additive nonlinear uncertain function satisfying lower triangular linear growth condition, and (b) the length of the PDE domain has to be restricted. We solve the stabilization problem despite the fact that all known backstepping transformation in the literature cannot decouple the PDE and the ODE subsystems. Such difficulty is due to the presence of a nonlinear uncertain term in the ODE system. This is done by introducing a new globally exponentially stable target system for which the PDE and ODE subsystems are strongly coupled. Finally, an example is given to illustrate the design procedure of the proposed method.