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In this work we study a dynamic contact problem between a thermoelastic mixture and a deformable obstacle. The classical normal compliance condition is used for modeling the contact. The variational formulation of this problem is written as a nonlinear coupled system of three parabolic variational equations. An existence and uniqueness result is proved using the Faedo-Galerkin method. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the decay of the discrete energy and the linear convergence of the algorithm are deduced. Finally, some numerical simulations are presented to show the convergence and the behavior of the solution.

Using matrix norm techniques, we give a unified convergence analysis of a general projection method for computing stationary density functions of Frobenius-Perron operators associated with piecewise C 2 and stretching transformations of the unit interval.

The weak and strong ergodic properties of nonautonomous linear ordinary differential equations are considered. It is shown that if the coefficient matrix function is bounded, essentially nonnegative and uniformly irreducible, then the normalized positive solutions are asymptotically equivalent to the Perron vectors of the strongly positive transition matrix at infinity (weak ergodicity). If, in addition, the coefficient matrix function is uniformly continuous, then the convergence of the normalized positive solutions to the same strongly positive limiting vector (strong ergodicity) is equivalent to the convergence of the Perron vectors of the coefficient matrices.

In the framework of an elliptic partial differential equation (PDE), certain boundary conditions and a set of points to approximate in a Lipschitz domain and arbitrary dimension, we use radial basis function (RBF) techniques for the construction and characterization of discrete PDE splines. We also show convergence and derive error estimates.

We study for the first time the ill-posed backward problem for a contaminated nonlinear predator-prey system whose velocities of migration depend on the total average populations in the considered space domain. We propose a new regularized problem for which we are able to prove its unique solvability in Theorem 1 . Moreover, under some mild assumptions on the true solution, we give useful and rigorous error estimates and convergence rates in both the L 2 - and H 1 -norms in Theorem 2 , Theorem 3 , respectively. Furthermore, numerical simulations are performed to illustrate the accuracy and stability of the regularized solution.

We derive new formulas for the price of the European call and put options in the Black-Scholes model, under the form of uniformly convergent series generalizing previously known approximations. We also provide precise boundaries for the convergence speed and apply the results to the calculation of hedge parameters (Greeks)., Comment: v4 (Some typos corrected + adding the DOI). To appear in Journal of Mathematical Analysis and Applications. arXiv admin note: text overlap with arXiv:1710.01141

In this study, we consider nonlinear partial differential equations (PDEs) and stochastic PDEs (SPDEs) with Neumann boundary conditions on domains that satisfy the Lions–Sznitmann–Saisho conditions. Using the convergence result based on the penalization approximation for stochastic differential equations with normal reflections on nonsmooth and nonconvex domains, we establish the existence and comparison principle for the viscosity solutions of nonlinear PDEs with the Neumann conditions associated with the optimal control problem. We also obtain for nonlinear SPDEs and backward SPDEs with Neumann condition the representations of the stochastic viscosity solutions.

We address the problem of the asymptotic reducibility of a system of analytic quasi-periodic linear functional differential equations extending the results of M.S. Bortei and V.I. Fodchuk [3] to the case of (possibly) complex eigenvalues. At the same time, we provide a simpler proof of the asymptotic reducibility that bypasses the method of accelerated convergence and it is instead based on more transparent KAM-type estimates.

In this paper, we establish a new estimate (including lower and upper bounds) for an important quantity involved in the convergence analysis of smoothed aggregation algebraic multigrid methods. The new upper bound improves the existing ones. And our upper bound is optimal.

In this study, we mainly consider the order reduction for the solution coefficient vectors of the classical continuous space-time finite element (CCSTFE) method for the two-dimensional (2D) unsteady Stokes equation. First, we construct the CCSTFE model for the 2D unsteady Stokes equation about vorticity–stream functions, and show the existence, stability, and convergence of the CCSTFE solutions. We then apply proper orthogonal decomposition (POD) to develop a reduced-order extrapolation continuous space-time finite element (ROECSTFE) model, as well as discussing the existence, stability, and convergence of the ROECSTFE solutions based on matrix analysis so the theoretical analysis is highly concise, and we use a flowchart to search for the ROECSTFE solutions. Finally, we provide two numerical examples to confirm the correctness of the theoretical results obtained. We also show that the computational results agree well with the theoretical results, thereby demonstrating the effectiveness and feasibility of the proposed ROECSTFE method. Therefore, both the theory and method are totally different from the existing POD reduced order methods.

In this paper, we study the full compressible magnetohydrodynamic (MHD) flows with the viscous stress tensor given by the Maxwell law in R 3 . First, local existence of solutions for general large initial data and global existence of solutions for small initial data are shown. Then we obtain the convergence of solutions for the system with the Maxwell law to the corresponding solution for the classical system with the Newtonian law when the relaxation time goes to 0.

A continuum steady-state model of oxygen transport in the brain is studied. The unique solvability of the corresponding nonlinear boundary-value problem is proven. The convergence of a simple iterative procedure for finding solutions of the boundary-value problem is established. Numerical experiments showing the consistency of the model are conducted.

We study the following fourth order quasilinear Schrodinger equations { Δ 2 u − Δ u + V ( x ) u − 1 2 u Δ ( u 2 ) = f ( u ) , u ∈ H 2 ( R N ) , where Δ 2 u = Δ ( Δ u ) and 3 ≤ N ≤ 6 . For some suitable assumptions on V ∈ C 1 ( R N ) and f ∈ C ( R ) , by developing a new Brezis-Lieb type convergence result, we prove the existence of Nehari-Pohozaev type ground state solutions. Our main results generalize and improve the existing ones, such as Chen et al. (2014) [10] , Liu and Zhao (2019) [26] , and some other related literature.

We consider a hemivariational inequality of elliptic type (HVI, for short) in a reflexive Banach space, prove its solvability and the compactness of its set of solutions. To this end we employ a surjectivity theorem for multivalued mappings that we use for the sum of a maximal monotone operator and a bounded pseudomonotone operator. Next, we introduce the concepts of strongly and weakly well-posedness in the generalized sense for the HVI and provide two characterizations for the strongly well-posedness, under different assumptions on the data. These characterizations are formulated in terms of the metric properties of the approximating sets. We also provide sufficient conditions which guarantee the weakly and strongly well-posedness in the generalized sense of the HVI. Finally, we consider two perturbations of the HVI for which we obtain convergence results in the sense of Kuratowski.

In the setting of hyperbolic spaces, we show that the convergence of Browder-type sequences and Halpern iterations respectively entail the convergence of their viscosity version with a Rakotch map. We also show that the convergence of a hybrid viscosity version of the Krasnoselskii-Mann iteration follows from the convergence of the Browder type sequence. Our results follow from proof-theoretic techniques (proof mining). From an analysis of theorems due to T. Suzuki, we extract a transformation of rates for the original Browder type and Halpern iterations into rates for the corresponding viscosity versions. We show that these transformations can be applied to earlier quantitative studies of these iterations. From an analysis of a theorem due to H.-K. Xu, N. Altwaijry and S. Chebbi, we obtain similar results. Finally, in uniformly convex Banach spaces we study a strong notion of accretive operator due to Brezis and Sibony and extract an uniform modulus of uniqueness for the property of being a zero point. In this context, we show that it is possible to obtain Cauchy rates for the Browder type and the Halpern iterations (and hence also for their viscosity versions).

We consider the convergence of solutions and cost functional in some optimal control problems arising in the study of the adsorption chemical phenomenon in which some microscopic reactant particles are placed over an internal manifold γ of the chemical reactor Ω. The chemical reaction is given by some Robin-type boundary condition on the boundary of the periodic set of particles. We consider the special case in which there is a critical relation between the coefficient of the reaction, the size of the particles and the dimension of the space. This gives rise to a “strange term”, which is not occurring for other scales, and thus the limit cost functional must be suitably defined. In a last section, we use this type of technique to prove a similar “energy convergence” result (improving the H 0 1 ( Ω ) -weak convergence) for the problem without control for the critical scale case.

In this present article we set forth with the new notion of I -αβ-statistical convergence which becomes more generalized version of I -statistical convergence. Successively to compare with the following important results of Balcerzak et al. (2020) [6] (i): Let I be an ideal such that I ⊆ Z = { A ⊆ N : lim n → ∞ ⁡ | A ∩ [ 1 , n ] | n = 0 } . Then I -statistical convergence coincides with statistical convergence, (ii): I -statistical convergence coincides with statistical convergence if I = I 1 n = { S ⊆ N : ∑ n ∈ S 1 n ∞ } , (iii): Let I be a maximal ideal. Then I -statistical convergence does not coincide with statistical convergence, we produce significant results that elucidate incongruity between I -αβ-statistical convergence and I -statistical convergence. Also, following the line of works of Aytar (2008) [4] , Savas et al. (2015) [40] and Ghosal et al. (2020) [23] , we propose two new notions, namely rough weighted I -αβ-statistical convergence and weighted I -αβ-statistical cluster points set over locally solid Riesz spaces. The core factor “degree of roughness” had been conventionally treated as a non-negative real number where as we invent the idea of neighborhood approximation. Finally, topological properties and relationships of the above mentioned notions are investigated.

Our goal is to study fractional powers of a cascade system of partial differential equations. We explicitly calculate the fractional powers of linear operators associated to this type of system and we discuss local solvability of the fractional equation with subcritical nonlinearity. As an example, a cascade system of Schrodinger equation is analyzed. A connection between the fractional system and the original system is established and we prove the convergence of the linear semigroups obtained by the fractional power operator to the original linear semigroup, as the power α approaches 1.

In this study, we investigate the convergence behavior as α → 2 of the solutions to stochastic differential equations (SDEs) with Lipschitz and Holder drifts, and driven by α-stable Levy processes. First, we prove that the solution of an SDE driven by an α-stable Levy process converges weakly to that of an SDE forced by Brownian motion in the Skorokhod space. We mainly employ Aldous' criterion and martingale characterization. Then, by using an abstract result given by Gyongy and Krylov, we prove that the solution of an SDE driven by the subordinated Brownian motion via an α / 2 -stable subordinator strongly converges to that of an SDE forced by Brownian motion. Based on uniform estimates of the mean exit times and related probabilistic representations, we also present an α-dependent result for fractional diffusion equations. Finally, given α ⁎ in ( 1 , 2 ) , we discuss the α-continuity of the solution to an SDE driven by an α-stable Levy process at α ⁎ .

We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay of the parameters involved in the uniformly perfect condition. The continuity may fail otherwise., Comment: 13 pages

In this article we apply proper splittings of matrices to develop an iterative process to approximate solutions of matrix equations of the form T X = W . Moreover, by using the partial order induced by positive semidefinite matrices, we obtain equivalent conditions to the convergence of this process. We also include some speed comparison results of the convergence of this method. In addition, for all matrix T we propose a proper splitting based on the polar decomposition of T.

A recent article of the first author presented a constructive proof for the Weierstrass approximation theorem for a wide class of weighted spaces of continuous functions defined on [ 0 , ∞ ) , using Chlodovsky's generalization of the Bernstein polynomial operators. Here, we investigate necessary and sufficient conditions related to the uniform boundedness of a sequence of such operators and to their convergence properties when the weight is a classical Freud weight of the form w ( x ) = e − x α , α ≥ 1 .

We prove that even irregular convergence of semigroups of operators implies similar convergence of mild solutions of the related semi-linear equations with Lipschitz continuous nonlinearity. This result is then applied to three models originating from mathematical biology: shadow systems, diffusions on thin layers, and dynamics of neurotransmitters, 21 pages, no figures. Minor revision: Some typos fixed

This study presents an efficient, accurate, effective and unconditionally stable time stepping scheme for the Darcy-Brinkman equations in double-diffusive convection. The stabilization within the proposed method uses the idea of stabilizing the curvature for velocity, temperature and concentration equations. Accuracy in time is proven and the convergence results for the fully discrete solutions of problem variables are given. Several numerical examples including a convergence study are provided that support the derived theoretical results and demonstrate the efficiency and the accuracy of the method. (C) 2018 Elsevier Inc. All rights reserved.

In this paper, we mainly utilize proper orthogonal decomposition (POD) to reduce the order for the coefficient vector of the classical Crank–Nicolson finite spectral element (CCNFSE) method of the two-dimensional (2D) non-stationary Boussinesq equations about vorticity-stream functions so that the reduced-order method maintains all the advantages of the CCNFSE method. Toward this end, we first establish a CCNFSE format with the second-order time accuracy for the two-dimensional (2D) non-stationary Boussinesq equations about vorticity-stream functions and analyze the existence, stability, and convergence of the CCNFSE solutions. And then, by POD, we establish a reduced-order extrapolated Crank–Nicolson finite spectral element (ROECNFSE) method and analyze the existence, stability, and convergence of the ROECNFSE solutions as well as offer the flowchart for seeking ROECNFSE solutions. Finally, we use two sets of numerical experiments to validate that the numerical computational consequences are accorded with the theoretical ones such that the effectiveness and feasibility of the ROECNFSE method are further verified. Both theory and method in this paper is new and completely different from the existing reduced-order methods.

Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. A series representation of the discrete fractional Laplace operator for positive non-integer powers is developed. Its convergence to a series representation of a known case of positive integer powers is proven as the power tends to the integer value. Furthermore, we show that the new representation for arbitrary real-valued positive powers of the discrete Laplace operator is consistent with existing theoretical results.

Maximal estimates for Schrodinger means and convergence almost everywhere of sequences of Schrodinger means are studied.

This work is concerned with various weighted four dimensional matrix summability methods in modular function spaces associating with generalized difference operator involving ( p , q ) -gamma function. Following very recent results of Kadak (2017) [20] , we first introduce a new type of difference operator on double function sequences and, also define modularly weighted A-statistical convergence and modularly statistically N ˜ A -summability by means of weighted four dimensional regular matrix A. We also present some important inclusion relations between newly proposed summability methods. Moreover, based on the concept of modularly statistically N ˜ A -summability, we prove a Korovkin type approximation theorem for functions of two variables in modular spaces. Finally, in order to show that our proposed method and its applications to approximation results are stronger than the existing methods, we display an illustrative example using bivariate ( p , q ) -Bernstein Kantorovich operators.

In this paper, we study on the local and semi-local convergence and dynamic of an iterative optimal two-step Steffensen's method. We study local convergence by using Lipschitz conditions and continuity of differentiable function F. We also obtain semi-local convergence by using majorizing sequences. By this approach, we can obtain the convergence of the method by minimum of auxiliary sequences and few initial conditions. Moreover, we analyze the stability of the method via Julia sets and parameter planes that is a novelty for Steffensen-type methods. We extend this method for obtaining the sign function of a matrix. Numerical examples are given to confirm our theoretical results.

We study the local convergence to equilibria, as time goes to infinity, of trajectories of semilinear wave systems with friction damping on the boundary and subject to nonlinear potential energy. Estimates for the speed of convergence are obtained in terms of the behavior of the nonlinear feedback close to the origin. As an example of application, we show that the trajectories of a sine-Gordon system with nonlinear boundary damping, approach equilibria at least polynomially.

We investigate the asymptotic behavior of solutions to a nonhomogeneous second order difference inclusion of monotone type and prove new results on the existence and approximation of zeroes of the operator. In particular, we introduce new weighted averages of the solutions and prove their convergence to a zero of the operator under different control conditions on the parameters.

In this paper, the state reconstruction of the wave equation with general viscosity and non-collocated observation and control are given from both the theoretical aspect and the numerical aspect. The forward-backward observers-based algorithm is utilized to solve this problem. The formula for calculating the initial value is derived. Moreover, the iterative sequence is also built for any given guess value and it is showed that it strongly converges to initial value. However, because the exponential stabilities of the error systems between the original systems and the forward∖backward observers play important roles in the involved algorithm, the exponential stability of some related system is firstly discussed. Furthermore, combining the theory of port-Hamiltonian system and the finite difference, the semi-discretization scheme of the finite difference with order reduction is given and the uniform exponential stability of the semi-discretization system is verified by the method paralleling to the continuous system. Finally, the convergence analysis of the finite difference scheme is given and the convergence of the solution of the mixed finite element scheme induced by the finite difference scheme with order reduction to the solution of the continuous counterpart is also presented.

We consider the 3D magnetohydrodynamic-alpha model driven by a multiplicative noise. The model arises in the modelling of turbulent flows in magneto fluids. We prove the existence and uniqueness of the probabilistic strong solution. We also propose and analyze a numerical scheme for the approximation of the unique strong solution. We prove the convergence in mean square of the scheme, which is a linear evolution equation with additive noise.

For scalar conservation laws posed on bounded domains in R d , the convergence of quasilinear parabolic viscous approximations to the entropy solution in the sense of Bardos-Leroux-Nedelec is established.

In this paper, a fourth-order finite difference scheme that preserves the dissipation of energy for solving the 2D Benjamin-Bona-Mahony-Burgers (BBMB) equation is proposed. The scheme is three-level in time and linear-implicit. The solvability, unconditional stability and convergence are rigorously proved by the discrete energy method. The rate of convergence is of O ( τ 2 + h 4 ) . Numerical examples are given to confirm the good accuracy and the effectiveness of the present scheme for handling the 2D homogeneous and inhomogeneous BBMB equations.

We consider a Lotka-Volterra system with both local and nonlocal intraspecific and interspecific competitions, where nonlocal competitions depend on both spatial and temporal effects in a general form. Firstly, global stability of two constant semi-trivial equilibria and global convergence of the coexistence equilibrium are derived by using the functional and energy method, which implies that strengths of nonlocal intraspecific competitions have great effects on these global dynamics but the nonlocal interspecific competitions not and extends global results of Gourley and Ruan (2003) [11] . Secondly, global attracting region of each constant semi-trivial equilibrium is limited by its environment capacity regardless of the distinction of local and nonlocal intraspecific competitions. Thirdly, in the weak competition case, the coexistence equilibrium becomes Turing unstable when the kernels are chosen as generally distributed delay functions in temporal and the nonlocal intraspecific competitions are suitably strong. Additionally, spatially homogeneous and inhomogeneous periodic solutions are found numerically.

The sufficient conditions of the almost sure exponential stability of the exact solution for the stochastic pantograph differential equation are considered, with a Khasminskii-type condition. The almost sure exponential stability of the numerical solutions by the Euler–Maruyama method and the backward Euler–Maruyama method is also discussed, based on the discrete semimartingale convergence theorem. We present the sufficient conditions for the stability of the Euler–Maruyama method, with one extra condition when compared with the exact solution. We show that the backward Euler–Maruyama method can share almost the same conditions for the almost sure exponential stability as the exact solution.

In this paper we study two optimal design problems associated to fractional Sobolev spaces Ws,p(Ω). Then we find a relationship between these two problems and finally we investigate the convergence when s↑1. Fil: Fernandez Bonder, Julian. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina Fil: Spedaletti, Juan Francisco. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Luis. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi". Universidad Nacional de San Luis. Facultad de Ciencias Físico, Matemáticas y Naturales. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi"; Argentina

We study strong (pathwise) approximation of Cox–Ingersoll–Ross processes. We propose a Milstein-type scheme that is suitably truncated close to zero, where the diffusion coefficient fails to be locally Lipschitz continuous. For this scheme we prove positive convergence rates for the full parameter range including the accessible boundary regime. The error criterion is given by the maximal L p -distance of the solution and its approximation on a compact interval. In the particular case of a squared Bessel process of dimension δ > 0 the convergence rate is given by min ⁡ ( 1 , δ ) / ( 2 p ) .

In this paper, we study the minimal Keller-Segel model with a logistic source and obtain quantitative and qualitative descriptions of the competition between logistic damping and other ingredient, especially, chemotactic aggregation to guarantee boundedness and convergence. More specifically, we establish how precisely strong a logistic source can prevent blow-up, and then we obtain an explicit relationship between logistic damping and other ingredient, especially, chemotactic aggregation so that convergences are ensured and their respective convergence rates are explicitly calculated out. Known results in the literature are completed and refined. Furthermore, our findings provide clues on how to produce blowup solutions for KS chemotaxis models with logistic sources., Comment: 34 pages, a second revised version; detected typos were amended

In this paper, we study the Cauchy problem for the hyperbolic p-system with time-gradually-degenerate damping term − 1 ( 1 + t ) λ u for 0 ⩽ λ 1 , and show that the damped p-system has a couple of global solutions uniquely, and such solutions tend time-asymptotically to the shifted nonlinear diffusion waves, which are the solutions of the corresponding nonlinear parabolic equation governed by the Darcy's law. We further derive the convergence rates when the initial perturbations are in L 2 . The approach adopted is the technical time-weighted energy method.

The classical concept of synchronization is usually related to the locking of the basic frequencies and instantaneous phases of regular oscillations, and this question is addressed by studying specific kinds of coupled systems. This work presents a different point of view. We do not study the convergence of coupled systems to a synchronized behaviour, but try to answer the following question: in a population of coupled differential systems, when each cell (subsystem) exhibits a periodic behaviour, is the whole trajectory periodic? We define generalized spatial automata, with reference to continuous spatial automata, by means of coupling maps and associated measures on the set of cells: the main idea is the fact that a cell interprets its own environment via the states of the whole population and according to its own state. A natural partition of periods is such that cells belong to the same class if their trajectories share a common period. We demonstrate that in a general case where cells belong to an a priori unstructured set, and their trajectories evolve in possibly distinct Banach spaces, the set of classes of periods is generally countable. In particular, when the set of cells is endowed with a Borelian structure, all the cells necessarily share a common period.

We study a nonlocal biharmonic MEMS equation with the fringing field. It arises in the Micro-Electro Mechanical System (MEMS) devices. First, we establish the local solution and extend it globally in time. Next, we discuss the dynamical properties of omega limit set. Especially, we show that the omega limit set is included in the set of the stationary solution. Finally, we consider the convergence rate to the stationary solution.

The viscous and inviscid aggregation equation with Newtonian potential models a number of different physical systems and has close analogs in 2D incompressible fluid mechanics. We consider a slight generalization of these equations in the whole space establishing well-posedness of the viscous and inviscid equations, spatial decay of the viscous solutions, and the convergence of viscous solutions to the inviscid solution as the viscosity goes to zero.

We give a sufficient condition that guarantees the convergence on some neighborhood of the origin of formal power series in C n .

There are generalizations and complementations of results from our joint papers with Filipow ( [15] and [16] ). We study relationships between ideal equal convergence and various kinds of ideal convergences of sequences of real functions. Furthermore we consider ideal version of the bounding number on sets from coideals.

This paper address the existence and uniqueness, in suitable functional spaces, of the solution of a non linear system of equations originating from the implicit time discretization of the coupled unsteady Navier-Stokes and Allen-Cahn equations. This result is based on the convergence analysis of an original stabilized fixed point algorithm, from which we also get a maximum principle. Numerical experiments are performed to illustrate the influence of the implicit formulation in comparison to a simpler approach based on a semi-explicit time scheme.

In this paper, we consider a generalized differential equation with feedback and time-varying delays and prove the existence of a unique pseudo compact almost automorphic solution that is exponentially attractive. We show our result by applying the Banach fixed point theorem and the properties of pseudo compact almost automorphic function. We apply our results and obtain new criteria for the existence and convergence dynamics of pseudo compact almost automorphic solutions for some models that involves sum of nonlinear terms that satisfy a general condition. These models are very general and include sums of unimodals terms with monotones, such as Nicholson, Mackey-Glass, Lasota-Wazeska and others. The results obtained are new and they recover, extend and improve recent works.

In this paper, the sufficient and necessary conditions for complete convergence and the Kolmogorov strong law of large numbers for weighted sums of m-extended negatively dependent random variables are presented. Some applications of the main results are also provided, including the weak and strong consistency of the least squares estimator in multiple linear regression models, strong consistency of conditional Value-at-risk estimator, and the asymptotics of the quasi-renewal counting process. Finally, some numerical simulations are carried out to confirm the theoretical results.

This paper develops a splitting-up method to solution of backward stochastic parabolic partial differential equations. Both splitting equations are simpler for the numerical computations. Two different approximation schemes are given, and their convergence results are proved. The first-order convergence rate is proved in the linear case.