When differential-algebraic equations of index 3 or higher are solved with backward differentiation formulas, the solution can have gross errors in the first few steps, even if the initial values are equal to the exact solution and even if the stepsize is kept constant. This raises the question of what are consistent initial values for the difference equations. Here we study how to change the e...

In this paper we study the linear mean-field backward stochastic differential equations (mean-field BSDE) of form (0.1) d Y ( t ) = − [ α 1 + β Z ∫ R 0 η , ζ K ν 2 E ] γ B N ̃ ∈ T ξ . where is unknown solution triplet, a Brownian motion, compensated Poisson random measure, independent We prove existence and uniqueness triplet such systems. Then give an explicit formula for first component by usi...

The aim of this study will be to design Parallel solver (PS) for oscillatory stiff systems ordinary differential equations (ODEs). PS constructed via a type specially transformed exponentially fitted multinomial approximant in accordance with the behaviour solution. method interpolation and collocation utilized. principal local truncation errors used derive suitable step size decide error toler...

We prove a weak rate of convergence fully discrete scheme for stochastic Cahn--Hilliard equation with additive noise, where the spectral Galerkin method is used in space and backward Euler time. Compared Allen--Cahn type partial differential equation, error analysis here much more sophisticated due to presence unbounded operator front nonlinear term. To address such issues, novel direct approac...

For a backward stochastic differential equation (BSDE, for short), when the generator is not progressively measurable, it might admit adapted solutions, shown by an example. However, Volterra integral equations (BSVIEs, generators are allowed to be anticipating. This gives, among other things, essential difference between BSDEs and BSVIEs. Under some proper conditions, well-posedness of such BS...