a quenching criterion for a multi dimensional parabolic problem due to a concentrated nonlinear source a multi dimensional parabolic first initial boundary value problem with a concentrated nonlinear source is studied a criterion for its solution to quench in a finite time t q everywhere on the concentrated nonlinear source only is given an upper bound for t q is also deduced for illustration an example is given

an adaptive splitting approach for the quenching solution of reaction diffusion equations over nonuniform grids the numerical solution of a nonlinear degenerate reaction diffusion equation of the quenching type is investigated while spatial derivatives are discretized over symmetric nonuniform meshes a peaceman rachford splitting method is employed to advance solutions of the semidiscretized system the temporal step is determined adaptively through a suitable arc length monitor function a criterion is derived to ensure that the numerical solution acquired preserves correctly the positivity and monotonicity of the analytical solution weak stability is proven in a von neumann sense via the norm computational examples are presented to illustrate our results

a splitting moving mesh method for reaction diffusion equations of quenching type this paper studies the numerical solution of multi dimensional nonlinear degenerate reaction diffusion differential equations with a singular force term over a rectangular domain the equations may generate strong quenching singularities our work focuses on a variable temporal step peaceman rachford splitting method with an adaptive moving mesh in space the temporal and spatial adaptation is implemented based on arc length type of estimations of the time derivative of the solution since the time derivative of the solution approaches infinity when the quenching occurs the multi dimensional problem is split into a few one dimensional problems and the splitting procedure can also be parallelized so that the computational time is significantly reduced the physical monotonicity of the solution and stability of this variable step moving mesh scheme are analyzed for the time away from the quenching as stability analysis may not be valid when it is very close to the quenching thus an exact linear problem is introduced to justify the stability near the quenching time finally we provide some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method for the quenching problem or other problems with point singularities we will also show the significant reduction in computational time required with parallel implementation of the algorithm on a computer with multi cpu

a fundamental mean square convergence theorem for sdes with locally lipschitz coefficients and its applications a version of the fundamental mean square convergence theorem is proved for stochastic differential equations sdes in which coefficients are allowed to grow polynomially at infinity and which satisfy a one sided lipschitz condition the theorem is illustrated on a number of particular numerical methods including a special balanced scheme and fully implicit methods the proposed special balanced scheme is explicit and its mean square order of convergence is 1 2 some numerical tests are presented

a stochastic model for internal hiv dynamics in this paper we analyse a stochastic model representing hiv internal virus dynamics the stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling we show that the model established in this paper possesses non negative solutions as this is essential in any population dynamics model we also carry out analysis on the asymptotic behaviour of the model we approximate one of the variables by a mean reverting process and find out the mean and variance of this process numerical simulations conclude the paper

moving mesh methods for problems with blow up in this paper we consider the numerical solution of pdes with blow up for which scaling invariance plays a natural role in describing the underlying solution structures it is a challenging numerical problem to capture the qualitative behaviour in the blow up region and the use of nonuniform meshes is essential we consider moving mesh methods for which the mesh is determined using so called moving mesh partial differential equations mmpdes specifically the underlying pde and the mmpde are solved for the blow up solution and the computational mesh simultaneously motivated by the desire for the mmpde to preserve the scaling invariance of the underlying problem we study the effect of different choices of mmpdes and monitor functions it is shown that for suitable ones the mmpde solution evolves towards a moving mesh which close to the blow up point automatically places the mesh points in such a manner that the ignition kernel which is well known to be a natural coordinate in describing the behaviour of blow up approaches a constant as t to t the blow up time several numerical examples are given to verify the theory for these mmpde methods and to illustrate their efficacy

complete blow up after tmax for the solution of a semilinear heat equation soit un sous ensemble ouvert borne de r n a frontiere lisse on considere le probleme u t u f u dans o t u 0 sur o t u x o u 0 x x ou f r r est localement de lipschitz non decroissante et f o 0

convergence and asymptotic stability of the explicit steklov method for stochastic differential equations in this paper we develop a new numerical method with asymptotic stability properties for solving stochastic differential equations sdes the foundations for the new solver are the steklov mean and an exact discretization for the deterministic version of the sdes strong consistency and convergence properties are demonstrated for the proposed method moreover a rigorous linear and nonlinear asymptotic stability analysis is carried out for the multiplicative case in a mean square sense and for the additive case in a path wise sense using the pullback limit in order to emphasize the characteristics of the steklov discretization we use as benchmarks the stochastic logistic equation and the langevin equation with a nonlinear potential of the brownian dynamics we show that the steklov method has mild stability requirements and allows long time simulations in several applications

semidiscretization in time of nonlinear parabolic equations with blowup of the solution in this paper the author considers the first boundary value problem for the nonlinear equation u t delta u m alpha u m in omega a smooth bounded domain in mathbb r n with the zero lateral boundary condition and with a positive initial condition m is supposed to be larger than one and alpha positive a scheme for the discretization in time of that problem is proposed it is proven that if the exact solution blows up in a finite time it is the same for the numerical solution estimates of the blow up time are obtained the stability of the method and the convergence for a class of initial conditions is proved

numerical simulation of singular solutions to the two dimensional cubic schrodinger equation numerical simulations of the cubic nonlinear schrodinger equation are presented in dimension n 2 the emphasis is on a detailed mechanism of blow up the numerical results indicate that the blow up is not restricted to the case where the problem is posed in the entire r2 space but also occurs with periodic boundary conditions the structure of the solution as the singular time t is approached has been investigated using examples with and without radial symmetry and or periodicity in most cases we observe that the amplitude develops something like the peaked self similar profile predicted by zakharov and synakh 38 which has a t t 2 3 maximum amplitude and a t t 2 3 width

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

strong convergence of the stopped euler maruyama method for nonlinear stochastic differential equations in this paper numerical methods for the nonlinear stochastic differential equations sdes with non global lipschitz drift coefficient are discussed the existing known results have only so far shown that the classical explicit euler maruyama em approximate solutions converge to the true solution in probability 22 23 more recently the authors in 16 proved that the classical em method will diverge in l 2 sense for the underlying sdes in this paper and those sdes with superlinearly growing coefficients these strongly indicate that the classical em method is not good enough for the highly nonlinear sdes however in this paper we introduce a modified em method using stopping time and show successfully that the discrete version of the modified em approximate solution converges to the true solution in the strong sense namely in l 2 with a order arbitrarily close to a half

continuation of blowup solutions of nonlinear heat equations in several space dimensions the possible continuation of solutions of the nonlinear heat equation in r n r ut u m u p with m 0 p 1 after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p thus for m p 2 we find a phenomenon of nontrivial continuation where the regionfx u x t 1g is bounded and propagates with finite speed this we call incomplete blowup for n 3 and p m n 2 n 2 we find solutions that blow up at finite t t and then become bounded again for t t otherwise we find that blowup is complete for a wide class of initial data in the analysis of the behavior for large p a list of critical exponents appears whose role is described we also discuss a number of related problems and equations we apply the same technique of analysis to the problem of continuation after the onset of extinction for example for the equation ut u m u p m 0

stability of a stochastic sir system abstract we propose a stochastic sir model with or without distributed time delay and we study the stability of disease free equilibrium the numerical simulation of the stochastic sir model shows that the introduction of noise modifies the threshold of system for an epidemic to occur and the threshold stochastic value is found

analysis of moving mesh partial differential equations with spatial smoothing two moving mesh partial differential equations mmpdes with spatial smoothing are derived based upon the equidistribution principle this smoothing technique is motivated by the robust moving mesh method of dorfi and drury j comput phys 69 1987 pp 175 195 it is shown that under weak conditions the basic property of no node crossing is preserved by the spatial smoothing and a local quasi uniformity property of the coordinate transformations determined by these mmpdes is proven it is also shown that discretizing the mmpdes using centered finite differences these basic properties are preserved

homogenization of high frequency wave propagation in linearly elastic layered media using a continuum irving kirkwood theory abstract this article presents an application of a recently developed continuum homogenization theory inspired by the classical work of irving and kirkwood to the homogenization of plane waves in layered linearly elastic media the theory explicitly accounts for the effects of microscale dynamics on the macroscopic definition of stress it is shown that for problems involving high frequency wave propagation the macroscopic stress predicted by the theory differs significantly from classical homogenized stress definitions the homogenization of plane waves is studied to illustrate key aspects and implications of the theory including the characteristics of the homogenized macroscopic stress and the influence of frequency on the determination of an intermediate asymptotic length scale in addition a method is proposed for predicting the homogenized stress field in a one dimensional bar subjected to a frequency dependent forced vibration using only knowledge of the boundary conditions and the material s dispersion solution furthermore it is shown that due to the linearity of the material the proposed method accurately predicts the homogenized stress for any time varying displacement or stress boundary condition that can be expressed as a sum of time periodic signals

the form of blow up for nonlinear parabolic equations on considere des equations paraboliques semi lineaires de la forme u t 2 u sf u ou f est positive et 0 ds f s est finie on etudie des problemes a 1 dimension dans la region semi infinie x o et on trouve le temps t

self similar subsolutions and blow up for nonlinear parabolic equations abstract for a wide class of nonlinear parabolic equations of the form u t u f u u we give finite time blow up results for large initial data blow up time estimates are also provided these results rely on a new method of comparison with suitable blowing up self similar subsolutions as a consequence we improve several known results on reaction diffusion equations on generalized burgers equations and on the equation of chipot and weissler this method can also apply to degenerate equations of porous medium type and provides a unified treatment for a large class of problems both semilinear and quasilinear

solving degenerate reaction diffusion equations via variable step peaceman rachford splitting this paper studies the numerical solution of two dimensional nonlinear degenerate reaction diffusion differential equations with singular forcing terms over rectangular domains the equations considered may generate strong quenching singularities this investigation focuses on a variable time step peaceman rachford splitting method for the aforementioned problem the time adaptation is implemented based on arc length estimations of the first time derivative of the solution the two dimensional problem is split into several one dimensional problems so that the computational cost is significantly reduced the monotonicity and localized linear stability of the variable step scheme are investigated we give some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method over existing methods for the quenching problem it is also shown that the numerical solution obtained preserves important properties of the physical solution of the given problem

a rescaling algorithm for the numerical calculation of blowing up solutions on presente un algorithme a invariance d echelle pour le calcul numerique des solutions a singularites explosives des equations d evolution non lineaires

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

blow up in a system of partial differential equations with conserved first integral part ii problems with convection a reaction diffusion convection equation with a nonlocal term is studied the nonlocal operator acts to conserve the spatial integral of the unknown function as time evolves the equations are parameterised by and for 1 the equation arises as a similarity solution of the navier stokes equations and the nonlocal term plays the role of pressure for 0 the equation is a nonlocal reaction diffusion problem the aim of the paper is to determine for which values of the parameter blow up occurs and to study its form in particular interest is focused on the three cases 1 2 and 1 it is observed that for any 0 1 2 nonuniform global blow up occurs if 1 2 1 then the blow up is global and uniform while for 1 the navier stokes equations there are exact solutions with initial data of arbitrarily large l l 2 and h 1 norms that decay to zero furthermore one of these exact solutions is proved to be nonlinearly stable in l 2 for arbitrarily large supremum norm an understanding of this transition from blow up behaviour to decay behaviour is achieved by a combination of analysis asymptotics and numerical techniques

blowup in a partial differential equation with conserved first integral a reaction diffusion equation with a nonlocal term is studied the nonlocal term acts to conserve the spatial integral of the unknown function as time evolves such equations give insight into biological and chemical problems where conservation properties predominate the aim of the paper is to understand how the conservation property affects the nature of blowup the equation studied has a trivial steady solution that is proved to be stable existence of nontrivial steady solutions is proved and their instability established numerically blowup is proved for sufficiently large initial data by using a comparison principle in fourier space the nature of the blowup is investigated by a combination of asymptotic and numerical calculations

boundedness of global solutions of nonlinear diffusion equations it is well known that there exist choices of u0 for which the corresponding solutions tend to zero as t cc and other choices for which the solutions blow up in finite time if we are interested in other types of behavior we may consider the possibility p u t u exists globally

stability of the peaceman rachford approximation abstract letaanbbe two self adjoint operators in a hilbert space it is assumed thataandbare bounded from below the following operator expressionp t 1 ta 1 1 tb 1 tb 1 1 ta called the peaceman rachford formula satisfies formally fortsmall the relationp t exp 2t a b o t2 ifaandbdo not commute it is not true in general thatp t is a stable approximation we give an example ofaandbsuch that p t b h the operator norm ofp t is not bounded by 1 o t sufficient conditions of stability are given they involve conditions on the commutators of a and b the proof relies on estimates on sandwiches i e products of a finite number of copies of a b 1 ta 1 1 tb 1 and elements of an algebra m of bounded operators inh

a homogenization method for thermomechanical continua using extensive physical quantities this article proposes a continuum thermomechanical homogenization method inspired by the irving kirkwood procedure relating the atomistic equations of motion to the balance laws of continuum mechanics this method yields expressions for the macroscopic stress and heat flux in terms of microscopic kinematic and kinetic quantities the resulting equation for macroscopic stress affords a rational comparison with the widely used hill mandel stress deformation condition while the one for heat flux reduces under certain assumptions to a hill mandel like condition involving heat flux and the gradient of temperature

novel formulations of microscopic boundary value problems in continuous multiscale finite element methods abstract this article explores the use of a homogenization method inspired by the classical irving kirkwood procedure in the continuous multiscale modeling of elastic solids within the context of the finite element method this homogenization method gives rise to a broad range of allowable boundary conditions for the rve which in turn yield a rich spectrum of estimates of the macroscopic stress response for two representative problems involving heterogeneous elastic bodies

asymptotic behaviour of the stochastic lotka volterra model this paper examines the asymptotic behaviour of the stochastic extension of a fundamentally important population process namely the lotka volterra model the stochastic version of this process appears to have far more intriguing properties than its deterministic counterpart indeed the fact that a potential deterministic population explosion can be prevented by the presence of even a tiny amount of environmental noise shows the high level of difference which exists between these two representations

the linear steklov method for sdes with non globally lipschitz coefficients we present an explicit numerical method for solving stochastic differential equations with non globally lipschitz coefficients a linear version of the steklov average under a split step formulation supports our new solver the linear steklov method converges strongly with a standard one half order also we present numerical evidence that the explicit linear steklov reproduces almost surely stability solutions with high accuracy for diverse application models even for stochastic differential systems with super linear diffusion coefficients

strong and weak divergence in finite time of euler s method for stochastic differential equations with non globally lipschitz continuous coefficients the stochastic euler scheme is known to converge to the exact solution of a stochastic differential equation sde with globally lipschitz continuous drift and diffusion coefficients recent results extend this convergence to coefficients that grow at most linearly for superlinearly growing coefficients finite time convergence in the strong mean square sense remains in this article we answer this question to the negative and prove for a large class of sdes with non globally lipschitz continuous coefficients that euler s approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point even worse the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

a simple adaptive grid method in two dimensions this paper gives an interpretation of the concept of equidistribution in the context of adaptive grid generation for multidimensional problems it is shown that the equidistribution principle cannot be satisfied throughout the domain of the problem and based on this recognition a local equidistribution principle is developed a discrete formulation is described for grid generation in two space dimensions and a smoothing mechanism is presented for improving mesh quality the adaptive grid method that is constructed contains three grid quality parameters numerical examples illustrate adaptive grid generation using a prescribed monitor function and grid generation for numerical solution of partial differential equations results show that the method produces high quality grids and that it is fairly insensitive to the choice of parameters

numerical simulation of blowup in nonlocal reaction diffusion equations using a moving mesh method in this paper we implement the moving mesh pde method for simulating the blowup in reaction diffusion equations with temporal and spacial nonlinear nonlocal terms by a time dependent transformation the physical equation is written into a lagrangian form with respect to the computational variables the time dependent transformation function satisfies a parabolic partial differential equation usually called moving mesh pde mmpde the transformed physical equation and mmpde are solved alternately by central finite difference method combined with a backward time stepping scheme the integration time steps are chosen to be adaptive to the blowup solution by employing a simple and efficient approach the monitor function in mmpdes plays a key role in the performance of the moving mesh pde method the dominance of equidistribution is utilized to select the monitor functions and a formal analysis is performed to check the principle a variety of numerical examples show that the blowup profiles can be expressed correctly in the computational coordinates and the blowup rates are determined by the tests

convergence and stability of the semi tamed euler scheme for stochastic differential equations with non lipschitz continuous coefficients recently explicit tamed schemes were proposed to approximate the sdes with the non lipschitz continuous coefficients this work proposes a semi tamed euler scheme which is also explicit to solve the sdes with the drift coefficient equipped with the lipschitz continuous part and non lipschitz continuous part it is shown that the semi tamed euler converges strongly with the standard order one half to the exact solution of the sde we also investigate the stability inheritance of the semi tamed euler schemes and reveal that this scheme does have advantage in reproducing the exponential mean square stability of the exact solution numerical experiments confirm the theoretical analysis

adaptive finite element methods for parabolic problems iv nonlinear problems we extend our program on adaptive finite element methods for parabolic problems to a class of nonlinear scalar problems we prove a posteriori error estimates design corresponding adaptive algorithms and present some numerical results

strong convergence and stability of implicit numerical methods for stochastic differential equations with non globally lipschitz continuous coefficients we are interested in the strong convergence and almost sure stability of euler maruyama em type approximations to the solutions of stochastic differential equations sdes with non linear and non lipschitzian coefficients motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence in addition we examine the globally almost surely asymptotic stability in this non linear setting for em type schemes in particular we present a stochastic counterpart of the discrete lasalle principle from which we deduce stability properties for numerical methods

on the critical exponent for reaction diffusion equations in this paper we study the initial boundary value problem for u t u h t u p with homogeneous dirichlet boundary conditions where h t t q for large t let s sup s positive solutions w of u t u such that mathop lim sup limits t to infty t s parallel w cdot t parallel infty infty then for p 1 q 1 s we show if p p there are global positive solutions that decay to zero uniformly for t if 1 p p then all nontrivial solutions blow up in finite time we determine p for some conical domains in r 2 and r 3 a similar result is derived for a bounded domain if h t e t for large t

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

blowup in diffusion equations a survey this paper deals with quasilinear reaction diffusion equations for which a solution local in time exists if the solution ceases to exist for some finite time we say that it blows up in contrast to linear equations blowup can occur even if the data are smooth and well defined for all times depending on the equation either the solution or some of its derivatives become singular we shall concentrate on those cases where the solution becomes unbounded in finite time this can occur in quasilinear equations if the heat source is strong enough there exist many theoretical studies on the question on the occurrence of blowup in this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup the asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power a better knowledge would be useful also for their numerical treatment thus not surprisingly the numerical analysis of this type of problems is still at a rather early stage the goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems

the truncated euler maruyama method for stochastic differential equations influenced by higham et al 2003 several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations sdes under the local lipschitz condition these numerical methods include the tamed euler maruyama em method the tamed milstein method the stopped em the backward em the backward forward em etc in this paper we will develop a new explicit method called the truncated em method for the nonlinear sde d x t f x t d t g x t d b t and establish the strong convergence theory under the local lipschitz condition plus the khasminskii type condition x t f x p 1 2 g x 2 k 1 x 2 the type of convergence specifically addressed in this paper is strong l q convergence for 2 q p and p is a parameter in the khasminskii type condition

numerical solutions to singular reaction diffusion equation over elliptical domains solid fuel ignition models for which the dynamics of the temperature is independent of the single species mass fraction attempt to follow the dynamics of an explosive event such models may take the form of singular degenerate reaction diffusion equations of the quenching type that is the temporal derivative blows up in finite time while the solution remains bounded theoretical and numerical investigations have proved difficult for even the simplest of geometries and mathematical degeneracies quenching domains are known to exist for piecewise smooth boundaries but often lack theoretical estimates rectangular geometries have been primarily studied here this acquired knowledge is utilized to determine new theoretical estimates to quenching domains for arbitrary piecewise smooth connected geometries elliptical domains are of primary interest and a peaceman rachford splitting algorithm is then developed that employs temporal adaptation and nonuniform grids rigorous numerical analysis ensures numerical solution monotonicity positivity and linear stability of the proposed algorithm simulation experiments are provided to illustrate the accomplishments

the blowup property of solutions to some diffusion equations with localized nonlinear reactions abstract in this paper we investigate the blowup property of solutions to the equation u t u u x 0 t where x0 is a fixed point in the domain we show that under certain conditions the solution blows up in finite time moreover we prove that the set of all blowup points is the whole region furthermore the growth rate of solutions near the blowup time is also derived finally the results are generalized to the following nonlocal reaction diffusion equation u t u u dx

semi discretization in time of a fast diffusion equation abstract in this paper we consider the first boundary value problem for the nonlinear concentration dependent diffusion equation u t um t 0 in a smooth bounded domain in r n with the zero lateral boundary condition and with a positive initial condition m is supposed to be between 0 and 1 then the solution decays to zero in some finite time t depending upon the initial data here we propose a scheme for the discretization in time of that problem we prove that the numerical solution is null after a finite number of time steps and we obtain the convergence of the method and estimates of the numerical extinction time

a stochastic model for the evolution of bone metastasis persistence and recovery abstract bone metastasis is a complex disease that modifies the natural dynamics of the bone remodeling process which is vital for maintaining a healthy bone structure nowadays there are deterministic differential models that describe the bone metastasis evolution by considering interaction mechanisms between cancer cells and basic bone cells such as osteoclasts and osteoblasts based on these deterministic models a stochastic differential model is proposed to describe the bone metastasis evolution in order to analyze the effect of environmental fluctuations inclusions conditions for the existence and uniqueness of a nonnegative solution of the proposed stochastic model are given to characterize whether the invasion of cancer persists or is eradicated finally we illustrate different dynamical behaviors of the stochastic solution by means of numerical simulations

the numerical study of blowup with application to a nonlinear schrodinger equation abstract we discuss the use of numerical methods in the study of the solutions of evolution problems which exhibit finite time unbounded growth we first examine a naive approach in which the growth rate of the numerical solution is accepted as an approximation of the true growth rate as we shall demonstrate for a radial nonlinear schrodinger equation this approach is inadequate since different discretizations exhibit different growth rates the spurious behaviour of discretizations in the neighbourhood of the singularity is discussed a reliable procedure for the estimation of the blowup parameters is considered which eliminates the discrepancies between different numerical methods

deciphering singularities by discrete methods we consider the problem of estimating numerically the parameters of singularities of solutions of differential equations we propose a novel approach which is based on discretizing the governing equation and timestepping in the complex domain some applications to ordinary and partial differential equations are discussed