利用带有改进算子矩阵的模块脉冲函数求非线性随机微分方程数值解.pdf

应用数学MATHEMATICA APPLICATA2023,36(4):1059-1068Numerical Solution of NonlinearStochastic Differential Equations byBlock Pulse Functions with ImprovedOperator MatrixJIANG Guo(姜国),LIU Fugang(刘富钢),CHEN Dan(陈丹)(School of Mathematics and Statistics,Hubei Normal University,Huangshi 435002,China)Abstract:This paper introduces an effective numerical method based on the blockpulse functions with improved operator matrix to solve the nonlinear stochastic differentialequations.The nonlinear stochastic differential equation is transformed into a set ofalgebraic equations by the improved operator matrix of block pulse function.Furthermore,we perform an error analysis and demonstrate that the method converges faster.Finally,numerical examples are used to support the method.Key words:Stochastic differential equation;Improved operator matrix;Block pulsefunctionCLC Number:O211.6AMS(2010)Subject Classification:60H20;45D99;65C30Document code:AArticle ID:1001-9847(2023)04-1059-101.IntroductionTheoretical foundation of stochastic differential equations(SDEs)was established in the1960s.With the rapid development of the theory of stochastic analysis,SDE had been widelyused in system science,engineering science and ecological science.However,it is difficultto obtain an exact solution for SDE.It makes sense to discuss numerical solutions to theequations110.Many authors have contributed in these areas.For instance,Akinbo et al.11introducedthe main concepts and techniques necessary for those wishing to perform numerical experi-ments involving SDEs.PEI et al.12have discussed the stochastic averaging for SDEs.Kloe-den and Platen13discussed the numerical solution of SDEs.JIANG and Schaufelberger14applied block pulse functions(BPFs)to control systems.Heydari et al.15illustrated theaccuracy and effectiveness of the generalized hat functions method.SANG et al.16proposedan effective numerical method to solve nonlinear stochastic It o-Volterra integral equations.On the other hand,Maleknejad et al.17introduced a method for solving stochastic Volter-ra integral equations,but the results are not very accurate.Therefore,we use BPFs withReceived date:2022-12-25Foundation item:Supported by the NSF Grant of Hubei Province(2022CFD042,2023AFD013)Biography:JIANG Guo,male,Han,Hubei,professor,major in stochastic integral equation.1060MATHEMATICA APPLICATA2023improved operator matrix to solve the following general nonlinear SDEs(1.1)or(1.2).Theadvantage of this method is that the calculation is simpler and the numerical solution is moreaccurate.dv(t)=F1(t,v(t)dt+F2(t,v(t)dW(t),v(0)=v0,t 0,1),(1.1)orv(t)=v0+t0F1(s,v(s)ds+t0F2(s,v(s)dW(s),0 s t 1,(1.2)where v(t)is unknown stochastic processes,v0is the initial value,F1(t,v)and F2(t,v)areknown analytic functions,and W(t)is standard Brownian motion.In Section 2,we consider the properties and definition of the BPFs,and show the im-proved operator matrix and the stochastic integral operator matrix of the BPFs.In Section 3,the numerical method of the nonlinear integral equation(1.2)is obtained.In Section 4,erroranalysis is presented.In Section 5,some numerical examples illustrate the method.Section6 is the conclusion.2.PreliminaryIn this section,the improved operator matrix and the stochastic integral operator matrixof BPFs are introduced.We define the BPFs asp(t)=1,(p 1)h t 0,p=1,4are positive constants and k1,k2 0,1).ThenT0E(?em(t)?2)dt=T0E(?v(t)vm(t)?2)dt O(h2),T 0,1).(4.4)ProofFor(4.3),we obtainE(?em(t)?2)3E(?v0 v0m?2)+E(?t0F1(s,v(s)F1(sm,vm(s)ds?2)+E(?t0F2(s,v(s)F2(sm,vm(s)dW(s)?2).No.4JIANG Guo,et al.：Numerical Solution of Nonlinear Stochastic Differential Equations1065By using the Lipschitz conditions and Cauchy-Schwartz inequality,we getE(?em(t)?2)3E(?v0 v0m?2)+E(t0?F1(s,v(s)F1(sm,vm(s)?2ds)+E(t0?F2(s,v(s)F2(sm,vm(s)?2ds)3?v0 v0m?2+t0E(1?s sm?+3?v(s)vm(s)?)2ds+t0E(2?s sm?+4?v(s)vm(s)?)2ds3?v0 v0m?2+221t0?s sm?2ds+223t0E(?em(s)?2)ds+222t0?s sm?2ds+224t0E(?em(s)?2)ds.Then,we can getE(?em(t)?2)3?v0 v0m?2+2(21+22)t0?s sm?2ds+6(23+24)t0E(?em(s)?2)ds,orE(?em(t)?2)(t)+t0E(?em(s)?2)ds,(t)=3?v0 v0m?2+2(21+22)t0?s sm?2ds,=6(23+24).Letg(t)=E(?em(t)?2).We getg(t)(t)+t0g()d,0,1).According to Gronwalls inequality,we getg(t)(t)+t0e(t)()d,t 0,1).ThenT0g(t)dt=T0E(?em(t)?2)dt T0(t)+t0e(t)()d)dt=T0(t)dt+T0t0e(t)()ddtT0(t)dt+eTT0t0()ddt=3T0?v0 v0m?2dt+6(21+22)T0t0?s sm?2dsdt+eT3T0t0?v0 v0m?2ddt+6(21+22)T0t00?s sm?2dsddt:=3I1+6(21+22)I2+eT3I3+6(21+22)I4,1066MATHEMATICA APPLICATA2023by using Lemma 4.1 and Lemma 4.2,we haveIp wph2.SoT0E(?em(t)?2)dt 3w1+6(21+22)w2+eT(3w3+6(21+22)w4)h2 O(h2),where wp,p=1,4,are independent nonnegative constants.The proof is completed.5.Some ExampleIn this section,we show some numerical examples.Example 5.113Consider the nonlinear SDEdv(t)=z2cos(v(t)sin3(v(t)dt zsin2(v(t)dW(t),t 0,1),the exact solution isv(t)=arccot(zW(t)+cot(v0).This example is solved for z=120and v0=120.The error standard deviations VS,error means VE,and confidence intervals for error mean of Example 5.1 with m=32 andm=64 are given in Tab.5.1 and Tab.5.2,respectively.The simulation result of the exactand approximate solution of the Example 5.1 are given in Fig.5.1 for m=32 and are shownin Fig.5.2 for m=64.The Legendre wavelet Galerkin method uses the integral operatormatrix of the general BPFs,and obtains the absolute errors of the approximate solution bynumerical examples.Compared with the method described in this paper,it can be seen fromthe error means of the table below that our accuracy rate(109)is higher than that of theLegendre wavelet Galerkin method(106)in 4.Tab.5.1When m=32,numerical results of Example 5.1tVEVS95%confidence intervalUpperLower1327.260847751093.630423871091.437647851081.4521695510102322.883569941091.441784971095.709468491095.7671398910113321.569868191087.849340951093.108339011083.1397363810104322.461167441081.230583721084.873111541084.9223348910105324.307197271082.153598631088.528250611088.614394551010Tab.5.2When m=64,numerical results of Example 5.1tVEVS95%confidence intervalUpperLower1642.659910251091.329955121095.266622291095.3198205010112647.536324741093.768162371091.492192291081.5072649410103641.640491401088.202457001093.248172971083.2809828010104642.084357421081.042178711084.127027701084.1687148510105642.483186651081.241593321084.916709571084.966373311010Example 5.26Consider the nonlinear SDEdv(t)=etsin(v(t)dt+etcos(v(t)dW(t),t 0,1).Let v0=3,the simulation results for m=32 and m=64 are severally given in Fig.5.3 and Fig.5.4.These two figs also show that the approximate solution fluctuates aroundthe mean orbit,where the mean solution is obtained by 104trajectories.No.4JIANG Guo,et al.：Numerical Solution of Nonlinear Stochastic Differential Equations106700.10.20.30.40.50.60.70.80.910.049950.050.050050.05010.050150.05020.05025Approximation solutionExact solutionFig.5.1When m=32,exact andapproximate solutions00.10.20.30.40.50.60.70.80.910.049950.050.050050.0501Approximation solutionExact solutionFig.5.2When m=64,exact andapproximate solutions00.10.20.30.40.50.60.70.80.91-3.2-3.15-3.1-3.05-3-2.95-2.9-2.85-2.8-2.75Approximation solutionMean solutionFig.5.3When m=32,mean andapproximate solutions00.10.20.30.40.50.60.70.80.91-3.4-3.3-3.2-3.1-3-2.9-2.8-2.7-2.6-2.5Approximation solutionMean solutionFig.5.4When m=64,mean andapproximate solutions6.ConclusionThis paper proposes a numerical method for solving nonlinear SDEs based on BPFs.Byusing the improved operator matrix,simulation results show that the solution of this methodis very close to the exact solution.Section 4 demonstrates an error analysis and comparesit with 2.The method of approximate solution convergence is faster.Some examples arelisted to verify the feasibility of the current method.References:1 HIGHAM D J.An algorithmic introduction to numerical simulation of stochastic differential equa-tionsJ.SIAM Rev.,2001,43(3):525-546.2 YUAN C,MAO X.A note on the rate of convergence of the EulerMaruyama method for stochasticdifferential 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Sys-temsM.Berlin:Springer-Verlag,1992.15 HEYDARI M H,HOOSHMANDASL M R,CATTANI C,et al.An efficient computational method forsolving nonlinear stochastic It o integral equations:Application for stochastic problems in physicsJ.J.Comput.Phys.,2015,283:148-168.16 SANG X Y,JIANG G,WU J H.Numerical solution of nonlinear stochastic It o-Volterra integralequations by block pulse functionsJ.Math.Appl.,2019,32(4):935-946.17 MALEKNEJAD K,KHODABIN M,ROSTAMI M.Numerical solution of stochastic Volterra integralequations by a stochastic operational matrix based on block pulse functionsJ.Math.Comput.Model.,2012,55(3-4):791-800.利用带有改进算子矩阵的模块脉冲函数求非线性随机微分方程数值解姜国,刘富钢,陈丹(湖北师范大学数学与统计学院,湖北 黄石 435002)摘要：本文介绍基于改进算子矩阵的模块脉冲函数求解非线性随机微分方程的有效数值方法.利用模块脉冲函数的积分算子矩阵将非线性随机积分方程转化为代数方程.此外,我们进行了误差分析,并证明该方法收敛更快.最后,通过数值算例对该方法进行了验证.关键词：随机微分方程;改进算子矩阵;模块脉冲函数