We consider the mean-square stability of the so-called improved split-step theta method for stochastic differential equations. First, we study the mean-square stability of the method for linear test equations with rea...We consider the mean-square stability of the so-called improved split-step theta method for stochastic differential equations. First, we study the mean-square stability of the method for linear test equations with real parameters. When 0 ≥ 3/2, the improved split-step theta methods can reproduce the mean-square stability of the linear test equations for any step sizes h 〉 0. Then, under a coupled condition on the drift and diffusion coefficients, we consider exponential mean-square stability of the method for nonlinear non-autonomous stochastic differential equations. Finally, the obtained results are supported by numerical experiments.展开更多
In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both...In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both stochastically C-stable and stochastically B-consistent,is convergent has been proved in a previous paper.In order to analyze the convergence of the split-step theta method(θ∈[1/2,1]),the stochastic C-stability and stochastic B-consistency under the condition of global monotonicity have been researched,and the rate of convergence 1/2 has been explored in this paper.It can be seen that the convergence does not require the drift function should satisfy the linear growth condition whenθ=1/2 Furthermore,the rate of the convergence of the split-step scheme for stochastic differential equations with additive noise has been researched and found to be 1.Finally,an example is given to illustrate the convergence with the theoretical results.展开更多
Through-thickness heterogeneity in creep properties of 7B50-T7451 aluminum alloy Friction Stir Welding(FSW)joints was investigated.Creep tests for three slices of the FSW joint were conducted at the temperature of 150...Through-thickness heterogeneity in creep properties of 7B50-T7451 aluminum alloy Friction Stir Welding(FSW)joints was investigated.Creep tests for three slices of the FSW joint were conducted at the temperature of 150-200℃ and applied stress of 60-225 MPa.The theta projection method was used to predict creep curves and minimum creep rate.The results show that the minimum creep rate increases and creep rupture life decreases with the increase of creep temperature and applied stress.Creep properties of the FSW joint deteriorate along the thickness direction from the top to the bottom.The threshold stress of all three slices of the FSW joint decreases with the increase of creep temperature and even disappears at 200℃ for the bottom slice.Creep activation energy approaches the activation energy of the lattice self-diffusion of aluminum.The value of true stress exponent for different slices is approximately equal to three.The predominant creep mechanism of the FSW joint is dislocation viscous glide by lattice self-diffusion.What is more,a constitutive model is established based on the theta method to accurately describe creep behavior ofdifferent slices of the FSW joint.展开更多
In this paper,we make use of stochastic theta method to study the existence of the numerical approximation of random periodic solution.We prove that the error between the exact random periodic solution and the approxi...In this paper,we make use of stochastic theta method to study the existence of the numerical approximation of random periodic solution.We prove that the error between the exact random periodic solution and the approximated one is at the 1/4 order time step in mean sense when the initial time tends to∞.展开更多
In this paper,we investigate the stability of the split-step theta(SST)method for a class of nonlinear regime-switching jump systems–neutral stochastic delay differential equations(NSDDEs)with Markov switching and ju...In this paper,we investigate the stability of the split-step theta(SST)method for a class of nonlinear regime-switching jump systems–neutral stochastic delay differential equations(NSDDEs)with Markov switching and jumps.As we know,there are few results on the stability of numerical solutions for NSDDEs with Markov switching and jumps.The purpose of this paper is to enrich conclusions in such respect.It first devotes to show that the trivial solution of the NSDDE with Markov switching and jumps is exponentially mean square stable and asymptotically mean square stable under some suitable conditions.If the drift coefficient also satisfies the linear growth condition,it then proves that the SST method applied to the NSDDE with Markov switching and jumps shares the same conclusions with the exact solution.Moreover,a numerical example is demonstrated to illustrate the obtained results.展开更多
In this paper,we investigate the theoretical and numerical analysis of the stochastic Volterra integro-differential equations(SVIDEs)driven by L´evy noise.The existence,uniqueness,boundedness and mean square expo...In this paper,we investigate the theoretical and numerical analysis of the stochastic Volterra integro-differential equations(SVIDEs)driven by L´evy noise.The existence,uniqueness,boundedness and mean square exponential stability of the analytic solutions for SVIDEs driven by L´evy noise are considered.The split-step theta method of SVIDEs driven by L´evy noise is proposed.The boundedness of the numerical solution and strong convergence are proved.Moreover,its mean square exponential stability is obtained.Some numerical examples are given to support the theoretical results.展开更多
For solving the stochastic differential equations driven by fractional Brownian motion,we present the modified split-step theta method by combining truncated Euler-Maruyama method with split-step theta method.For the ...For solving the stochastic differential equations driven by fractional Brownian motion,we present the modified split-step theta method by combining truncated Euler-Maruyama method with split-step theta method.For the problem under a locally Lipschitz condition and a linear growth condition,we analyze the strong convergence and the exponential stability of the proposed method.Moreover,for the stochastic delay differential equations with locally Lipschitz drift condition and globally Lipschitz diffusion condition,we give the order of convergence.Finally,numerical experiments are done to confirm the theoretical conclusions.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos. 91130003 and 11371157)the Scientific Research Innovation Team of the University “Aviation Industry Economy” (Grant No. 2016TD02)
文摘We consider the mean-square stability of the so-called improved split-step theta method for stochastic differential equations. First, we study the mean-square stability of the method for linear test equations with real parameters. When 0 ≥ 3/2, the improved split-step theta methods can reproduce the mean-square stability of the linear test equations for any step sizes h 〉 0. Then, under a coupled condition on the drift and diffusion coefficients, we consider exponential mean-square stability of the method for nonlinear non-autonomous stochastic differential equations. Finally, the obtained results are supported by numerical experiments.
基金Supported by the National Natural Science Foundation of China (Grant No. 12301521)the Natural Science Foundation of Shanxi Province (Grant No. 20210302124081)。
文摘In this paper,the convergence of the split-step theta method for stochastic differential equations is analyzed using stochastic C-stability and stochastic B-consistency.The fact that the numerical scheme,which is both stochastically C-stable and stochastically B-consistent,is convergent has been proved in a previous paper.In order to analyze the convergence of the split-step theta method(θ∈[1/2,1]),the stochastic C-stability and stochastic B-consistency under the condition of global monotonicity have been researched,and the rate of convergence 1/2 has been explored in this paper.It can be seen that the convergence does not require the drift function should satisfy the linear growth condition whenθ=1/2 Furthermore,the rate of the convergence of the split-step scheme for stochastic differential equations with additive noise has been researched and found to be 1.Finally,an example is given to illustrate the convergence with the theoretical results.
基金financially supported by the National Natural Science Foundation of China(No.52075450)the Fundamental Research Funds for the Central Universities,China(No.D5000220503).
文摘Through-thickness heterogeneity in creep properties of 7B50-T7451 aluminum alloy Friction Stir Welding(FSW)joints was investigated.Creep tests for three slices of the FSW joint were conducted at the temperature of 150-200℃ and applied stress of 60-225 MPa.The theta projection method was used to predict creep curves and minimum creep rate.The results show that the minimum creep rate increases and creep rupture life decreases with the increase of creep temperature and applied stress.Creep properties of the FSW joint deteriorate along the thickness direction from the top to the bottom.The threshold stress of all three slices of the FSW joint decreases with the increase of creep temperature and even disappears at 200℃ for the bottom slice.Creep activation energy approaches the activation energy of the lattice self-diffusion of aluminum.The value of true stress exponent for different slices is approximately equal to three.The predominant creep mechanism of the FSW joint is dislocation viscous glide by lattice self-diffusion.What is more,a constitutive model is established based on the theta method to accurately describe creep behavior ofdifferent slices of the FSW joint.
基金supported by the National Natural Science Foundation of China (No.11871184,11701127)by the Natural Science Foundation of Hainan Province(Grant No.117096)
文摘In this paper,we make use of stochastic theta method to study the existence of the numerical approximation of random periodic solution.We prove that the error between the exact random periodic solution and the approximated one is at the 1/4 order time step in mean sense when the initial time tends to∞.
基金This work is partially supported by the National Natural Science Foundation of China(Nos.1190139&11671149,11871225)the Natural Science Foundation of Guangdong Province(No.2017A030312006).
文摘In this paper,we investigate the stability of the split-step theta(SST)method for a class of nonlinear regime-switching jump systems–neutral stochastic delay differential equations(NSDDEs)with Markov switching and jumps.As we know,there are few results on the stability of numerical solutions for NSDDEs with Markov switching and jumps.The purpose of this paper is to enrich conclusions in such respect.It first devotes to show that the trivial solution of the NSDDE with Markov switching and jumps is exponentially mean square stable and asymptotically mean square stable under some suitable conditions.If the drift coefficient also satisfies the linear growth condition,it then proves that the SST method applied to the NSDDE with Markov switching and jumps shares the same conclusions with the exact solution.Moreover,a numerical example is demonstrated to illustrate the obtained results.
基金supported by the Natural Science Foundation of Heilongjiang Province(Grant No.LH2022A020).
文摘In this paper,we investigate the theoretical and numerical analysis of the stochastic Volterra integro-differential equations(SVIDEs)driven by L´evy noise.The existence,uniqueness,boundedness and mean square exponential stability of the analytic solutions for SVIDEs driven by L´evy noise are considered.The split-step theta method of SVIDEs driven by L´evy noise is proposed.The boundedness of the numerical solution and strong convergence are proved.Moreover,its mean square exponential stability is obtained.Some numerical examples are given to support the theoretical results.
基金supported by the National Natural Science Foundation of China(Project No.12071100)Funds for the Central Universities(Project No.2022FRFK060019).
文摘For solving the stochastic differential equations driven by fractional Brownian motion,we present the modified split-step theta method by combining truncated Euler-Maruyama method with split-step theta method.For the problem under a locally Lipschitz condition and a linear growth condition,we analyze the strong convergence and the exponential stability of the proposed method.Moreover,for the stochastic delay differential equations with locally Lipschitz drift condition and globally Lipschitz diffusion condition,we give the order of convergence.Finally,numerical experiments are done to confirm the theoretical conclusions.
