With the urgent need to resolve complex behaviors in nonlinear evolution equations,this study makes a contribution by establishing the local existence of solutions for Cauchy problems associated with equations of mixe...With the urgent need to resolve complex behaviors in nonlinear evolution equations,this study makes a contribution by establishing the local existence of solutions for Cauchy problems associated with equations of mixed types.Our primary contribution is the establishment of solution existence,illuminating the dynamics of these complex equations.To tackle this challenging problem,we construct an approximate solution sequence and apply the contraction mapping principle to rigorously prove local solution existence.Our results significantly advance the understanding of nonlinear evolution equations of mixed types.Furthermore,they provide a versatile,powerful approach for tackling analogous challenges across physics,engineering,and applied mathematics,making this work a valuable reference for researchers in these fields.展开更多
We study the distribution limit of a class of stochastic evolution equation driven by an additive-stable Non-Gaussian process in the case of α∈(1,2).We prove that,under suitable conditions,the law of the solution co...We study the distribution limit of a class of stochastic evolution equation driven by an additive-stable Non-Gaussian process in the case of α∈(1,2).We prove that,under suitable conditions,the law of the solution converges weakly to the law of a stochastic evolution equation with an additive Gaussian process.展开更多
Rocks will suffer different degree of damage under freeze-thaw(FT)cycles,which seriously threatens the long-term stability of rock engineering in cold regions.In order to study the mechanism of rock FT damage,energy c...Rocks will suffer different degree of damage under freeze-thaw(FT)cycles,which seriously threatens the long-term stability of rock engineering in cold regions.In order to study the mechanism of rock FT damage,energy calculation method and energy self-inhibition model are introduced to explore their energy characteristics in this paper.The applicability of the energy self-inhibition model was verified by combining the data of FT cycles and uniaxial compression tests of intact and pre-cracked sandstone samples,as well as published reference data.In addition,the energy evolution characteristics of FT damaged rocks were discussed accordingly.The results indicate that the energy self-inhibition model perfectly characterizes the energy accumulation characteristics of FT damaged rocks under uniaxial compression before the peak strength and the energy dissipation characteristics before microcrack unstable growth stage.Taking the FT damaged cyan sandstone sample as an example,it has gone through two stages dominated by energy dissipation mechanism and energy accumulation mechanism,and the energy rate curve of the pre-cracked sample shows a fall-rise phenomenon when approaching failure.Based on the published reference data,it was found that the peak total input energy and energy storage limit conform to an exponential FT decay model,with corresponding decay constants ranging from 0.0021 to 0.1370 and 0.0018 to 0.1945,respectively.Finally,a linear energy storage equation for FT damaged rocks was proposed,and its high reliability and applicability were verified by combining published reference data,the energy storage coefficient of different types of rocks ranged from 0.823 to 0.992,showing a negative exponential relationship with the initial UCS(uniaxial compressive strength).In summary,the mechanism by which FT weakens the mechanical properties of rocks has been revealed from an energy perspective in this paper,which can provide reference for related issues in cold regions.展开更多
概率密度演化方法(probability density evolution equation,PDEM)为非线性随机结构的动力响应分析提供了新的途径.通过PDEM获得结构响应概率密度函数(probability density function,PDF)的关键步骤是求解广义概率密度演化方程(generali...概率密度演化方法(probability density evolution equation,PDEM)为非线性随机结构的动力响应分析提供了新的途径.通过PDEM获得结构响应概率密度函数(probability density function,PDF)的关键步骤是求解广义概率密度演化方程(generalized probability density evolution equation,GDEE).对于GDEE的求解通常采用有限差分法,然而,由于GDEE是初始条件间断的变系数一阶双曲偏微分方程,通过有限差分法求解GDEE可能会面临网格敏感性问题、数值色散和数值耗散现象.文章从全局逼近的角度出发,基于Chebyshev拟谱法为GDEE构造了全局插值格式,解决了数值色散、数值耗散以及网格敏感性问题.考虑GDEE的系数在每个时间步长均为常数,推导了GDEE在每一个时间步长内时域上的序列矩阵指数解.由于序列矩阵指数解形式上是解析的,从而很好地克服了数值稳定性问题.两个数值算例表明,通过Chebyshev拟谱法结合时域的序列矩阵指数解求解GDEE得到的结果与精确解以及Monte Carlo模拟的结果非常吻合,且数值耗散和数值色散现象几乎可以忽略.此外,拟谱法具有高效的收敛性且序列矩阵指数解不受CFL (Courant-Friedrichs-Lewy)条件的限制,因此该方法具有良好的数值稳定性和计算效率.展开更多
基金Supported by the National Natural Science Foundation of China(12201368,62376252)Key Project of Natural Science Foundation of Zhejiang Province(LZ22F030003)Zhejiang Province Leading Geese Plan(2024C02G1123882,2024C01SA100795).
文摘With the urgent need to resolve complex behaviors in nonlinear evolution equations,this study makes a contribution by establishing the local existence of solutions for Cauchy problems associated with equations of mixed types.Our primary contribution is the establishment of solution existence,illuminating the dynamics of these complex equations.To tackle this challenging problem,we construct an approximate solution sequence and apply the contraction mapping principle to rigorously prove local solution existence.Our results significantly advance the understanding of nonlinear evolution equations of mixed types.Furthermore,they provide a versatile,powerful approach for tackling analogous challenges across physics,engineering,and applied mathematics,making this work a valuable reference for researchers in these fields.
基金Supported by the Science and Technology Research Projects of Hubei Provincial Department of Education(B2022077)。
文摘We study the distribution limit of a class of stochastic evolution equation driven by an additive-stable Non-Gaussian process in the case of α∈(1,2).We prove that,under suitable conditions,the law of the solution converges weakly to the law of a stochastic evolution equation with an additive Gaussian process.
基金Project(52174088)supported by the National Natural Science Foundation of ChinaProject(104972024JYS0007)supported by the Independent Innovation Research Fund Graduate Free Exploration,Wuhan University of Technology,China。
文摘Rocks will suffer different degree of damage under freeze-thaw(FT)cycles,which seriously threatens the long-term stability of rock engineering in cold regions.In order to study the mechanism of rock FT damage,energy calculation method and energy self-inhibition model are introduced to explore their energy characteristics in this paper.The applicability of the energy self-inhibition model was verified by combining the data of FT cycles and uniaxial compression tests of intact and pre-cracked sandstone samples,as well as published reference data.In addition,the energy evolution characteristics of FT damaged rocks were discussed accordingly.The results indicate that the energy self-inhibition model perfectly characterizes the energy accumulation characteristics of FT damaged rocks under uniaxial compression before the peak strength and the energy dissipation characteristics before microcrack unstable growth stage.Taking the FT damaged cyan sandstone sample as an example,it has gone through two stages dominated by energy dissipation mechanism and energy accumulation mechanism,and the energy rate curve of the pre-cracked sample shows a fall-rise phenomenon when approaching failure.Based on the published reference data,it was found that the peak total input energy and energy storage limit conform to an exponential FT decay model,with corresponding decay constants ranging from 0.0021 to 0.1370 and 0.0018 to 0.1945,respectively.Finally,a linear energy storage equation for FT damaged rocks was proposed,and its high reliability and applicability were verified by combining published reference data,the energy storage coefficient of different types of rocks ranged from 0.823 to 0.992,showing a negative exponential relationship with the initial UCS(uniaxial compressive strength).In summary,the mechanism by which FT weakens the mechanical properties of rocks has been revealed from an energy perspective in this paper,which can provide reference for related issues in cold regions.
文摘概率密度演化方法(probability density evolution equation,PDEM)为非线性随机结构的动力响应分析提供了新的途径.通过PDEM获得结构响应概率密度函数(probability density function,PDF)的关键步骤是求解广义概率密度演化方程(generalized probability density evolution equation,GDEE).对于GDEE的求解通常采用有限差分法,然而,由于GDEE是初始条件间断的变系数一阶双曲偏微分方程,通过有限差分法求解GDEE可能会面临网格敏感性问题、数值色散和数值耗散现象.文章从全局逼近的角度出发,基于Chebyshev拟谱法为GDEE构造了全局插值格式,解决了数值色散、数值耗散以及网格敏感性问题.考虑GDEE的系数在每个时间步长均为常数,推导了GDEE在每一个时间步长内时域上的序列矩阵指数解.由于序列矩阵指数解形式上是解析的,从而很好地克服了数值稳定性问题.两个数值算例表明,通过Chebyshev拟谱法结合时域的序列矩阵指数解求解GDEE得到的结果与精确解以及Monte Carlo模拟的结果非常吻合,且数值耗散和数值色散现象几乎可以忽略.此外,拟谱法具有高效的收敛性且序列矩阵指数解不受CFL (Courant-Friedrichs-Lewy)条件的限制,因此该方法具有良好的数值稳定性和计算效率.
