Some Numerical Extrapolation Methods for the Fractional Sub-diffusion Equation and Fractional Wave Equation Based on the L1 Formula

With the help of the asymptotic expansion for the classic Li formula and based on the L1-type compact difference scheme,we propose a temporal Richardson extrapolation method for the fractional sub-diffusion equation.Three extrapolation formulas are presented,whose temporal convergence orders in L_(∞)-norm are proved to be 2,3-α,and 4-2α,respectively,where 0<α<1.Similarly,by the method of order reduction,an extrapola-tion method is constructed for the fractional wave equation including two extrapolation formulas,which achieve temporal 4-γ and 6-2γ order in L_(∞)-norm,respectively,where1<γ<2.Combining the derived extrapolation methods with the fast algorithm for Caputo fractional derivative based on the sum-of-exponential approximation,the fast extrapolation methods are obtained which reduce the computational complexity significantly while keep-ing the accuracy.Several numerical experiments confirm the theoretical results.

Temporal Second-order Scheme for a Hidden-memory Variable Order Time Fractional Diffusion Equation with an Initial Singularity

In this work,a novel time-stepping L1 formula is developed for a hidden-memory variable-order Caputo’s fractional derivative with an initial singularity.This formula can obtain second-order accuracy and an error estimate is analyzed strictly.As an application,a fully discrete difference scheme is established for the initial-boundary value problem of a hidden-memory variable-order time fractional diffusion model.Numerical experiments are provided to support our theoretical results.

Permafrost degradation along the Qinghai-Tibet Highway from 1995 to 2020

Permafrost degradation significantly affects engineering infrastructure,hydrologic processes,landscape and geomorphic processes,ecosystems and carbon cycling in cold regions.The permafrost degradation along the Qinghai–Tibet Highway(QTH)on the Qinghai–Tibet Plateau,China,introduces an adverse effect on the deformation of the highway subgrade.At present,observation of a long series of ground temperatures is lacking.From 1995 to 2020,a monitoring system of ground temperature in 10 natural sites along QTH was built and maintained.Ground temperatures at different depths were continuously observed semi-monthly.In this study,permafrost changes along QTH were quantitatively investigated based on these records.The main results showed that both the permafrost table depth(PTD)and ground temperature at different depths exhibited an increasing trend from 1995 to 2020 with widespread spatiotemporal differences.The higher the annual mean and range of PTD were,the higher the increase rate in PTD.The increase rates in PTDs in the warm permafrost regions were 6.18 cm per year larger than those in the cold ones.Overall,the increase rates in ground temperature decreased with the increase in depth at each site.At different depths,the smaller the mean annual ground temperature(MAGT)was,the larger the increase rate in the permafrost temperatures.The larger the range of ground temperatures was,the bigger the increase rate in the permafrost temperatures.At a depth of 6.0 m,the increase rate in the ground temperature in cold permafrost regions was twice that in warm permafrost regions.Information on the magnitudes and differences in permafrost degradation along QTH is necessary for the design of effective adaption strategies for engineering construction and environment protection in permafrost regions under climatic warming.

Conservative Three-Level Linearized Finite Difference Schemes for the Fisher Equation and Its Maximum Error Estimates

A three-level linearized difference scheme for solving the Fisher equation is firstly proposed in this work.It has the good property of discrete conservative energy.By the discrete energy analysis and mathematical induction method,it is proved to be uniquely solvable and unconditionally convergent with the secondorder accuracy in both time and space.Then another three-level linearized compact difference scheme is derived along with its discrete energy conservation law,unique solvability and unconditional convergence of order two in time and four in space.The resultant schemes preserve the maximum bound principle.The analysis techniques for convergence used in this paper also work for the Euler scheme,the Crank-Nicolson scheme and others.Numerical experiments are carried out to verify the computational efficiency,conservative law and the maximum bound principle of the proposed difference schemes.

Development and high-throughput genotyping of substitution lines carting the chromosome segments of indica 9311 in the background of japonica Nipponbare

Chromosome segment substitution lines（CSSLs） are useful for the precise mapping of quantitative trait loci（QTLs） and dissection of the genetic basis of complex traits.In this study,two whole-genome sequenced rice cultivars,the japonica Nipponbare and indica 9311 were used as recipient and donor,respectively.A population with 57 CSSLs was developed after crossing and back-crossing assisted by molecular markers, and genotypes were identified using a high-throughput resequencing strategy.Detailed graphical genotypes of 38 lines were constructed based on resequencing data.These CSSLs had a total of 95 substituted segments derived from indica 9311,with an average of about 2.5 segments per CSSL and eight segments per chromosome,and covered about 87.4%of the rice whole genome.A multiple linear regression QTL analysis mapped four QTLs for 1000-grain weight.The largest-effect QTL was located in a region on chromosome 5 that contained a cloned major QTL GW5/qSW5 for grain size in rice.These CSSLs with a background of Nipponbare may provide powerful tools for future whole-genome

The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space.For the approximation to this equation with the initial and boundary value conditions using the finite difference method,the difficulty is how to construct matched finite difference schemes at all the inner grid points.In this paper,two finite difference schemes are constructed for the problem.The accuracy is second-order in time and first-order in space.The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme.The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme.The con-servation,boundedness,stability,convergence of these schemes are discussed and analyzed by the energy method together with other techniques.The two-level non-linear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent.Some numerical examples illustrate the efficiency of the proposed finite difference schemes.

A Second-Order Three-Level Difference Scheme for a Magneto-Thermo-Elasticity Model

This article deals with the numerical solution to the magneto-thermoelasticity model,which is a system of the third order partial differential equations.By introducing a new function,the model is transformed into a system of the second order generalized hyperbolic equations.A priori estimate with the conservation for the problem is established.Then a three-level finite difference scheme is derived.The unique solvability,unconditional stability and second-order convergence in L∞-norm of the difference scheme are proved.One numerical example is presented to demonstrate the accuracy and efficiency of the proposed method.