Reduced-order finite element method based on POD for fractional Tricomi-type equation

The reduced-order finite element method （FEM） based on a proper orthogo- nal decomposition （POD） theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general FEM. It can significantly save mem- ory space and effectively relieve the computing load due to its reconstruction of POD basis functions. Furthermore, the reduced-order finite element （FE） scheme is shown to be un- conditionally stable, and error estimation is derived in detail. Two numerical examples are presented to show the feasibility and effectiveness of the method for time fractional differential equations （FDEs）.

Design and experimental validation of sliding knife notch-type disc opener for a no-till combine harvester cum seed drill

One major constraint in developing a combine harvester cum seed drill(CHCSD)is the limited space available for the attachment of no-till furrow openers.Investigating the straw stubble cutting and furrow opening performance of small-sized disc furrow openers is critically important for the development and the optimization of the no-till seeding assemblies.A set of down-scaled sliding knife notch-type disc opener(SKO),modified notch-type disc opener(MNO)and smooth-type disc opener(SDO),with upper size limit of 160 mm,were designed and fabricated for field testing.Experimental results show that for small-sized disc furrow openers of the same type,a larger disc diameter yields a greater furrow depth and width.For small-sized disc furrow openers with an identical diameter,SKOs yield a greater furrow depth and width than MNOs,with SDOs yielding the lowest values.The measured furrow depth and width for an SKO with diameter of 160 mm are 3.52 cm and 3.56 cm,respectively,meeting the no-tillage furrowing requirements for a CHCSD.The highest stubble cover rate of 62.5%is obtained for the SKO with diameter of 160 mm.Therefore,this opener has greater ability to remove stubble from the furrow and seeding band than the other designs.Finally,experimental results of the stubble cutting rates confirm that,for irregularly placed straw and residue,the stubble cutting rate of the SKO with diameter of 160 mm is the highest at 61.7%.

A NEW CHARACTERISTIC EXPANDED MIXED METHOD FOR SOBOLEV EQUATION WITH CONVECTION TERM

In this paper,a new numerical method based on a new expanded mixed scheme and the characteristic method is developed and discussed for Sobolev equation with convection term.The hyperbolic part d(x)∂u/∂t+c(x,t)·∇u is handled by the characteristic method and the diffusion term∇·(a(x,t)∇u+b(x,t)∇ut)is approximated by the new expanded mixed method,whose gradient belongs to the simple square integrable(L^(2)(Ω))^(2)space instead of the classical H(div;Ω)space.For a priori error estimates,some important lemmas based on the new expanded mixed projection are introduced.An optimal priori error estimates in L^(2)-norm for the scalar unknown u and a priori error estimates in(L^(2))^(2)-norm for its gradientλ,and its fluxσ(the coefficients times the negative gradient)are derived.In particular,an optimal priori error estimate in H1-norm for the scalar unknown u is obtained.

Application of low-dimensional finite element method to fractional diffusion equation

Classical finite element method(FEM)has been applied to solve some fractional differential equations,but its scheme has too many degrees of freedom.In this paper,a low-dimensional FEM,whose number of basis functions is reduced by the theory of proper orthogonal decomposition(POD)technique,is proposed for the time fractional diffusion equation in two-dimensional space.The presented method has the properties of low dimensions and high accuracy so that the amount of computation is decreased and the calculation time is saved.Moreover,error estimation of the method is obtained.Numerical example is given to illustrate the feasibility and validity of the low-dimensional FEM in comparison with traditional FEM for the time fractional differential equations.