In this series of three papers we study singularly perturbed (SP) boundaryvalue problems for equations of eiliptic and parabolic type- For small values ofthe perturbation parameter parabolic boundary and interior laye...In this series of three papers we study singularly perturbed (SP) boundaryvalue problems for equations of eiliptic and parabolic type- For small values ofthe perturbation parameter parabolic boundary and interior layers appear in theseproblems. If classical discretisation methods are used, the solution of the finitedifference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, edapted grids,we can construct difference schemes that allow apprcximation of the solution andthe normalised diffusive flux uniformly with respect to the small parameter.We also consider singularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite differenceschemes, the solution of which converges ε-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. We compare the results with those obtained by the adapted approach. Re-sults of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, andthen we consider respectively (i) Problems for SP parabolic equations, for whichthe solution and the normalised diffusive fluxes are required; (ii) Problems for SPelliptic equations with boundary conditions of Dirichlet, Neumann and Robin type;(iii) Problems for SP parabolic equation with discontinuous boundary conditions.展开更多
The biwave maps are a class of fourth order hyperbolic equations.In this paper,we are interested in the solution formulas of the linear homogeneous biwave equations.Based on the solution formulas and the weighted ener...The biwave maps are a class of fourth order hyperbolic equations.In this paper,we are interested in the solution formulas of the linear homogeneous biwave equations.Based on the solution formulas and the weighted energy estimate,we can obtain the L∞(R^(n))−WN,1(R^(n))and L∞(R^(n))−WN,2(R^(n))estimates,respectively.By our results,we find that the biwave maps enjoy some different properties compared with the standard wave equations.展开更多
文摘In this series of three papers we study singularly perturbed (SP) boundaryvalue problems for equations of eiliptic and parabolic type- For small values ofthe perturbation parameter parabolic boundary and interior layers appear in theseproblems. If classical discretisation methods are used, the solution of the finitedifference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, edapted grids,we can construct difference schemes that allow apprcximation of the solution andthe normalised diffusive flux uniformly with respect to the small parameter.We also consider singularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite differenceschemes, the solution of which converges ε-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. We compare the results with those obtained by the adapted approach. Re-sults of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, andthen we consider respectively (i) Problems for SP parabolic equations, for whichthe solution and the normalised diffusive fluxes are required; (ii) Problems for SPelliptic equations with boundary conditions of Dirichlet, Neumann and Robin type;(iii) Problems for SP parabolic equation with discontinuous boundary conditions.
基金the Zhejiang Provincial Outstanding Youth Science Foundation(Grant No.LR22A010004)the Natural Science Foundation of Zhejiang Province(Grant No.LY20A010026)the National Natural Science Foundation of China(Grant Nos.12071435 and 11871212).
文摘The biwave maps are a class of fourth order hyperbolic equations.In this paper,we are interested in the solution formulas of the linear homogeneous biwave equations.Based on the solution formulas and the weighted energy estimate,we can obtain the L∞(R^(n))−WN,1(R^(n))and L∞(R^(n))−WN,2(R^(n))estimates,respectively.By our results,we find that the biwave maps enjoy some different properties compared with the standard wave equations.
