JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)

Local discontinuous Galerkin method and numerical simulation of semiconductor drift-diffusion model

XIAO Hong-dan, LIU Yun-xian

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(4): 1-7. doi:10.6040/j.issn.1671-9352.0.2021.808

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This paper considers the local discontinuous Galerkin(LDG)method for one-dimensional and two-dimensional problems of semiconductor drift-diffusion(DD)model, and performs numerical simulations. When simulating a one-dimensional problem, fine meshes are used in the parts where the concentration changes sharply, and coarse meshes are used in the places where the concentration changes gently, and compared with the numerical simulation of uniform meshes, it realizes the purpose of saving space and dividing the number of elements and speeding up the running speed under non-uniform division. When simulating two-dimensional problems, a combination of Dirichlet and Neumann boundaries is used. Numerical results verify the stability of the LDG method.

An efficient spectral approximation for the transmission eigenvalue problem in spherical domains

REN Shi-xian, AN Jing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(4): 8-15. doi:10.6040/j.issn.1671-9352.0.2021.778

Abstract ( 317 )

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In order to solve the interior transmission eigenvalue problems in a spherical region, an effective spectral approximation method is proposed. Firstly, a product type Sobolev space is defined, and the corresponding approximation space is constructed by using a class of orthogonal polynomials on the unit sphere. Then, by introducing an auxiliary function, the original problem is transformed into an equivalent fourth-order mixed scheme, and the variational form and discrete form of the fourth-order mixed scheme are derived. Moreover, by using the approximation property of the projection operator and Babuška-Osborn theory, the error estimation of approximation solution is proved. Finally, the implementation process of the algorithm is described in detail, and some numerical examples are given to verify the convergence and high accuracy of the algorithm.

Compact difference schemes for the fourth-order parabolic equations with the third Dirichlet boundary

HUANG Yu, GAO Guang-hua

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(4): 16-28. doi:10.6040/j.issn.1671-9352.0.2022.274

Abstract ( 354 )

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Based on some techniques involving the weighted average and high order Hermite interpolation, several useful differentiation formulae for approximating the fourth-order derivatives are derived along with the truncation error analyses. Then three high order compact difference schemes are proposed to solve the initial-boundary value problem of the fourth-order parabolic equations with the third Dirichlet boundary conditions. The unconditional stability is proved by the Fourier analysis method. Numerical experiments are carried out. The major difference of the proposed three schemes lies in the different numerical treatment of spatial derivatives near the boundary. The global accuracy of all presented schemes can attain the order of two in time and four in space.

Barycentric Lagrange interpolation collocation method for solving nonlinear pseudo-parabolic equations

QU Jin-zheng, LI Jin, SU Xiao-ning

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(4): 29-39. doi:10.6040/j.issn.1671-9352.0.2022.352

Abstract ( 300 )

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Barycentric Lagrange interpolation collocation method for solving a class of nonlinear pseudo-parabolic equations is proposed. Firstly, barycentric Lagrange interpolation is introduced and the expression of differential matrix is given. Secondly, direct linearized iterative scheme, partial linearized iterative scheme, Newton linearized iterative scheme for solving nonlinear pseudo-parabolic equation are constructed. Thirdly, unknown functions and initial-boundary value conditions are approximated by barycentric Lagrange interpolation function, discrete equation is obtained by using collocation method, then the matrix equation is obtained. Finally, numerical examples show that the barycentric Lagrange interpolation collocation method has the advantages of high precision and high efficiency.

Cubic B-spline finite element method for parabolic optimal control problems

DU Fang-fang, SUN Tong-jun

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(4): 40-48. doi:10.6040/j.issn.1671-9352.0.2022.280

Abstract ( 367 )

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A cubic B-spline finite element method is proposed for optimal control problems governed by a class of fourth-order nonlinear parabolic equations. The state and co-state variables are discretized by piecewise cubic B-spline continuous functions which have better smoothness and the control variable is approximated by piecewise constant functions. The numerical solutions of the state and co-state variables thus obtained are second-order continuously differentiable. A fully discrete scheme of the optimality system is established and solved by an iterative method. Finally, some numerical examples are presented to verify the effectivity of the proposed method.

Real-valued implicit Lagrangian for the stochastic linear second-order cone complementarity problem

WANG Guo-xin, NIU Yu-jun

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(4): 49-54. doi:10.6040/j.issn.1671-9352.0.2021.777

Abstract ( 311 )

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In order to study the solution of stochastic second-order cone complementarity problem, this paper studies the stochastic linear second-order cone complementarity problem by the real-valued implicit Lagrangian function. By using the real-valued implicit Lagrangian for symmetric cone complementarity problems and the expected residual minimization formulation for stochastic problems, the existence of solutions of the obtained problems is discussed. Because the objective function of the expected residual minimization formulation contains mathematical expectation, the problem is approximated by using the Monte Carlo method. It is proved that the optimal solution sequence of the approximate problems converges to the optimal solution of the expected residual minimization problem according to probability 1, and the stable point sequence of the approximate problems converges to the stable point of the expected residual minimization problem with probability 1, which can provide a new method for solving stochastic second order cone complementarity problems.

A new method for solving quaternion linear system

FAN Xue-ling, LI Ying, ZHAO Jian-li, LIU Zhi-hong

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(4): 55-64. doi:10.6040/j.issn.1671-9352.0.2022.013

Abstract ( 309 )

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The circulant solution of quaternion Stein equation is solved by using semi-tensor product of matrices and H-representation method. First, some new conclusions about the semi-tensor product of quaternion matrices are presented. The quaternion Stein equation is transformed into a matrix equation with independent variables by using this conclusions. Then, the sufficient and necessary conditions for the existence of the circulant solution and the general solution expression of the original system are given by using H-representation of the circulant matrix and classical matrix theory. Finally, the effectiveness of the algorithm is verified by the corresponding numerical algorithm, and the method is applied to solve quaternion Stein equations in linear time-varying systems.

Multiplicity of positive solutions for elastic beam equations under inhomogeneous boundary conditions

SUN Xiao-yue

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(4): 65-73. doi:10.6040/j.issn.1671-9352.0.2022.052

Abstract ( 259 )

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This paper studies the existence of multiple positive solutions for elastic beam equations simply supported at both ends with inhomogeneous boundary conditions{Y ⁽⁴⁾(x)=f(x,y), x∈(0,1),y(0)=0, y(1)=b, y″(0)=0, y″(1)=0,where f∈C(［0,1］×［0,∞),［0,∞)), b>0, and f(x,s) is a monotone increasing function with respect to s for a fixed x∈［0,1］. Under appropriate conditions, there exists b^*>0 such that the problem has at least two positive solutions for 0<b<b^*, at least one positive solution for b=b^*, and no positive solution for b>b^*. The proof of the main results is based on the upper and lower solution method and topological degree theory.

Number of positive solutions for mean curvature problem with convex-concave nonlinearity

XU Jing, GAO Hong-liang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(4): 74-81. doi:10.6040/j.issn.1671-9352.0.2022.157

Abstract ( 267 )

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This paper considers the exact multiplicity and bifurcation curves of positive solutions for the prescribed mean curvature problem in one-dimensional Minkowski space in the form of{-((u')/((1-u'²)^1/2))'=λf(u), x∈(-L,L),u(-L)=0=u(L)where λ,L are positive parameters, f∈C1［0,∞)∩C2(0,∞)satisfies f(0)=0, and f(u)>0,u∈(0,L)and f is convex-concave on (0,L). By using a detailed analysis of the time map, it is obtained that the above problem has zero, exactly one or exactly two positive solutions according to different ranges of λ in two different cases.

Existence of positive solutions for a class of semipositone second order Neumann boundary value problems

LEI Xiang-bing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(4): 82-88. doi:10.6040/j.issn.1671-9352.0.2022.222

Abstract ( 256 )

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This paper considers the existence of positive solutions of second order Neumann boundary value problems{u″(t)+a(t)u(t)=λf(t,u(t)), 0<t<1,u'(0)=u'(1)=0,where λ is a positive parameter, a∈C［0,1］ and 0<a(t)<(π²)/4, f∈C(［0,1］×R⁺,R)and f(t,0)<0. The proof of that there exists a positive constant λ₀ such that the problem has one positive solution for 0<λ<λ₀. The proof of the main results is based on topological degree theory.

Existence of positive solutions for a class of second order semipositone problems

SHI Xuan-rong

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(4): 89-96. doi:10.6040/j.issn.1671-9352.0.2022.261

Abstract ( 283 )

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The existence of positive solutions for the second order semipositone problem{-u″(t)=λh(t)f(u(t)), t∈(0,1),αu(0)-b(u'(0))u'(0)=0, c(u(1))u(1)+δu'(1)=0 is studied, where λ is a positive parameter,α,δ>0 are constants,b,c∈C(［0,∞),［0,∞)),h∈C(［0,1］,［0,∞)), f∈C(［0,∞),R), f >-M(M>0)and f_∞:=lim_x→∞(f(x))/x=∞。The proof of the main theorems is based on fixed point theorem of Krasnoselskii.

Existence of solutions for the Kirchhoff type Schrödinger-Bopp-Podolsky system with indefinite potentials

TANG Li-qin, WANG Li, WANG Jun

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(4): 97-103. doi:10.6040/j.issn.1671-9352.0.2022.506

Abstract ( 238 )

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This paper is devoted to the Kirchhoff type Schrödinger-Bopp-Podolsky system. It considers the case where the potential V is indefinite so that the Schrödinger operator -Δ+V possesses a finite-dimensional negative space. The authors obtain nontrivial solutions for the Kirchhoff type Schrödinger-Bopp-Podolsky system via Morse theory.

Regularity of weak solutions for obstacle problems with natural growth

ZHAO Song, KANG Di, XU Xiu-juan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2023, 58(4): 104-110. doi:10.6040/j.issn.1671-9352.0.2022.268

Abstract ( 232 )

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The regularity of weak solutions of obstacle problem for nonhomogeneous A-harmonic equations on natural growth are studied. By proving the Caccippoli inequality of weak solution for the obstacle problem of this equation, inverse Hölder inequality is deduced and Gehring lemma is used to prove its local integrability. Finally, the properties at zero are deduced by using the essential zero properties.

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