In this article, we first propose the Riemann-Hilbert problem for uniformly elliptic complex equations of first order and its well-posed-ness in multiply connected domains.Then we give the integral representation of s...In this article, we first propose the Riemann-Hilbert problem for uniformly elliptic complex equations of first order and its well-posed-ness in multiply connected domains.Then we give the integral representation of solutions for modified Riemann-Hilbert problem of the complex equations. Moreover we shall obtain a priori estimates of solutions of the modified Riemann-Hilbert problem and verify its solvability. Finally the solvability results of the original boundary value problem can be obtained.展开更多
This paper is dedicated to implementing and presenting numerical algorithms for solving some linear and nonlinear even-order two-point boundary value problems.For this purpose,we establish new explicit formulas for th...This paper is dedicated to implementing and presenting numerical algorithms for solving some linear and nonlinear even-order two-point boundary value problems.For this purpose,we establish new explicit formulas for the high-order derivatives of certain two classes of Jacobi polynomials in terms of their corresponding Jacobi polynomials.These two classes generalize the two celebrated non-symmetric classes of polynomials,namely,Chebyshev polynomials of third-and fourth-kinds.The idea of the derivation of such formulas is essentially based on making use of the power series representations and inversion formulas of these classes of polynomials.The derived formulas serve in converting the even-order linear differential equations with their boundary conditions into linear systems that can be efficiently solved.Furthermore,and based on the first-order derivatives formula of certain Jacobi polynomials,the operational matrix of derivatives is extracted and employed to present another algorithm to treat both linear and nonlinear two-point boundary value problems based on the application of the collocation method.Convergence analysis of the proposed expansions is investigated.Some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11171349)the Science Foundation of Hebei Province(Grant No.A2010000346)
文摘In this article, we first propose the Riemann-Hilbert problem for uniformly elliptic complex equations of first order and its well-posed-ness in multiply connected domains.Then we give the integral representation of solutions for modified Riemann-Hilbert problem of the complex equations. Moreover we shall obtain a priori estimates of solutions of the modified Riemann-Hilbert problem and verify its solvability. Finally the solvability results of the original boundary value problem can be obtained.
文摘This paper is dedicated to implementing and presenting numerical algorithms for solving some linear and nonlinear even-order two-point boundary value problems.For this purpose,we establish new explicit formulas for the high-order derivatives of certain two classes of Jacobi polynomials in terms of their corresponding Jacobi polynomials.These two classes generalize the two celebrated non-symmetric classes of polynomials,namely,Chebyshev polynomials of third-and fourth-kinds.The idea of the derivation of such formulas is essentially based on making use of the power series representations and inversion formulas of these classes of polynomials.The derived formulas serve in converting the even-order linear differential equations with their boundary conditions into linear systems that can be efficiently solved.Furthermore,and based on the first-order derivatives formula of certain Jacobi polynomials,the operational matrix of derivatives is extracted and employed to present another algorithm to treat both linear and nonlinear two-point boundary value problems based on the application of the collocation method.Convergence analysis of the proposed expansions is investigated.Some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.
