共查询到20条相似文献,搜索用时 15 毫秒
1.
Christopher C. Tisdell 《International Journal of Mathematical Education in Science & Technology》2017,48(5):794-801
Knowing an equation has a unique solution is important from both a modelling and theoretical point of view. For over 70 years, the approach to learning and teaching ‘well posedness’ of initial value problems (IVPs) for second- and higher-order ordinary differential equations has involved transforming the problem and its analysis to a first-order system of equations. We show that this excursion is unnecessary and present a direct approach regarding second- and higher-order problems that does not require an understanding of systems. 相似文献
2.
Manabu Naito 《Journal of Mathematical Analysis and Applications》2011,381(1):315-327
In this paper second order quasilinear ordinary differential equations are considered, and a necessary and sufficient condition for the existence of a slowly growing positive solution is established. Moreover, the precise asymptotic forms as t→∞ of slowly growing positive solutions and slowly decaying positive solutions are obtained. 相似文献
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《Journal of the Egyptian Mathematical Society》2014,22(2):227-237
An algorithm for approximating solutions to 2nd-order linear differential equations with polynomial coefficients in B-polynomials (Bernstein polynomial basis) subject to Dirichlet conditions is introduced. The algorithm expands the desired solution in terms of B-polynomials over a closed interval [0, 1] and then makes use of the orthonormal relation of B-polynomials with its dual basis to determine the expansion coefficients to construct a solution. Matrix formulation is used throughout the entire procedure. However, accuracy and efficiency are dependent on the size of the set of B-polynomials, and the procedure is much simpler compared to orthogonal polynomials for solving differential equations. The current procedure is implemented to solve five linear equations and one first-order nonlinear equation, and excellent agreement is found between the exact and approximate solutions. In addition, the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques. This procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations. 相似文献
5.
We aim at demonstrating a novel theorem on the derivation of energy integrals for linear second-order ordinary differential equations with variable coefficients. Namely, in this context, we will present a possible and consistent method to overcome the traditional difficulty of deriving energy integrals for Lagrangian functions that explicitly exhibit the independent variable. Our theorem is such that it appropriately governs the arbitrariness of the variable coefficients in order to have energy integrals ensured. In view of the theoretical framework in which the theorem will be embedded, we will also demonstrate that it can be applied as a mathematical method to solve linear second-order ordinary differential equations with variable coefficients. These results are expected to have a generalized fundamental character. 相似文献
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Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations 总被引:1,自引:0,他引:1
In this paper, we study the three-point boundary value problems for systems of nonlinear second order ordinary differential equations of the form Under some conditions, we show the existence and multiplicity of positive solutions of the above problem by applying the fixed point index theory in cones. 相似文献
8.
In this paper, we investigate the existence of multiple solutions to a second‐order Dirichlet boundary‐value problem with impulsive effects. The proof is based on critical point theorems. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
9.
A. V. Pokrovskii 《Mathematical Notes》1998,64(2):220-229
We consider generalized mean value theorems for solutions of linear differential equations with constant coefficients and zero right-hand side which satisfy the following homogeneity condition with respect to a given vectorM with positive integer components: for each partial derivative occurring in the equation, the inner product of the vector composed of the orders of this derivative in each variable by the vectorM is independent of the derivative. The main results of this paper generalize the well-known Zalcman theorem. Some corollaries are given.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 260–272, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01366. 相似文献
10.
某类线性微分方程亚纯解的增长性 总被引:1,自引:0,他引:1
陈玉 《纯粹数学与应用数学》2009,25(2):261-267
研究了一类亚纯函数系数的线性微分方程的解的增长性问题,得到了齐次和非齐线性微分方程亚纯解的增长级、超级、二级不同零点收敛指数的精确估计. 相似文献
11.
Kamel Al-Khaled Mohamed Ali Hajji 《Journal of Mathematical Analysis and Applications》2008,344(2):1165-1175
The objectives of this paper are twofold. Firstly, to prove the existence of an approximate solution in the mean for some nonlinear differential equations, we also investigate the behavior of the class of solutions which may be associated with the differential equation. Secondly, we aim to implement the homotopy perturbation method (HPM) to find analytic solutions for strongly nonlinear differential equations. 相似文献
12.
Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations 总被引:3,自引:0,他引:3
John R. Graef Chuanxi Qian Bo Yang 《Proceedings of the American Mathematical Society》2003,131(2):577-585
In this paper, the authors consider the boundary value problem
and give sufficient conditions for the existence of any number of symmetric positive solutions of (E)-(B). The relationships between the results in this paper and some recent work by Henderson and Thompson (Proc. Amer. Math. Soc. 128 (2000), 2373-2379) are discussed.
and give sufficient conditions for the existence of any number of symmetric positive solutions of (E)-(B). The relationships between the results in this paper and some recent work by Henderson and Thompson (Proc. Amer. Math. Soc. 128 (2000), 2373-2379) are discussed.
13.
Existence of solutions of boundary value problems for second order functional differential equations
In this paper, we deal with the boundary value problem
14.
In this paper,we consider the singular boundary value problems for second order quasilinear ordinary differential equations and prove existence and convergence on second order perturbation terms.The result is applied to solve the Riemann problem for 2×2 hyperbolic conservation laws,which is a partial differential equation arising in applied mathematical area. 相似文献
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K. L. Teo 《Journal of Optimization Theory and Applications》1979,29(3):437-481
In this paper, we consider a class of systems governed by time-delayed, second-order, linear, parabolic partial differential equations with first boundary conditions. The existence and uniqueness of solutions of this class of systems are established in Theorem 3.2. A necessary condition for optimality for the corresponding controlled system is presented in Theorem 5.1. For the proof of this theorem, we develop several preparatory results in Sections 2, 3, and 4. 相似文献
17.
In this paper we present a technique to study the existence of rational solutions for systems of differential equations —
for an ordinary differential equation, in particular. The method is relatively straightforward; it is based on a rationality
characterisation that involves matrix Padé approximants. It is important to note that, when the solution is rational, we use
formal power series “without taking into account” their circle of convergence; at the end of this paper we justify this. We
expound the theory for systems of linear first-order ordinary differential equations in the general case. However, the main
ideas are applied in numerical resolution of partial differential equations.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
18.
A. Lomtatidze S. Mukhigulashvili J. remr 《Mathematical and Computer Modelling》2008,47(11-12):1292-1313
On the rectangle , the problem on the existence and uniqueness of a nonnegative solution of the characteristic initial value problem for the equation is considered, where is a linear bounded operator and . 相似文献
19.
A. A. Alikhanov A. M. Berezgov M. X. Shkhanukov-Lafishev 《Computational Mathematics and Mathematical Physics》2008,48(9):1581-1590
Boundary value problems for loaded ordinary and partial differential equations are considered. A priori bounds are obtained for solutions to differential and difference equations. These bounds imply the stability and convergence of difference schemes for the equations under consideration. 相似文献
20.
In this paper, a new technique of homotopy analysis method (HAM) is proposed for solving high‐order nonlinear initial value problems. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM) by transform the nth‐order nonlinear differential equation to a system of n first‐order equations. Second‐ and third‐ order problems are solved as illustration examples of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献