Value Problems For Differential Equations Research Articles

无穷区间上奇异边值问题正解的存在性

Banach空间中的微分方程理论是非线性泛函分析的重要分支，奇异方程边值问题是微分方程学科的组成部分，处于微分方程理论和线性及非线性泛函分析的交叉结合点上，广泛存在于弹簧的振动、梁的非弹性振动、种群生态系统等自然界的各种数学模型中，本文主要应用锥上的不动点理论，通过建立特殊的空间和范数，在非线性项f奇异的条件下，讨论了无穷区间上一类微分方程边值问题解的存在性，获得了方程至少存在一个正解的结论。本文的结果在一定程度上推广了奇异和非奇异条件下的许多已知结果。 The theory of differential equations in Banach spaces is an important branch of nonlinear analysis. The boundary value problem of differential equation is the component of the differential equation subject, which is in the intersection of differential equation theory and linear and nonlinear functional analysis. It exists widely in various mathematical models of nature, such as spring vibration, inelastic vibration of beams and population ecosystem. In this paper, by using the fixed point theory in the cone with a special norm and space, the authors discuss the existence of positive solutions for a class of boundary value problems of differential equation on the infinite interval and obtain that the equation has at least one positive solution. The results improve many known results including singular and non-singular cases to a certain extent.

Caputo-type fractional boundary value problems for differential equations and inclusions with multiple fractional derivatives

Caputo-type fractional boundary value problems for differential equations and inclusions with multiple fractional derivatives

The Exact and Approximate Solutions of Some Boundary Value Problems in Domains with Angular Points of the Boundary

On the basis of the exact solution of biharmonic problems of elasticity theory in a half-strip one possible reason is shown of those problems that arise when an approximate or numerical approaches leading the solution of boundary value problems to infinite systems of linear algebraic equations. Construction of exact solutions of some boundary value problems for differential equations in partial derivatives is not possible without their extensions to Riemann surfaces. Moreover, each of the boundary value problem corresponds to its Riemann surface. This fact is important to consider when developing an effective approximate and numerical methods of solving boundary value problems.

Monotone iterative technique for the initial value problem for differential equations with non-instantaneous impulses

An algorithm for constructing two monotone sequences of upper and lower solutions of the initial value problem for a scalar nonnlinear differential equation with non-instantaneous impulses is given. The impulses start abruptly at some points and their action continue on given finite intervals. We prove that the functional sequences are convergent and their limits are minimal and maximal solutions of the considered problem. An example is given to illustrate the results.

Solvability of Fractional Differential Equation with Nonlocal Boundary Conditions at Resonance

By using the Mawhin’s continuation theorem, we establish a sufficient condition for the existence of at least one solution of a boundary value problem for differential equation of fractional order. The new point is that we proved Fredholm property of the differential operator directly without going through the construction of projectors. Moreover, our work shows the way to construct the general continuous projector Q for both two cases of \(\dim \mathrm {K}\mathrm {e}\mathrm {r}\,L \in \{1, 2\}\).

Positive solution of a system of integral equations with applications to boundary value problems of differential equations

In this paper, by using the Guo-Krasnoselskii theorem, we investigate the existence and nonexistence of positive solutions of a system of integral equation with parameters which can be seen as an effective generalization of various types of systems of boundary value problems for differential equation on continuous interval and time scales or fractional differential equations. We give a general approach of positive solutions to cover various systems of boundary value problems in a unified way, which avoids treating these problems on a case-by-case basis. Under some growth conditions imposed on the nonlinear term, we obtain explicit ranges of values of parameters with which the problem has a positive solution and has no positive solution, respectively. By giving some examples, we will show how our results may be applied to consider existence of positive solutions to a variety of system of boundary value problems of differential equations, differential equations on time scales or fractional differential equations.

Similar Constructing Method for Solving the Boundary Value Problem of the Composite Thomson Equation

Based on the theory of similar structure of solution of the boundary value problem of differential equation, this paper studies the solution of a class of boundary value problem of the composite Thomson equation, After analysis? it is indicate that the solution with a form of continued fraction product is obtained by introducing functions of guide solution, constructing similar kernel functions and assembling coefficients of the left non-homogeneous boundary condition, functions of guide solution and similar kernel functions in this paper. Then a new method is obtained for solving the composite boundary value problem——Similar Constructive Method. Furthermore, the function solution is clearly illustrated by the chart which is carried out by the corresponding program in MTLAB. Thus, the method is not only simple and effective for solving the complicated boundary value problem of differential equation, but also is a kind of innovation of ideological.

Quantum algorithms and the finite element method

The finite element method is used to approximately solve boundary value problems for differential equations. The method discretises the parameter space and finds an approximate solution by solving a large system of linear equations. Here we investigate the extent to which the finite element method can be accelerated using an efficient quantum algorithm for solving linear equations. We consider the representative general question of approximately computing a linear functional of the solution to a boundary value problem, and compare the quantum algorithm's theoretical performance with that of a standard classical algorithm -- the conjugate gradient method. Prior work had claimed that the quantum algorithm could be exponentially faster, but did not determine the overall classical and quantum runtimes required to achieve a predetermined solution accuracy. Taking this into account, we find that the quantum algorithm can achieve a polynomial speedup, the extent of which grows with the dimension of the partial differential equation. In addition, we give evidence that no improvement of the quantum algorithm could lead to a super-polynomial speedup when the dimension is fixed and the solution satisfies certain smoothness properties.

On well-posed solvability of boundary value problems for equations with fractional derivatives in hyper-weight spaces of continuous functions on

In the paper, we establish uniformly well-posed solvability of boundary value problems for equations , in the Banach space where the operator A is a generator of semi-group U(x) of the class with the estimate , , M does not depend on x. The results are applied to the investigation of well-posed solvability of boundary value problems for differential equations with fractional derivatives in weight spaces of continuous functions and investigation of problems of heat-and-mass-transfer and of radio physics.

Difference schemes for generalized solutions of some elliptic differential equations, I

It is known that many applied problems are reduced to boundary value problems for differential equations with non-regular data. There are some works devoted to the construction of difference schemes and the estimation of approximate solutions for these problems [1-3]. In this work the difference schemes for generalized solutions of some elliptic differential equations are constructed. Here we first consider the partial differential equations with the right-hand side defined by a linear functional, for example, by the Dirac distribution δ.

On Two Birkhoff-Type Interpolations with First- and Second-Order Derivative

In this paper, we consider two interpolations of Birkhoff-type with integer-order derivative. The Birkhoff interpolation is related with collocation method for the corresponding initial or boundary value problems of differential equations. The solvability of the interpolation problems is proved. For Gauss-type interpolating points, error of interpolation approximation is deduced. Also, we give efficient algorithms to implement the concerned interpolations.

ASYMPTOTIC-NUMERICAL METHOD FOR SINGULARLY PERTURBED DIFFERENTIAL DIFFERENCE EQUATIONS OF MIXED-TYPE

A computational method for solving singularly perturbed boundary value problem of differential equation with shift arguments of mixed type is presented. When shift arguments are sufficiently small (o(ε)), most of the existing method in the literature used Taylor's expansion to approximate the shift term. This procedure may lead to a bad approximation when the delay argument is of O(ε). The main idea for this work is to deal with constant shift arguments, which are independent of ε. In the present method, we construct the formally asymptotic solution of the problem using the method of composite expansion. The reduced problem is solved numerically by using operator compact implicit method, and the second problem is solved analytically. Error estimate is derived by using the maximum norm. Numerical examples are provided to support the theoretical results and to show the efficiency of the proposed method.

Fuzzy fractional initial value problem

We consider initial value problems for differential equations of fractional order with uncertainty and present the theory and some numerical methods to solve such type of problems under generalized differentiability conditions. The main tool is Banac

On one class of persymmetric matrices generated by boundary value problems for differential equations of fractional order

In this paper, we consider the matrix generated by boundary value problems for differential equations of fractional order. In particular, we show that the eigenvalues of these matrices are simple and real.

Dirichlet boundary value problem for differential equations involving dry friction

Sufficient conditions in terms of growth restrictions are given for the solvability of the Dirichlet boundary value problem to forced nonlinear differential equations involving the combination of viscous and dry frictions. Explicit estimates of solutions and their derivatives allow us to restrict ourselves to a sufficiently large neighbourhood of the origin, when formulating these effective conditions. In this way, the behaviour of nonlinearities outside of this neighbourhood can be quite arbitrary. In order to get optimal solvability criteria, the problems with one-term and complete linear differential operators will be treated separately by means of various Green’s functions. The obtained results are compared with some of their analogies of the other authors.

Nonlocal initial value problems for differential equations with Hilfer fractional derivative

In this paper, we discuss the existence of solutions to nonlocal initial value problem for differential equations with Hilfer fractional derivative. To begin with, we establish an equivalent mixed type integral equation for our problem. Then, we utilize basic properties of Hilfer fractional calculus and fixed point methods to derive three fundamental existence results in the weighted space of continuous functions. Some examples are given to illustrate our results.

Devising direct regular algorithms of computational potential theory with collocation points inside the solution domain

Our attempts to systemize and classify the most popular direct collocation algorithms of computational potential theory have resulted in distinguishing two families of algorithms that have not yet been properly researched or used in computational practice. The first family of algorithms is the method of boundary elements with collocation points inside the solution area, while the second one is a direct regular method of discrete singularities with collocation points inside and outside the solution area. Using decomposition of the desired function in the Taylor series along the normal to the border in the vicinity of collocation, we have modified the considered algorithms and obtained significant advantages in comparison with the respective traditional algorithms. The presented analysis was based on the plane boundary value problems for the Laplace equation. Advantages of the suggested algorithms result from their regular structure that allows to exclude the computation of singular integrals and further program implementation. In addition, the structure permits integration over the real (non-approximated) boundary, which improves the computation accuracy. Eventually, the obtained boundary integral equations are regular integral equations of the second type, which secures their stable solution. The conclusion is proved by several series of test computation with the use three selected analytically set test functions. The suggested approach can be easily applied to solving broad classes of boundary value problems for differential equations in mathematical physics. The suggested algorithms can be mainly applied in computational mechanics.

The Similar Structure Method for Solving the Radial Seepage Model of Fractal Composite Reservoir with Double-porosity

This paper introduces the fractal theory into composite reservoir with double-porosity, and establishes the radial seepage model of fractal composite reservoir with double-porosity. Based on the similar structure theory of solutions for the boundary value problem of differential equation, the similar structure expression of solutions can be obtained in Laplace space. The similar structure theory of solution which avoids the complex calculation is used to solve the model, meanwhile, the similar structure expression of solution reflects the influence of different parameters for the bottom pressure.

О корректной разрешимости некоторых начально-краевых задач для дифференциальных уравнений с вырождением

О корректной разрешимости некоторых начально-краевых задач для дифференциальных уравнений с вырождением

A Study on Solving the Seepage Flow Model of Three-Area Composite Reservoir

This paper studies the seepage flow mathematical model of three-area composite reservoir under three kinds of outer boundary conditions (infinite boundary, constant pressure boundary and closed boundary), in which influences of well-bore storage and skin factor are not taken into consideration. On the basic of theory of similar structure of solution of boundary value problem of differential equation, this paper obtain the solution of the seepage flow model of three-area composite reservoir. The study is not only conducive to further analyze the inherent law of the solution and solve corresponding application problems, but also easy to compile corresponding analysis software.