Value Problems For Differential Equations Research Articles

Least-squares solution of a class of optimal space guidance problems via Theory of Connections

In this paper, we apply a newly developed method to solve boundary value problems for differential equations to solve optimal space guidance problems in a fast and accurate fashion. The method relies on the least-squares solution of differential equations via orthogonal polynomials expansion and constrained expression as derived via Theory of Connection (ToC). The application of the optimal control theory to derive the first order necessary conditions for optimality, yields a Two-Point Boundary Value Problem (TPBVP) that must be solved to find state and costate. Combining orthogonal polynomials expansion and ToC, we solve the TPBVP for a class of optimal guidance problems including energy-optimal landing on planetary bodies and fixed-time optimal intercept for a target-interceptor scenario. The performance analysis in terms of accuracy shows the potential of the proposed methodology as applied to optimal guidance problems.

Existence and Stability Results for Boundary Value Problem for Differential Equation with ψ -Hilfer Fractional Derivative

Existence and Stability Results for Boundary Value Problem for Differential Equation with ψ -Hilfer Fractional Derivative

Terminal Value Problem for Differential Equations with Hilfer–Katugampola Fractional Derivative

We present in this work the existence results and uniqueness of solutions for a class of boundary value problems of terminal type for fractional differential equations with the Hilfer–Katugampola fractional derivative. The reasoning is mainly based upon different types of classical fixed point theory such as the Banach contraction principle and Krasnoselskii’s fixed point theorem. We illustrate our main findings, with a particular case example included to show the applicability of our outcomes.

Unique Existence of Solutions to Initial Value Problem for Differential Equation of Variable-Order

Unique Existence of Solutions to Initial Value Problem for Differential Equation of Variable-Order

Unique Existence Result of Approximate Solution to Initial Value Problem for Fractional Differential Equation of Variable Order Involving the Derivative Arguments on the Half-Axis

The semigroup properties of the Riemann–Liouville fractional integral have played a key role in dealing with the existence of solutions to differential equations of fractional order. Based on some results of some experts’, we know that the Riemann–Liouville variable order fractional integral does not have semigroup property, thus the transform between the variable order fractional integral and derivative is not clear. These judgments bring us extreme difficulties in considering the existence of solutions of variable order fractional differential equations. In this work, we will introduce the concept of approximate solution to an initial value problem for differential equations of variable order involving the derivative argument on half-axis. Then, by our discussion and analysis, we investigate the unique existence of approximate solution to this initial value problem for differential equation of variable order involving the derivative argument on half-axis. Finally, we give examples to illustrate our results.

RETRACTED ARTICLE: Existence of solutions of nonlocal initial value problems for differential equations with Hilfer–Katugampola fractional derivative

RETRACTED ARTICLE: Existence of solutions of nonlocal initial value problems for differential equations with Hilfer–Katugampola fractional derivative

A hybrid collocation method for fractional initial value problems

This paper is concerned with the application of a hybrid collocation method to a class of initial value problems for differential equations of fractional order. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with a weakly singular kernel. Then, the Volterra integral equation is converted to a fixed point problem. A hybrid collocation algorithm is developed to solve the fixed point problem, and the optimal order of convergence of the proposed algorithm is obtained. Two numerical experiments are conducted to demonstrate the efficiency of the hybrid collocation algorithm.

Nonlinear analysis for the polygonal element

As an important method for solving boundary value problems of differential equations, the finite element method (FEM) has been widely used in the fields of engineering and academic research. For two dimensional problems, the traditional finite element method mainly adopts triangular and quadrilateral elements, but the triangular element is constant strain element, its accuracy is low, the poor adaptability of quadrilateral element with complex geometry. The polygon element is more flexible and convenient in the discrete complex geometric model. Some interpolation functions of the polygon element were introduced. And some analysis was given. The numerical calculation accuracy and related features of different interpolation function were studied. The damage analysis for the koyna dam was given by using the polygonal element polygonal element of Wachspress interpolation function. The damage result is very similar to that by using Xfem, which shows the calculation accuracy of this method is very high.

Integral Inequality of Hardy-Type on Time Scales

Inequalities are essential in the study of Mathematics and are useful tools in the theory of analysis. They have been playing a critical role in the study of the existence and uniqueness properties of solutions of initial and boundary value problems for differential equations as well as difference equations with their bounds. In this paper, we obtain new integral inequalities mainly by using some known inequalities. Various generalizations of Hardy's inequality are special cases of the results therein.

Математичне моделювання коливних процесів у необмеженому кусково-однорідному клиновидному суцільному циліндрі

The theory of boundary value problems for differential equations with partial derivatives develops intensively and its results are important for the development of many sections of mathematics. Its achievements are applied in the mathematical modeling of various processes and phenomenon of physics, mechanics, biology, medicine, economics, engineering. It is well known that the complexity of a boundary-value problem significantly depends on the coefficients of equations and the geometry of domain in which the problem is considered. Properties of solutions of boundary value problems for linear, quasilinear, and some classes of nonlinear equations in single-connected domains have been studied in sufficient detail. However, many important applied problems of thermal physics, thermomechanics, theory of elasticity, theory of electrical circuits, theory of vibrations lead to boundary value problems for differential equations with partial derivatives not only in homogeneous domains when the coefficients of the equations are continuous, but also in piecewise homogeneous and inhomogeneous domains when the coefficients of the equations are piecewise continuous. In this article the exact analytical solutions of mathematical models of oscillating processes (hyperbolic initial-boundary problem of conjugation) for unlimited piecewise-homogeneous wedge-shaped solid cylinder are obtained by means of the method of integral and hybrid integral transforms, in combination with the method of main solutions (influence matrices and Green's matrices). The obtained solutions are of algorithmic character, continuously depend on the parameters and data of problem and can be used in further theoretical research and in practical engineering calculations of real processes which are modeled by hyperbolic boundary-value problems that are described by a cylindrical coordinate system (problems of acoustics, hydrodynamics, the theory of vibrations of mechanical systems).

Existence and stability results for nonlocal initial value problems for differential equations with Hilfer fractional derivative

In this paper, we establish sufficient conditions for the existence and stability of solutions for a class of nonlocal initial value problems for differential equations with Hilfer's fractional derivative, The arguments are based upon the Banach contraction principle. Two examples are included to show the applicability of our results.

General Embedding Theorem

The quotient space of an arbitrary Banach space B by any subspace B1 ⊂ B equipped with the norm \(||b|B/{B_1}|| = \mathop {\inf }\limits_{f \in B/{B_1}} ||f|b||\) is considered. In the case where the infimum is a minimum, i.e., it is attained at some element, a formula for this element is presented. The proof is based on restating the original problem in dual spaces with the help of corresponding Legendre transforms. Although the original problem is nonlinear, it is found that its formulation in dual spaces is always linear and solvable. The results are applied to the general theory of boundary value problems for differential equations of mathematical physics.

The nonlocal problem for the differential-operator equation of the even order with the involution

In this paper, the problem with boundary nonself-adjoint conditions for a differential-operator equations of the order $2n$ with involution is studied. Spectral properties of operator of the problem is investigated.
 By analogy of separation of variables the nonlocal problem for the differential-operator equation of the even order is reduced to a sequence $ \{L_{k}\}_{k=1}^{\infty}$ of operators of boundary value problems for ordinary differential equations of even order. It is established that each element $L_{k}$, of this sequence, is an isospectral perturbation of the self-adjoint operator $L_{0,k}$ of the boundary value problem for some linear differential equation of order 2n.
 We construct a commutative group of transformation operators whose elements reflect the system $V(L_{0,k})$ of the eigenfunctions of the operator $L_{0,k}$ in the system $V(L_{k})$ of the eigenfunctions of the operators $L_{k}$. The eigenfunctions of the operator $L$ of the boundary value problem for a differential equation with involution are obtained as the result of the action of some specially constructed operator on eigenfunctions of the sequence of operators $L_{0,k}.$
 The conditions under which the system of eigenfunctions of operator $L$ the studied problem is a Riesz basis is established.

Mathematical modeling of the stressed-deformed state of circular arches of specialized cranes

We present the possibilities to calculate the stressed-deformed state of circular arches under various conditions of bearing based on the algorithm of a numerical-analytical variant of the method of boundary elements and application of the MATLAB programming and calculation environment. The exact solution to the problem of flat deformation of a circular rod is shown, considering the deformations of bending and stretching-compression, which has not so far been implemented in professional packages of the finite element method. We have constructed solving equations for the boundary value problems on flat deformation of circular arches under various bearing conditions. An example is given for calculating the circular arch SDS employing BEM; by using the MATLAB environment, we represented the results numerically and visually in the form of diagrams. It was established that the boundary element method in the calculation of circular arches has the simplest algorithm logic among other methods and it allows obtaining accurate and reliable results of the stressed-deformed state of crane structures with specialized designation. BEM presented in the given work could be successfully applied to solving boundary value problems for differential equations with variable coefficients. The structure in this case should be discretized while the BEM algorithm does not change. Additional advantage is the minimal requirements to variable coefficients of the differential equation. They may possess first-order discontinuities, breakpoints, and an arbitrary set of continuous functions, which significantly expands the range of problems to be solved. It is obvious that the boundary element method enables fulfillment of the increased requirements to calculation results, which therefore renders relevance to the present study, as well as scientific and practical value for professionals involved in the design of crane structures

Boundary value problems for differential equations with fractional derivatives

ABSTRACTThis paper is devoted to the spectral analysis of the operators generated by differential equations of second order with fractional derivatives in lower terms and boundary conditions of Sturm–Liouville type.

Boundary value problems for parabolic equations of high order with a changing time direction

We analyze the solvability of boundary value problems for differential equations of high order with a changing direction of time. It is shown that a generalized solution exists in Sobolev spaces. Correctness of boundary value problems with a complete matrix of conjugation conditions is investigated. For such problems, it is shown that the Hölder classes of their solutions essentially depend on both the type of gluing conditions and on the non-integer index Hölder space.

Dirichlet boundary value problem for differential equation with ϕ-Laplacian and state-dependent impulses

Abstract The paper deals with the boundary value problem for differential equation with ϕ-Laplacian and state-dependent impulses of the form ϕ ( z ′ ( t ) ) ′ = f ( t , z ( t ) , z ′ ( t ) ) for a.e. t ∈ [ 0 , T ] ⊂ R , Δ z ′ ( t ) = M ( z ( t ) , z ′ ( t − ) ) , t = γ ( z ( t ) ) , z ( 0 ) = z ( T ) = 0. $$\begin{array}{} \left(\phi(z'(t))\right)' = f(t,z(t),z'(t))\qquad \text{ for a.e. } t\in [0,T]\subset\mathbb R,\\ \Delta z'(t) = M(z(t),z'(t-)),\qquad t=\gamma (z(t)),\\ z(0) = z(T) = 0. \end{array} $$ Here, T > 0, ϕ : ℝ → ℝ is an increasing homeomorphism, ϕ(ℝ) = ℝ, ϕ(0) = 0, f : [0, T] × ℝ2 → ℝ satisfies Carathéodory conditions, M : ℝ → ℝ is continuous and γ : ℝ → (0, T) is continuous, Δ z′(t) = z′(t+) − z′(t−). Sufficient conditions for the existence of at least one solution to this problem having no pulsation behaviour are provided.

The uniqueness result of solutions to initial value problems of differential equations of variable-order

This paper is concerned with the existence and uniqueness of solution to an initial value problem for a differential equation of variable-order. The results are obtained by means of fixed point theorem. The obtained results are illustrated with the aid of examples.

Models and methods for analysis of temperature and thermomechanical fields in the bodies of complex shape in specialized intelligent system

The design of axisymmetric bodies of complex form, taking into account temperature and thermomechanical fields, is the object of research. To model these fields need to solve boundary value problems for differential equations with partial derivatives. Structural (R-functions) and regional structural methods are used to solve the boundary value problems of thermal conductivity and thermoelasticity. CAD, meeting modern requirements, should include systems based on knowledge. Also, improving the efficiency of CAD is achieved thanks to the development of methods of solution of thermal conductivity and thermoelasticity tasks in the bodies of complex form. Using created structures, we receive more stable and accurate solution of the problem of analysis of thermal and thermomechanical fields in the considered objects. The specialized intelligent system is created. It contains a database of knowledge: a database of geometric forms of designed objects and methods of solving of relevant problems; a database of rules which allow to automate the choice of method and structure of solutions, equations of boundary of its areas; other information, required to increase the degree of automation of computer modelling. Created system creates the program on RL language for some programming system (PS), which solves the problem. Created system is production system. The pictures of temperature and thermomechanical fields are studied with the help of created system in axisymmetric bodies of complex form – pistons of internal combustion engine (ICE), homogeneous and consisting of composite materials. The impact of insert material, form and location on the picture of the temperature and thermomechanical fields in these ICE pistons is researched. These researches help to design these objects in order to reduce the temperature in the critical points of the object. This improves the thermal regimes in these objects, increase their strength, reliability and durability.

Numerical Solution of Two-Dimensional Advection–Diffusion Equation Using Generalized Integral Representation Method

Generalized integral representation method (GIRM) is designed to numerically solve initial and boundary value problems for differential equations. In this work, we develop numerical schemes based on 1- and 2-step GIRMs for evaluation of the two-dimensional problem of advective diffusion in an infinite domain. Accurate approximate solutions are obtained in both cases of GIRM and compared to the exact ones. The derivation of GIRM is straightforward and implementation is simple.