Cauchy Problem For Differential Equations Research Articles

Convergence Criteria for Fixed Point Problems and Differential Equations

We consider a Cauchy problem for differential equations in a Hilbert space X. The problem is stated in a time interval I, which can be finite or infinite. We use a fixed point argument for history-dependent operators to prove the unique solvability of the problem. Then, we establish convergence criteria for both a general fixed point problem and the corresponding Cauchy problem. These criteria provide the necessary and sufficient conditions on a sequence {un}, which guarantee its convergence to the solution of the corresponding problem, in the space of both continuous and continuously differentiable functions. We then specify our results in the study of a particular differential equation governed by two nonlinear operators. Finally, we provide an application in viscoelasticity and give a mechanical interpretation of the corresponding convergence result.

Численный метод решения задачи Коши для одного дифференциального уравнения с дробной производной Капуто

The Cauchy problem for differential equations with fractional derivatives is used in many spheres of science and technology. It was the reason for the development of various methods for its solution, both analytic and approximate ones. The search of an exact solution of differential equations with fractional derivatives by analytic methods is a complex and ineffective task; for this reason, an attempt to solve the considered problem approximately is undertaken in this paper. gated on the segment [0, T]. The method of finite differences which is relatively primary to implement is used for the numerical solution. A difference scheme approximating the Cauchy problem with the first order is constructed on a uniform grid. The difference problem is studied for stability and convergence with a fixed value of the function α(t). It is shown that the numerical solution of the problem converges to the exact solution in the first order. Explicit recurrent formulas for the numerical solution are obtained. A computational experiment upon analysis of the numerical solution of the Cauchy problem is carried out. It is shown on the basis of the computational experiment that if we take the average value for α(t), the first order exactness takes place.

МЯГКИЕ ВЫЧИСЛЕНИЯ И SMALL DATA В ЭКОНОМЕТРИЧЕСКОМ ИССЛЕДОВАНИИ НАЦИОНАЛЬНЫХ ЦЕЛЕЙ РАЗВИТИЯ

The aim of the work was to model the mechanism of the influence of an exogenous resource variable on the predicted indicator of the national development goal, in which one model was selected from an infinite family of hypothetical resource allocation models within a year, reflecting the uncertainty inherent in soft computing. The study of problems based on small sample data in the age of big data is substantiated by the fact that there are a number of longitudinal tasks of longterm research in which, after 5–6 years of testing any methods, it is necessary to obtain data on their effectiveness and predictability in general. At the same time, the data slice occurs strictly at the end of the year as a control and summary measurement. In such cases, special econometric specifications of models are required that take into account the "physics" of relationships. This situation is similar to the situation in mechanics, when small data are compensated by the laws of motion and play the role of initial data in the Cauchy problem for differential equations – when the law itself is known and does not need to be estimated using big data.

The Cauchy Problem for Differential Equations with Piecewise Smooth Characteristics

We study the Cauchy problem for an equation with first order partial derivatives and two independent variables. One of coefficients of partial derivatives is discontinuous. Hence characteristics are piecewise smooth curves and the solution to the Cauchy problem (understood in some generalized sense) has specific properties. In particular, the solutions is not defined in some domain and is discontinuous and unextendable in some other domain.

Fractional Derivatives and Cauchy Problem for Differential Equations of Fractional Order

Abstract Editorial Note: This is a paper by M.M. Djrbashian and A.B. Nersesian of 1968, that was published in Russian. There is a constant interest to Djrbashian’s contributions to the topic of fractional calculus and theory of Mittag-Leffler function. Unfortunately, his works were published in Russian and thus, are not easy accessible and not enough popular. Therefore, we invited hS. Rogosin and M. Dubatovskaya to prepare the survey paper in this same issue of “FCAA” and also to translate and edit the present paper in English. On behalf of Editorial Board and fractional calculus’ community, we express to them our thanks for this hard work, including also retyping, mentioning some typos, etc. Authors’ Summary: The concept of fractional integro-differentiation has found a number of applications in earlier papers of the present authors. With this paper we begin the publication of our results in the field of boundary problems for differential operators of fractional order.

Mkhitar Djrbashian and His Contribution to Fractional Calculus

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

The Parabolic Transform and Some Singular Integral Evolution Equations

Some singular integral evolution equations with wide class of closed operators are studied in Banach space. The considered integral equations are investigated without the existence of the resolvent of the closed operators. Also, some non-linear singular evolution equations are studied. An abstract parabolic transform is constructed to study the solutions of the considered ill-posed problems. Applications to fractional evolution equations and Hilfer fractional evolution equations are given. All the results can be applied to general singular integro-differential equations. The Fourier Transform plays an important role in constructing solutions of the Cauchy problems for parabolic and hyperbolic partial differential equations. This means that the Fourier transform is suitable but under conditions on the characteristic forms of the partial differential operators. Also, the Laplace transform plays an important role in studying the Cauchy problem for abstract differential equations in Banach space. But in this case, we need the existence of the resolvent of the considered abstract operators. This note is devoted to exploring the Cauchy problem for general singular integro-partial differential equations without conditions on the characteristic forms and also to study general singular integral evolution equations. Our approach is based on applying the new parabolic transform. This transform generalizes the methods developed within the regularization theory of ill-posed problems.

On the exact and approximate solutions of several differential equations with variational derivatives of the first and second orders

In this paper, we consider the problem of the exact and approximate solutions of certain differential equations with variational derivatives of the first and second orders. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations with the first variational derivatives are given. An interpolation method for solving ordinary differential equations with variational derivatives is demonstrated. The general scheme of an approximate solution of the Cauchy problem for nonlinear differential equations with variational derivatives of the first order, based on the use of the operator interpolation apparatus, is presented. The exact solution of the differential equation of the hyperbolic type with variational derivatives, similar to the classical Dalamber solution, is obtained. The Hermite interpolation problem with the conditions of coincidence in the nodes of the interpolated and interpolation functionals, as well as their variational derivatives of the first and second orders, is considered for functionals defined on the sets of differentiable functions. The found explicit representation of the solution of the given interpolation problem is based on an arbitrary Chebyshev system of functions. This solution is generalized for the case of interpolation of functionals on one out of two variables and applied to construct an approximate solution of the Cauchy problem for the differential equation of the hyperbolic type with variational derivatives. The description of the material is illustrated by numerous examples.

DENSENESS OF SETS OF CAUCHY PROBLEMS WITHHOUT SOLUTIONS AND WITH NONUNIQUE SOLUTIONS IN THE SET OF ALL CAUCHY PROBLEMS

When finding solutions of differential equations it is necessary to take into account the theorems on innovation and unity of solutions of equations. In case of non-fulfillment of the conditions of these theorems, the methods of finding solutions of the studied equations used in computational mathematics may give erroneous results. It should also be borne in mind that the Cauchy problem for differential equations may have no solutions or have an infinite number of solutions. The author presents two statements obtained by the author about the denseness of sets of the Cauchy problem without solutions (in the case of infinite-dimensional Banach space) and with many solutions (in the case of an arbitrary Banach space) in the set of all Cauchy problems. Using two examples of the Cauchy problem for differential equations, the imperfection of some methods of computational mathematics for finding solutions of the studied equations is shown.

Random Walks and Measures on Hilbert Space that are Invariant with Respect to Shifts and Rotations

We study random walks in a Hilbert space H and their applications to representations of solutions to Cauchy problems for differential equations whose initial conditions are number-valued functions on the Hilbert space H. Examples of such representations of solutions to various evolution equations in the case of a finite-dimensional space H are given. Measures on a Hilbert space that are invariant with respect to shifts are considered for constructing such representations in infinite-dimensional Hilbert spaces. According to a theorem of A. Weil, there is no Lebesgue measure on an infinite-dimensional Hilbert space. We study a finitely additive analog of the Lebesgue measure, namely, a nonnegative, finitely additive measure λ defined on the minimal ring of subsets of an infinite-dimensional Hilbert space H containing all infinite-dimensional rectangles whose products of sides converge absolutely; this measure is invariant with respect to shifts and rotations in the Hilbert space H. We also consider finitely additive analogs of the Lebesgue measure on the spaces lp, 1 ≤ p ≤ ∞, and introduce the Hilbert space \( \mathcal{H} \) of complex-valued functions on the Hilbert space H that are square integrable with respect to a shift-invariant measure λ. We also obtain representations of solutions to the Cauchy problem for the diffusion equation in the space H and the Schrodinger equation with the coordinate space H by means of iterations of the mathematical expectations of random shift operators in the Hilbert space \( \mathcal{H} \).

On the nonlinear $$\varvec{\varPsi }$$ Ψ -Hilfer fractional differential equations

We consider the nonlinear Cauchy problem for $ \Psi $- Hilfer fractional differential equations and investigate the existence, interval of existence and uniqueness of solution in the weighted space of functions. The continuous dependence of solutions on initial conditions is proved via Weissinger fixed point theorem. Picard's successive approximation method has been developed to solve nonlinear Cauchy problem for differential equations with $ \Psi $- Hilfer fractional derivative and an estimation have been obtained for the error bound. Further, by Picard's successive approximation, we derive the representation formula for the solution of linear Cauchy problem for $ \Psi $-Hilfer fractional differential equation with constant coefficient and variable coefficient in terms of Mittag-Leffler function and Generalized (Kilbas-Saigo) Mittag-Leffler function.

Existence and uniqueness of solutions to fractional differential equations in the frame of generalized Caputo fractional derivatives

The generalized Caputo fractional derivative is a name attributed to the Caputo version of the generalized fractional derivative introduced in Jarad et al. (J. Nonlinear Sci. Appl. 10:2607–2619, 2017). Depending on the value of ρ in the limiting case, the generality of the derivative is that it gives birth to two different fractional derivatives. However, the existence and uniqueness of solutions to fractional differential equations with generalized Caputo fractional derivatives have not been proven. In this paper, Cauchy problems for differential equations with the above derivative in the space of continuously differentiable functions are studied. Nonlinear Volterra type integral equations of the second kind corresponding to the Cauchy problem are presented. Using Banach fixed point theorem, the existence and uniqueness of solution to the considered Cauchy problem is proven based on the results obtained.

Averaging of random walks and shift-invariant measures on a Hilbert space

We study random walks in a Hilbert space H and representations using them of solutions of the Cauchy problem for differential equations whose initial conditions are numerical functions on H. We construct a finitely additive analogue of the Lebesgue measure: a nonnegative finitely additive measure λ that is defined on a minimal subset ring of an infinite-dimensional Hilbert space H containing all infinite-dimensional rectangles with absolutely converging products of the side lengths and is invariant under shifts and rotations in H. We define the Hilbert space H of equivalence classes of complex-valued functions on H that are square integrable with respect to a shift-invariant measure λ. Using averaging of the shift operator in H over random vectors in H with a distribution given by a one-parameter semigroup (with respect to convolution) of Gaussian measures on H, we define a one-parameter semigroup of contracting self-adjoint transformations on H, whose generator is called the diffusion operator. We obtain a representation of solutions of the Cauchy problem for the Schrodinger equation whose Hamiltonian is the diffusion operator.

A new semi-analytical approach for numerical solving of cauchy problem for differential equations with delay

In the paper, we present new semi-analytical approach for FDE?s consisting in combination of the method of steps and a technique called differential transformation method (DTM). This approach reduces the original Cauchy problem for delayed or neutral differential equation to Cauchy problem for ordinary differential equation for which DTM is convenient and efficient method. Moreover, there is no need of any symbolic calculations or initial approximation guesstimates in contrast to methods like the homotopy analysis method, the homotopy perturbation method, the variational iteration method or the Adomian decomposition method. The efficiency of the proposed method is shown on certain classes of FDE?s with multiple constant delays including FDE of neutral type. We also compare it to the current approach of using DTM and the Adomian decomposition method where Cauchy problem is not well posed.

Weighted Cauchy Problem for Differential Equations with Deviating Arguments

For higher-order nonlinear differential equations with deviating arguments and with non-integrable singularities with respect to the time variable, we establish sharp sufficient conditions for the Cauchy problem to be solvable and well posed.

Heat balance equation of air solar collector with heat accumulator

The aim of the paper is to find the mathematical dependence of temperature of heat-receiving surface of heat accumulator of a solar collector on its operating time under conditions of variable external factors. In this case, variable solar activity throughout the day is considered as the key external factor. Air solar collector with heat accumulator is a basic element of solar power plants intended, for example, for grain drying, water heating, natural ventilation systems of livestock houses etc. By the example of operation of a drum solar grain dryer with water heat accumulator, the differential equation of heat balance of solar collector is obtained. The equation takes into account the following components of heat balance: amount of heat coming into solar collector with atmospheric air; amount of heat coming from solar energy and absorbed by heat-receiving surface of water accumulator; amount of heat taken away by drying agent (warmed-up atmospheric air) after heat exchange with heat-receiving surface; amount of heat for heating of accumulator walls; amount of heat for heating of water in accumulator; external heat loss. On the basis of available experimental data, it is assumed that water temperature in accumulator is directly proportional to the temperature of its walls, and the enthalpy of atmospheric air is proportional to the flow density of solar energy. Required dependence of temperature of heat-receiving surface of a heat accumulator is found by solving the Cauchy problem for differential equation of heat balance of solar collector. The obtained exponential expression connects the parameters of variable external factors with design and technological parameters of a solar collector. This allows to model the output thermal performance of solar power plants used in agriculture depending on various external conditions.

On the well-posedness of the Cauchy problem for differential equations with distributed prehistory considering delay function perturbations

Theorems on the continuous dependence of the solution on perturbations of the initial data and the right-hand side of equation are proved. Under initial data we understand the collection of initial moment, of delay function and initial function. Perturbations of the right-hand side of equation are small in the integral sense.

О разрешимости задачи коши для дифференциального уравнения с дробной производной Капуто

О разрешимости задачи коши для дифференциального уравнения с дробной производной Капуто

Multidimensional Generalized Erdélyi-Kober Operator and its Application to Solving Cauchy Problems for Differential Equations with Singular Coefficients

Abstract In this paper we investigate some properties of the n-dimensional generalized Erdélyi-Kober operator with the Bessel function in the kernel and its applications in the case n = 3 to solving a Cauchy problem for four- dimensional differential equation of hyperbolic type with singular coefficients.

Sumudu Transform Method For Solving Fractional Differential Equations And Fractional Diffusion-wave Equation

In this paper, we obtain the solutions of a cauchy problems for differential equations with the Caputo fractional derivative and the solution of fractional Diffusion-Wave equation by using Sumudu transform techniques. The results presented here are in compact and elegant expressed in term of Mittag-Leffler function and generalized Mittag-Leffler function which are suitable for numerical computation.