Problems For Differential Equations Research Articles

The nonlocal solvability conditions for a system with constant terms and coefficients of the variable t

We consider the Cauchy problem for a system of quasilinear differential equations with constantterms and coefficients of the variablet.We investigate the solvability of the Cauchy problem fora system of quasilinear differential equations with constant terms and coefficients of the variabletusing the additional argument method. A theorem on the existence and uniqueness of the localsolution of the Cauchy problem for a system of quasilinear differential equations with constantterms and coefficients of the variabletis formulated. We obtain sufficient conditions for theexistence and uniqueness of a nonlocal solution of the Cauchy problem in original coordinates fora system of quasilinear differential equations with constant terms and coefficients of the variablet.A theorem on the existence and uniqueness of the nonlocal solution of the Cauchy problem fora system of quasilinear differential equations with constant terms and coefficients of the variabletis formulated. A theorem on the existence and uniqueness of the nonlocal solution of the Cauchyproblem for a system of quasilinear differential equations with constant terms and coefficients ofthe variabletis proved. The proof of the nonlocal solvability of the Cauchy problem for a systemof quasilinear differential equations with constant terms and coefficients of the variabletrelies onglobal estimates.

On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems

<abstract><p>The main aim of this paper is to study the Cauchy problem for nonlinear differential equations of fractional order containing the weighted Riemann-Liouville fractional derivative of a function with respect to another function. The equivalence of this problem and a nonlinear Volterra-type integral equation of the second kind have been presented. In addition, the existence and uniqueness of the solution to the considered Cauchy problem are proved using Banach's fixed point theorem and the method of successive approximations. Finally, we obtain a new estimate of the weighted Riemann-Liouville fractional derivative of a function with respect to functions at their extreme points. With the assistance of the estimate obtained, we develop the comparison theorems of fractional differential inequalities, strict as well as nonstrict, involving weighted Riemann-Liouville differential operators of a function with respect to functions of order $ \delta $, $ 0 < \delta < 1 $.</p></abstract>

Uniqueness of Solutions to Boundary Value Problems of Higher Order Fractional Differential Equations

Uniqueness of Solutions to Boundary Value Problems of Higher Order Fractional Differential Equations

Algorithmic implementation of an exact three-point difference scheme for a certain class of singular Sturm–Liouville problems

In this article, we present a new algorithmic implementation of exact three-point difference schemes for a certain class of singular Sturm–Liouville problems. We demonstrate that computing the coefficients of the exact scheme at any grid node xj requires solving two auxiliary Cauchy problems for the second-order linear ordinary differential equations: one problem on the interval [xj−1,xj] (forward) and one problem on the interval [xj,xj+1] (backward). We have also proven the coefficient stability theorem for the exact three-point difference scheme.

Sequence-to-Sequence Stacked Gate Recurrent Unit Networks for Approximating the Forward Problem of Partial Differential Equations

Sequence-to-Sequence Stacked Gate Recurrent Unit Networks for Approximating the Forward Problem of Partial Differential Equations

A Finite Difference Technique for Numerical Solution of the Boundary Value Problem in ODEs of Order Three

In the article, we study the approximate numerical solution to the boundary value problem in ordinary differential equations. In the present article, a third-order two-point boundary value problem is considered for discussion. We developed a second order accurate finite difference method for the approximate numerical solution of the considered problem. We took a special boundary condition; we did not find this boundary condition in the literature. We have discussed the standard convergence analysis of the proposed method. Numerical experiments on linear, nonlinear, and obstacle problems approve the order of accuracy and efficiency of the method.

Well-posedness and order preservation for neutral type stochastic differential equations of infinite delay with jumps

<abstract><p>In this work, we are concerned with the order preservation problem for multidimensional neutral type stochastic differential equations of infinite delay with jumps under non-Lipschitz conditions. By using a truncated Euler-Maruyama scheme and adopting an approximation argument, we have developed the well-posedness of solutions for a class of stochastic functional differential equations which allow the length of memory to be infinite, and the coefficients to be non-Lipschitz and even unbounded. Moreover, we have extended some existing conclusions on order preservation for stochastic systems to a more general case. A pair of examples have been constructed to demonstrate that the order preservation need not hold whenever the diffusion term contains a delay term, although the jump-diffusion coefficient could contain a delay term.</p></abstract>

МОДЕЛЬ АДСОРБЦИОННОЙ ОЧИСТКИ СТОКОВ С УЧЕТОМ ХИМИЧЕСКОЙ РЕАКЦИИ

The article poses and solves the general problem of the dynamics of adsorption from a liquid in a fixed layer of adsorbent, taking into account a first-order chemical reaction and a linear adsorption isotherm. The adsorption model is based on the mass balance equations of the target component for an infinitesimal layer element and adsorption kinetics, written as a system of linear first-order partial differential equations with corresponding boundary conditions. The problem is solved by the Laplace transform method. This made it possible to reduce the joint solution of a system of equations in the originals to an independent solution in images of the Cauchy problem for an ordinary differential equation and an algebraic equation. As a result, expressions were obtained for the concentrations of the target component in the adsorbent layer and in the flow at any time and in any section of the layer. This made it possible to simulate a practical problem of the adsorption of chloroform by zeolites for the treatment of wastewater from a real swimming pool and obtain interesting graphic materials based on the solution and the developed algorithm. The results of the solution can be translated into numbers by programming in any available high-level algorithmic language. It is shown that the shape of the curves of changes in the concentrations of the target component in the adsorbent and adsorbent over the thickness of the adsorbent layer and over time does not change, although the numerical values of these parameters change in a fairly wide range. It is also noted that in the range of concentration changes considered in the article, the concentration of the adsorbent initially decreases sharply, and then smoothly along the entire length of the fixed adsorbent layer decreases slightly. It has been revealed that the use of methods, techniques and theorems of the Laplace transform method in relation to the problem of adsorption dynamics makes it possible to deeply penetrate the mechanism of the problem being solved, which is not possible with other solution methods. This allows, with a known set of initial data, to find the change in the concentrations of the target component in both phases and select the required thickness of the adsorbent layer.

Linear-quadratic optimal control problem for mean-field stochastic differential equations with a type of random coefficients

Linear-quadratic optimal control problem for mean-field stochastic differential equations with a type of random coefficients

Initial value problems for fractional p-Laplacian equations with singularity

<abstract><p>We have studied initial value problems for Caputo fractional differential equations with singular nonlinearities involving the p-Laplacian operator. We have given a precise mathematical analysis of the equivalence of the fractional differential equations and Volterra integral equations studied in this paper. A theorem for the global existence of the solution was proven. In addition, an example was given at the end of the article as an application of the results found in this paper.</p></abstract>

Existence of nodal solutions to some nonlinear boundary value problems for ordinary differential equations of fourth order

Existence of nodal solutions to some nonlinear boundary value problems for ordinary differential equations of fourth order

New uniqueness criterion for Cauchy problems of Caputo fractional multi-term differential equations

New uniqueness criterion for Cauchy problems of Caputo fractional multi-term differential equations

Inverse problem for abstract delay differential equation with impulsive effects

Inverse problem for abstract delay differential equation with impulsive effects

AVERAGING METHOD FOR MULTI-POINT BOUNDARY VALUE PROBLEMS OF SET-VALUED FUNCTIONAL DIFFERENTIAL EQUATIONS

AVERAGING METHOD FOR MULTI-POINT BOUNDARY VALUE PROBLEMS OF SET-VALUED FUNCTIONAL DIFFERENTIAL EQUATIONS

Uniqueness of Solutions for Boundary Value Problems of Nonlinear Coupled Fractional Differential Equations with Integral Boundary Conditions

Uniqueness of Solutions for Boundary Value Problems of Nonlinear Coupled Fractional Differential Equations with Integral Boundary Conditions

New results for fractional ordinary differential equations in fuzzy metric space

<abstract><p>In this paper, we primarily focused on the existence and uniqueness of the initial value problem for fractional order fuzzy ordinary differential equations in a fuzzy metric space. First, definitions and relevant properties of the Gamma function and Beta function within a fuzzy metric space were provided. Second, by employing the principle of fuzzy compression mapping and Choquet integral of fuzzy numerical functions, we established the existence and uniqueness of solutions to initial value problems for fuzzy ordinary differential equations. Finally, several examples were presented to demonstrate the validity of our obtained results.</p></abstract>

A study of the multipoint boundary value problem for nonlinear differential equations of arbitrary fractional order based on the fixed point theorem for composite nonlinear operators

Abstract The existence and distinctiveness of solutions, as well as the iterative approximations of the distinctive answer to the initial boundary value issue of nonlinear differential equations, may be solved with the help of the theory of nonlinear operators, which offers a strong theoretical guarantee and a fundamental tool. In this work, we examine the fixed point theory for composites nonlinear operators, which may offer a fresh approach to solving linear differential equations having arbitrary order fractional multipoint boundary value problems. The existing result of the boundary value issue’s solution is confirmed by the fixed point theorem for composites nonlinear operators, and the four-point boundary value problem for fractional-order linear equations of the Riemann-Liouville type is examined. A group of fractional-order nonlinear differential equations with deviating quantities are examined to examine the presence of an unusual positive solution for this boundary value problem. The existence of this unique positive solution is determined by the use of both the mixed monotone operator and the combined nonlinear operator fixed point theorem. Ultimately, the nonlinear Bagley-Torvik equation with a fractional order variable coefficient four-point boundary value problem is converted into a Fredholm-Hammerstein integral the formula of the second type with weakly singular or continuous kernels, and the fixed point principle is used to demonstrate the uniqueness of the solution in the space of continuous functions. Approximate solutions are provided for the second class of nonlinear Fredholm-Hammerstein integral equations with weakly singular kernels. The outcomes demonstrate the applicability and validity of the fixed point theorem to composite nonlinear operators, offering a fresh approach to solving the multipoint boundary value issue for nonlinear differential equations of any fractional order.

Об одной начально-краевой задаче, возникающей в динамике сжимаемой идеальной стратифицированной жидкости

In this paper, we investigate the problem on small motions of a compressible ideal stratified fluid in a bounded domain. The problem is studied on the base of approach connected with application of so-called operator matrices theory, as well as abstract differential operator equations. For this purpose, Hilbert spaces and some of their subspaces are introduced. The original initial-boundary value problem reduces to the Cauchy problem for a second-order differential operator equation in the orthogonal sum of some Hilbert spaces. Further, an equation with a closed operator is associated with the resulting equation. On this basis, sufficient conditions for the existence of a solution to the corresponding problem are found.

Solution approximation of fractional boundary value problems and convergence analysis using AA-iterative scheme

<abstract><p>Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature.</p></abstract>

Solutions of initial and boundary value problems using invariant curves

<p>The purpose of this study is to investigate the solutions of initial and boundary value problems of ordinary differential equations by employing Lie symmetry generators. In this investigation, it shown that invariant curves, which obtained by symmetry generators, also be utilized to find solutions to initial and boundary value problems. A method, involving invariant curves, presented to find solutions to initial and boundary value problems. Solutions to many linear and nonlinear initial and boundary value problems discussed by applying the proposed method.</p>