Problems For Differential Equations Research Articles

A basic theory for initial value problems of first order ordinary differential equations with Lp-Carathéodory functions and applications

A basic theory on the existence and uniqueness of solutions and local solutions for initial value problems (IVPs) of first order ordinary differential equations is established, where the nonlinearities are Lp-Carathéodory functions. The basic theory contains three parts: (i) the existence and uniqueness of solutions in C([a,b],[c,d]); (ii) the existence and uniqueness of local solutions in C([a,a+h1(p)];[c,d]), where the expression of the number h1(p) will be given and is a computable constant, and (iii) the existence and uniqueness of solutions and nonnegative solutions by establishing a new Bihari inequality. The classical results dealt with the above part (ii) when f is continuous or an L1-Carathéodory function, and only obtained the existence of a number h such that the IVPs had local solutions in C[a,a+h]. Our methodology is to use the results obtained on (i) to study the results mentioned in (ii) when p∈[1,p] while the classical results used the Schauder fixed point theorems directly to obtain the existence and uniqueness of local solutions when f is continuous or p=1. Our results in parts (i) and (iii) are new. The results in part (ii) are new when p∈(1,∞] and improve the classical results when p=1. Our results will enrich the classical basic theory. As applications of the basic theory, the existence of solutions and nonnegative local solutions for logistic type population models with heterogeneous environments is studied, where the local rates of changes in the population density are Lp-Carathéodory functions.

Variable linear transformation improved physics-informed neural networks to solve thin-layer flow problems

Physics-informed neural networks (PINNs) have attracted wide attention due to their ability to seamlessly embed the learning process with physical laws and their considerable success in solving forward and inverse differential equation (DE) problems. While most studies are improving the learning process and network architecture of PINNs, less attention has been paid to the modification of the DE system, which may play an important role in addressing some limitations of PINNs. One of the simplest modifications that can be implemented to all DE systems is the variable linear transformation (VLT). Therefore, in this work, we propose the VLT-PINNs that solve the DE systems of the linear-transformed variables instead of the original ones. To clearly illustrate the importance of prior knowledge in determining the VLT parameters, we choose the thin-layer flow problems as our focus. Ten related cases were tested, including the jet flows, wake flows, mixing layers, boundary layers and Kovasznay flows. Based on the principle of normalization and for a better match of the DE system to the preference of NNs, we identify three principles for determining the VLT parameters: magnitude normalization for dependent variables (principle 1), local normalization for independent variables (principle 2), and appropriate scaling for physics-related parameters in inverse problems (principle 3). The VLT-PINNs with the VLT parameters suggested by the proposed principles show excellent performance over all the test cases, while the results are quite poor with the VLT parameters suggested by traditional linear transformations, such as nondimensionalization and global normalization. Comparison studies also show that only under the constraints of the VLT principles can we obtain satisfactory results. Besides, we find tanh is more appropriate as the activation function than sin for thin-layer flow problems, from both posteriori results and priori analyses with physical intuition. We highlight that our VLT method is an attempt to combine the three advantages of accuracy, universality and simplicity, and hope that it can provide new insights into the better integration of prior knowledge, physical intuition and the nature of NNs. The code for this paper is available on https://github.com/CAME-THU/VLT-PINN.

Monotone iterative method for boundary value problem with linear condition of first order delay differential equation

In this paper we discuss the boundary value problem for a first order delay differential equation of the type, \(y'(t) + \lambda y(t) = f(t, y(t - r))\). We prove the existence of solution between weakly coupled lower and upper solution by assuming \(f\) to be a non-decreasing function in the second coordinate. Further, we use this existence result to establish monotone iterative method, where we obtain increasing as well as decreasing sequence of functions whose limits are a solution of the boundary value problem. The sequence of functions obtained are solutions of some defined boundary value problem with linear condition of linear delay differential equation.

A Study on Convergence and Region of Absolute Stability of Implicit Two – Step Multiderivative Method for Solving Stiff Initial Value Problems of First Order Ordinary Differential Equations.

This study investigated and related the convergence and region of absolute stability of an implicit two - step multiderivative method, used in solving sampled stiff initial value problem of first order ordinary differential equation. It adopted the general implicit multiderivative linear multistep method at step - number (k) = 2 with derivative order (l) varied from 1 - 6 to develop six different variants of the method. Boundary locus method was adopted to determine the intervals of absolute stability which were plotted on the complex plane to show the regions of absolute stability. The variant methods were used to solve sampled stiff initial value problem of first order ordinary differential equation. The resulting numerical solutions were compared with the exact solution to determine accuracy and convergence of the methods. The study showed that two - step first derivative, two - step second derivative, two - step third derivative, two - step fifth derivative method and two - step sixth derivative methods yielded more accurate and convergent results with wider regions of absolute stability than two - step fourth derivative.

Inverse problems for stochastic partial differential equations: Some progresses and open problems

The aim of this paper is to survey some recent works on inverse problems for stochastic partial differential equations (SPDEs for short), which were solved by means of Carleman estimate. For simplicity, we focus on two typical SPDEs, i.e, stochastic hyperbolic equations and stochastic parabolic equations. Our particular concerns are those inverse problems which are genuinely stochastic and therefore cannot be reduced to any deterministic ones. Also, some open problems are presented.

Leighton–Wintner type oscillation criteria for second-order differential equations with p ( t ) -Laplacian

This paper deals with the oscillation problems for nonlinear differential equations of the form ( r ( t ) | x ′ | p ( t ) − 2 x ′ ) ′ + c ( t ) f ( x ) = 0 involving p ( t ) -Laplacian. The Leighton–Wintner type oscillation criteria are established without any conditions on the limit of p ( t ) . In addition, we discuss the applications to partial differential equations. Some examples are given to illustrate our results.

Existence Result for Fractional Differential Equation on Unbounded Domain

In this article, we establish certain sufficient conditions to show the existence of solutions of boundary value problem for fractional differential equations on the half-line in a Fréchet space. The main result is based on Tykhonoff fixed point theorem combining with a suitable measure of non-compactness. An example is given to illustrate our approach.

The non-classical optimality condition in the hybrid control problem of hyperbolic and ordinary differential equations with delay

The non-classical optimality condition in the hybrid control problem of hyperbolic and ordinary differential equations with delay

Positive solutions for integral boundary value problems of nonlinear fractional differential equations

In this paper, we consider integral boundary value problems of nonlinear fractional differential equations. Existence results of positive solutions for the problem are obtained based on the Guo-Krasnoselskii theorem and the Five functional fixed point theorem. Simple examples follow the main results in successive sections.

Exact solution of the Susceptible–Exposed–Infectious–Recovered–Deceased (SEIRD) epidemic model

An exact solution of an initial value problem for the Susceptible–Exposed–Infectious–Recovered–Deceased (SEIRD) epidemic model is derived, and various properties of the exact solution are obtained. It is shown that the parametric form of the exact solution satisfies some linear differential system including a positive solution of an Abel differential equation of the second kind. In this paper Abel differential equations play an important role in establishing the exact solution of the SEIRD differential system, in particular the number of infected individuals can be represented in a simple form by using a positive solution of an initial value problem for an Abel differential equation. Uniqueness of positive solutions of an initial value problem to SEIRD differential system is also investigated, and it is shown that the exact solution is a unique solution in the class of positive solutions.

Legendre operational differential matrix for solving ordinary differential equations

In this paper, we used the Legendre operational differential matrix method based on the Tau method to find the approximate analytical solutions to the initial value problems and boundary value problems of ordinary differential equations. This method allows the solution of the ordinary differential equation to be computed in the form of an infinite series in which the components can be easily calculated. We introduced a comparison between the approximate solution that we computed and the exact solution of the selected problem, as we found the absolute error. According to the numerical results, the series of solutions we found are accurate and very close to the exact analytical solutions.

Two-point boundary value problems for 4th order ordinary differential equations

The new optimal efficient sufficient conditions are established for solvability and uniqueness of a solution of the linear and nonlinear fourth order ordinary differential equations u^(4)(t) = p(t)u(t) + q(t) for t ∈ [a,b], u^(4)(t) = p(t)u(t) + f(t, u(t)) for t ∈ [a,b], under the following two-point boundary conditions u^(i)(a) = 0, u^(i)(b) = 0 (i = 0, 1), and u^(i)(a) = 0 (i = 0, 1, 2), u(b) = 0, where p ∈ L([a,b]; R) is a nonconstant sign function and f ∈ K([a,b] × R; R).

Initial value problems of first order fractional differential equations via monotone iterative techniques

Initial value problems of first order fractional differential equations via monotone iterative techniques

Some new methods for studying boundary value problems for general partial differential equations

Some new methods for studying boundary value problems for general partial differential equations

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.