Problems For Differential Equations Research Articles

A Three-step Hybrid Block Method for Direct Integration of Third Order Ordinary Differential Equations

This article presents a novel continuous numerical method designed for the numerical integration of general third-order initial value problems (IVPs) of ordinary differential equations (ODEs). A combination of power series and exponential function was formulated for the purpose of collocation and interpolation at nodal and off-nodal points to generate system of linear equations necessary for the method. The resulting hybrid linear multistep method was implemented using block mode approach. Consistency, stability, and convergence of the method were verified using established criteria. The developed method was applied directly to solve linear and nonlinear third order ODEs without reducing them to systems of first-order equations. Computational results demonstrated better accuracy when compared with existing numerical methods in the literature.

Solvability of Boundary Value Problems for Differential Equations Combining Ordinary and Fractional Derivatives of Non-Autonomous Variable Order

This study introduces a novel approach for investigating the solvability of boundary value problems for differential equations that incorporate both ordinary and fractional derivatives, specifically within the context of non-autonomous variable order. Unlike traditional methods in the literature, which often rely on generalized intervals and piecewise constant functions, we propose a new fractional operator better suited for this problem. We analyze the existence and uniqueness of solutions, establishing the conditions necessary for these properties to hold using the Krasnoselskii fixed-point theorem and Banach’s contraction principle. Our study also addresses the Ulam–Hyers stability of the proposed problems, examining how variations in boundary conditions influence the solution dynamics. To support our theoretical framework, we provide numerical examples that not only validate our findings but also demonstrate the practical applicability of these mixed derivative equations across various scientific domains. Additionally, concepts such as symmetry may offer further insights into the behavior of solutions. This research contributes to a deeper understanding of complex differential equations and their implications in real-world scenarios.

Piecewise Analytical Approximation Methods for Initial-Value Problems of Nonlinear Ordinary Differential Equations

Piecewise analytical solutions to scalar, nonlinear, first-order, ordinary differential equations based on the second-order Taylor series expansion of their right-hand sides that result in Riccati’s equations are presented. Closed-form solutions are obtained if the dependence of the right-hand side on the independent variable is not considered; otherwise, the solution is given by convergent series. Discrete solutions also based on the second-order Taylor series expansion of the right-hand side and the discretization of the independent variable that result in algebraic quadratic equations are also reported. Both the piecewise analytical and discrete methods are applied to two singularly perturbed initial-value problems and the results are compared with the exact solution and those of linearization procedures, and implicit and explicit Taylor’s methods. It is shown that the accuracy of piecewise analytical techniques depends on the number of terms kept in the series expansion of the solution, whereas that of the discrete methods depends on the location where the coefficients are evaluated. For Riccati equations with constant coefficients, the piecewise analytical method presented here provides the exact solution; it also provides the exact solution for linear, first-order ordinary differential equations with constant coefficients.

A Rigorous Integrator and Global Existence for Higher-Dimensional Semilinear Parabolic PDEs via Semigroup Theory

In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations on hyper-rectangular domains via semigroup theory and computer-assisted proofs. Once a numerical candidate for the solution is obtained via a finite dimensional projection, Chebyshev series expansions are used to solve the linearized equations about the approximation from which a solution map operator is constructed. Using the solution operator (which exists from semigroup theory), we define an infinite dimensional contraction operator whose unique fixed point together with its rigorous bounds provide the local inclusion of the solution. Applying this technique for multiple time steps leads to constructive proofs of existence of solutions over long time intervals. As applications, we study the 3D/2D Swift–Hohenberg, where we combine our method with explicit constructions of trapping regions to prove global existence of solutions of initial value problems converging asymptotically to nontrivial equilibria. A second application consists of the 2D Ohta–Kawasaki equation, providing a framework for handling derivatives in nonlinear terms.

ЗАДАЧА ПРОДОВЖЕННЯ ПЛОСКОМЕРИДІАННОГО МАГНІТОСТАТИЧНОГО ПОЛЯ З ПЛОСКОЇ ГРАНИЧНОЇ ПОВЕРХНІ ФЕРОГМАГНЕТИКА

The problem of axisymmetric steady magnetic field continuation from flat boundary surface of ferromagnetic is formulated for magnetic flux and scalar potential. Boundary conditions for magnetic flux are not classical because on boundary magnetic flux is unknown and its normal derivative equal zero. The formulation for scalar potential is Cauchy's problem for elliptical partial differential equation. Analytical solutions of the problem are obtained by method of partial solutions, which depend on parameter continuously, and Hankel's integral transformation. It is shown that there are similar properties in problems of axisymmetric fields continuation from flat boundaries of ideal conductor for magnetic field and conductor for steady electric field. It is fixed that field lines, which bound unknown profile, are determined directly by solution of Cauchy's problem for one from two functions. Equipotential lines are calculated to determine of electromagnetic pole profile. References 7, figures 2, table 1.

A method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations: application of the Kolmogorov-Arnold theorem

The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations.

On A Novel Neutrosophic Numerical Method for Solving Some Neutrosophic Boundary Value Problems

In this paper, we study a novel numerical method for finding the neutrosophic numerical solutions to some neutrosophic boundary values problems in differential equations of high orders. The proposed method based on neutrosophic numerical collocations of higher degree polynomials as an approximation to solve the problems. In addition, we provide many mathematical proofs about the existence of the solutions with many different examples and numerical tables that clarify the validity of the proposed method.

Analysis of Cauchy problems for variable-order derivatives with Mittag-Leffler kernel

In this paper, the Cauchy problem for variable-order fractional differential equations incorporating the Mittag-Leffler kernel is explored. The variable-order derivative is modeled as a bounded function that adapts to the underlying dynamics of the system. The existence of a solution by utilizing a fixed-point theorem along with an iterative series that converges to the precise solution is established. The uniqueness of the solution is guaranteed by enforcing conditions like generalized Lipschitz continuity and linear growth conditions. This study contributes to the broader understanding of fractional calculus and its applications in complex systems where classical models are insufficient.

Quasi-Periodic Orbits of a Difference Equation with Varied Coefficients

In this paper, we study the stability of the equilibrium point and investigate the periodicity properties of a class of difference equations [Formula: see text], [Formula: see text] whose parameters and initial values are positive. In particular, [Formula: see text] is a two-periodic coefficient with respect to n. Moreover, we depict the phase transition of orbits and analyze the deviation of the quasi-periodic orbits from some periodic orbits related to the arithmetic property of [Formula: see text], including Diophantine and Liouvillean numbers. We associate the response solution problem of differential equations with the fixed point problem of the area-preserving mapping related to multiperiodic difference equations, and pose some future work. The techniques are based on Moser’s twist mapping theorem, reversibility and some invariant structures embedded in the system.

Uniform asymptotic expansion of the solution for the initial value problem with a piecewise constant argument

The article is devoted to the study of a singularly perturbed initial problem for a linear differential equation with a piecewise constant argument second-order for a small parameter. This paper is considered the asymptotic expansion of the solution to the Cauchy problem for singularly perturbed differential equations with piecewise-constant argument. The initial value problem for first order linear differential equations with piecewise-constant argument was obtained that determined the regular members. The Cauchy problems for linear nonhomogeneous differential equations with a constant coefficient were obtained, which determined the boundary layer terms. An asymptotic estimate for the remainder term of the solution of the Cauchy problem was obtained. Using the remainder term, we construct a uniform asymptotic solution with accuracy O(εN+1) on the θi ≤ t ≤ θi+1, i = 0, p segment of the singularly perturbed Cauchy problem with a piecewise constant argument.

A generalized existence theorem of two barriers reflected backward doubly stochastic differential equation

We deal with double reflected backward doubly stochastic differential equations (DRBDSDEs) with a generalized growth generator f and a square integrable terminal datum. We introduce a new local conditions on the generator f where is left-Lipischitz in y and Lipischitz in z then we show that they ensure the existence of solutions. Our work go beyond the previous results on reflected backward doubly stochastic differential equations with two barriers. Although we are focused on double reflected case, our comparaison theorem is new for double reflected backward doubly stochastic differential equations too in the case where the generator f is Lipischitiz in y and z, we show that we can compare not only the components Y’s but also the non-decreasing processes K+’s of the solutions. As application, the reflected backward doubly stochastic differential equations with two barriers provide connections with the following obstacle problem of semilinear stochastic partial differential equations (SPDEs) with generalized growth gernerator.

Вычисление цен опционов в модели Хестона с помощью искусственных нейронных сетей

The article considers numerical methods for pricing options in the Heston model. The main attention is paid to solving the Cauchy problem for the partial differential equation for option prices using artificial neural networks. The possibility of approximating the solution using feedforward neural networks with one hidden layer and sigmoid activation function is substantiated. The advantage of the proposed method is the ability to explicitly calculate loss functions when choosing the logistic regression as an activation function. The loss function of neural networks tracks errors in the execution of equations for option prices and approximation of initial conditions. For comparison method of solving the boundary value problem, Monte-Carlo method and method based on recurrent formulas on a binary tree are used. Computational experiments show that even with a moderate number of neurons in the hidden mode, the neural network approximates the prices of European in-the-money options quite well.

Falkner hybrid block methods for second-order IVPs: A novel approach to enhancing accuracy and stability properties

Second-order initial value problems (IVPs) in ordinary differential equations (ODEs) are ubiquitous in various fields, including physics, engineering, and economics. However, their numerical integration poses significant challenges, particularly when dealing with oscillatory or stiff problems. This article introduces a novel Falkner hybrid block method for the numerical integration of second-order IVPs in ODEs. The newly developed method is of order six with a large interval of absolute stability and is implemented using a fixed step size technique. The numerical experiments show the accuracy of our methods when compared with Falkner linear multistep methods, block methods, and other hybrid codes proposed in the scientific literature. This innovative approach demonstrates improved accuracy and stability in solving second-order IVPs, making it a valuable tool for researchers and practitioners.

EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS TO MIXED-TYPE STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY THE FRACTIONAL BROWNIAN MOTIONS WITH HURST INDICES 𝐻 > 1/4

In this paper there is investigated the problem of unique solvability of the Caushy problem for the mixed type stochastic differential equation driven by the standard Brownian motion and fractional Brownian motions with Hurst indices 𝐻 > 1/4. We prove a theorem on the existence and uniqueness of strong solutions to the mentioned mixed type stochastic differential equations.

EXISTENCE AND UNIQUENESS OF SOLUTIONS OF NONLINEAR FUNCTIONAL INTEGRAL ITOˆ EQUATIONS

A new class of Ito^ integral equations is considered, which contains many classical problems, for example, the Cauchy problem for differential equations of integer and fractional order with and without stochastic perturbations, as well as some less known and little-studied types of equations that have been introduced recently. The purpose of the study is to find sufficiently general conditions that guarantee the existence and the uniqueness of solutions to such equations, taking into account their specific features. The article therefore proposes to use a special generalized Lipschitz condition, which, due to its flexibility, allows one to obtain effective solvability criteria in terms of the right-hand sides of equations. Numerous examples are considered, covering in particular Ito^ differential equations of fractional order with aftereffect and without aftereffect, equations with fractional Wiener processes, Ito^ equations with several time scales, as well as their generalizations.

Conjunction Problems for a Pseudo-hyperbolic Equation of the Fourth Order with Discontinuous Coefficients

The relevance of the study is justified by proving the correctness of the conjugation problem for a pseudo-hyperbolic fourth-order equation with discontinuous coefficients. In this work, the methods of Green's function and integral equations are employed to demonstrate the existence and uniqueness of solutions for the first boundary value problem of the pseudo-hyperbolic fourth-order partial differential equation. The obtained results can be applied in the education of students and graduate students in mathematical disciplines.

Existence and uniqueness of generalized normal solutions to first order fractional differential equations and applications

We study the existence and uniqueness of continuous generalized normal solutions to initial value problems of first order fractional differential equations. We use the Banach contraction principle and the Weissinger fixed point theorem to obtain our results. We assume that the absolute values of the nonlinearities have upper bound functions in a subspace of continuous functions. As an example, the results are applied to equations with nonlinearities arising in logistic type population models with heterogeneous environments, and to population models of Ricker type. For more information see https://ejde.math.txstate.edu/Volumes/2024/81/abstr.html

Quickest Change-point Detection Problems for Multidimensional Wiener Processes

We study the quickest change-point (disorder) detection problems for an observable multidimensional Wiener process with the constantly correlated components changing their drift rates at certain unobservable random (change-point) times. These problems seek to determine the times of alarms which should be as close as possible to the unknown change-point times at which some of the components have changed their drift rates. The optimal stopping times of alarm are shown to be the first times at which the appropriate posterior probability processes exit certain regions restricted by the stopping boundaries. We characterise the value functions and optimal boundaries as unique solutions to the associated free-boundary problems for partial differential equations. It is observed that the optimal stopping boundaries can also be uniquely specified by means of the equivalent nonlinear Fredholm integral equations in the class of continuous functions of bounded variation. We also provide estimates for the value functions and boundaries which are solutions to the appropriately constructed ordinary differential free-boundary problems.

Three quarter - step block hybrid algorithm for the numerical solution of first order initial value problems of ordinary differential equations

Three quarter - step block hybrid algorithm for the numerical solution of first order initial value problems of ordinary differential equations

Poincaré-Perron problem for high order differential equations in the class of almost periodic type functions

We address the Poincaré-Perron's classical problem of approximation for high order linear differential equations in the class of almost periodic type functions, extending the results for a second order linear differential equation in [23]. We obtain explicit formulae for solutions of these equations, for any fixed order n≥3, by studying a Riccati type equation associated with the logarithmic derivative of a solution. Moreover, we provide sufficient conditions to ensure the existence of a fundamental system of solutions. The fixed point Banach argument allows us to find almost periodic and asymptotically almost periodic solutions to this Riccati type equation. A decomposition property of the perturbations induces a decomposition on the Riccati type equation and its solutions. In particular, by using this decomposition we obtain asymptotically almost periodic and also p-almost periodic solutions to the Riccati type equation. We illustrate our results for a third order linear differential equation.