Terminal Value Problem Research Articles

A shooting-Newton procedure for solving fractional terminal value problems

In this paper we consider the numerical solution of fractional terminal value problems: namely, terminal value problems for fractional differential equations. In particular, the proposed method uses a Newton-type iteration which is particularly efficient when coupled with a recently-introduced step-by-step procedure for solving fractional initial value problems, i.e., initial value problems for fractional differential equations. As a result, the method is able to produce spectrally accurate solutions of fractional terminal value problems. Some numerical tests are reported to make evidence of its effectiveness.

Approximate solution of a zero-sum linear-quadratic differential game by the auxiliary parameter method: analytical/numerical study

ABSTRACT A zero-sum finite horizon linear-quadratic differential game is considered. First, the terminal-value problem for the game-theoretic Riccati matrix differential equation, associated with the considered game by the solvability conditions, is analysed. This problem may not have the solution in the entire time-interval of the game's duration. Using the method of auxiliary parameter, a sufficient condition (in the terms of the game's data) for the existence of the solution to the aforementioned terminal-value problem in the entire time-interval of the game's duration is derived. An approximate analytical solution to this problem also is obtained as a partial sum of some infinite series of functions arising in the method of auxiliary parameter. Based on this approximate solution, suboptimal state-feedback controls of the players are formally designed and justified. It is shown that the pair of these controls constitutes an approximate-saddle point in the considered game. The theoretical results of the article are illustrated by their application to approximate solution of the problem of pursuit-evasion engagement between two flying vehicles.

Topological derivatives for shell structure optimization considering crash loadcases with material nonlinearities and large deformations

The topological derivatives for shell structures in dynamic loadcases, which are often used in lightweight designs, are developed. These sensitivities consider the highly nonlinear material behavior and large deformations of the structure. The aim of the investigation is the support of an efficient topology optimization for crashloaded structures. The considered functionals can in general be described by displacements, velocities and accelerations. In particular, the semi-analytical sensitivity calculations for the internal deformation energy and for single point displacements in arbitrary directions in the shell structure are presented. These functionals cover the most important functions for the development of crash structures. Using the material derivative in combination with the adjoint method, a terminal value problem has to be solved. Depending on whether first differentiation and then discretization in the time domain is performed or first discretization and then differentiation, a separate solution scheme for the adjoint is derived. Both schemes are presented in a comparable description. For the numerical evaluation of the topological derivatives, the basic equations for large deformed shell elements are included as well as a phenomenological material interpolation, which represents the tangential material behavior resulting from plastic strain and isotropic hardening. All assumptions made for the derivation are described in detail and their influences are discussed. The elaborated equations and the procedure for the calculation of the topological derivatives are applied, discussed and checked for plausibility on two clearly defined loading conditions for the internal deformation energy and single point displacements on different locations.

Initial Value and Terminal Value Problems for Distributed Order Fractional Diffusion Equations

Initial Value and Terminal Value Problems for Distributed Order Fractional Diffusion Equations

A numerical method for solutions of tempered fractional differential equations

The main objective of this paper is to develop a technique for approximating the numerical solution of certain class of tempered fractional differential equations. The proposed technique is based on the Newton–Cotes quadrature formula, and it is established by using generalized Lagrange interpolation polynomials. The product integration technique is used to develop a formula for the left and right-sided tempered fractional integrals. Next, by using these formulas, a method is developed for solving a general class of tempered fractional differential equations. Furthermore, three distinct classes of initial value problems and one of the terminal value problems for tempered fractional differential equations are taken into consideration and each class is illustrated by a numerical experiment to demonstrate the validity and effectiveness of the proposed methodology. Moreover, upper bound of error in the numerical approximations have been provided.

On the nonlocal hybrid $ ({\mathsf{k}}, {\rm{\mathsf{φ}}}) $-Hilfer inverse problem with delay and anticipation

<p>This paper focused on establishing results regarding the existence of solutions for a class of nonlocal terminal value problems involving hybrid implicit nonlinear fractional differential equations with the $ ({\mathsf{k}}, {\rm{\mathsf{φ}}}) $-Hilfer fractional derivative, which includes both finite delay and anticipation arguments. Our analysis was based on the Banach fixed point technique, and the Schauder and Krasnoselskii fixed point theorems. Moreover, illustrative examples were considered to support our new results.</p>

New regularization and error estimate on terminal value problem for elliptic equations

New regularization and error estimate on terminal value problem for elliptic equations

On terminal value problem for fractional superdiffusive of Sobolev equation type

On terminal value problem for fractional superdiffusive of Sobolev equation type

Robust Solution of the Multi-Model Singular Linear-Quadratic Optimal Control Problem: Regularization Approach

We consider a finite horizon multi-model linear-quadratic optimal control problem. For this problem, we treat the case where the problem’s functional does not contain a control function. The latter means that the problem under consideration is a singular optimal control problem. To solve this problem, we associate it with a new optimal control problem for the same multi-model system. The functional in this new problem is the sum of the original functional and an integral of the square of the Euclidean norm of the vector-valued control with a small positive weighting coefficient. Thus, the new problem is regular. Moreover, it is a multi-model cheap control problem. Using the solvability conditions (Robust Maximum Principle), the solution of this cheap control problem is reduced to the solution of the following three problems: (i) a terminal-value problem for an extended matrix Riccati type differential equation; (ii) an initial-value problem for an extended vector linear differential equation; (iii) a nonlinear optimization (mathematical programming) problem. We analyze an asymptotic behavior of these problems. Using this asymptotic analysis, we design the minimizing sequence of state-feedback controls for the original multi-model singular optimal control problem, and obtain the infimum of the functional of this problem. We illustrate the theoretical results with an academic example.

A New Approach to Shooting Methods for Terminal Value Problems of Fractional Differential Equations

For terminal value problems of fractional differential equations of order alpha in (0,1) that use Caputo derivatives, shooting methods are a well developed and investigated approach. Based on recently established analytic properties of such problems, we develop a new technique to select the required initial values that solves such shooting problems quickly and accurately. Numerical experiments indicate that this new proportional secting technique converges very quickly and accurately to the solution. Run time measurements indicate a speedup factor of between 4 and 10 when compared to the standard bisection method.

Continuum Sensitivity Analysis for Shape Optimization of Transient Eddy Current Systems

This study proposes a shape sensitivity analysis for transient eddy current systems. The continuum sensitivity analysis is a deterministic optimization technique for designing the shape and topology of electromagnetic systems. Despite substantial research on the sensitivity analysis, most of the work has focused on static and steady-state issues. However, some problems require transient analysis. In this paper, the shape sensitivity formula for a transient eddy current system was derived by the continuum approach using the adjoint variable technique and material derivative concept. In the sensitivity analysis using the derived sensitivity formula, the state variable equation is an initial-value problem to be solved forward in time. Conversely, the adjoint variable equation is a terminal-value problem to be solved backward in time. Three numerical examples were tested to demonstrate the feasibility and usability of the derived sensitivity formula.

Right fractional calculus to inverse-time chaotic maps and asymptotic stability analysis

Inverse or terminal value problems of fractional differential equations become popular recently. But memory effects or non-locality of fractional operators cause many difficulties for theoretical analysis. This study suggests a right fractional calculus method for inverse problem modelling and proposes a concept of inverse-time fractional chaotic maps. First, a simple right fractional linear differential equation's terminal value problem and the solutions are investigated. Then, some basics of the right discrete fractional calculus are introduced and the idea is extended to the discrete case. Right fractional sum equations are derived and numerical schemes are provided for dynamical analysis. Discrete chaos does exist in an inverse-time fractional logistic map and Hénon map, respectively. Their local stability conditions are given. It can be concluded that this right fractional calculus method is simpler but more efficient than the left one.

A numerical method for fractional Sturm–Liouville problems involving the Cauchy–Euler operators

The objective of this paper is to establish a numerical scheme for the solutions of fractional Sturm–Liouville problems involving Cauchy–Euler operators. The proposed scheme is based on the normalized Φ-Legendre functions. The Sturm–Liouville operators in this work are chosen to be a combination of the left Caputo and right Riemann–Liouville type fractional Cauchy–Euler operators. The choice of fractional operators is inspired by the structure of the classical Sturm–Liouville operator. Some important properties of eigenvalues and eigenfunctions corresponding to a class of generalized fractional Sturm–Liouville operators are also investigated. In addition to the fractional Sturm–Liouville problems, we propose schemes for solutions of some terminal and boundary value problems. Furthermore, we derive the upper bounds of the errors in approximations of derivatives of the unknown functions in terms of normalized Φ-Legendre functions. The accuracy of the proposed schemes is illustrated by numerical examples.

A streamlined numerical method to treat fractional nonlinear terminal value problems with multiple delays appearing in biomathematics

In this study, a computational matrix-collocation method based on the Lagrange interpolation polynomial is specifically streamlined to treat the fractional nonlinear terminal value problems with multiple delays, such as the Hutchinson, the Wazewska-Czyzewska and the Lasota models in biomathematics. To do this, the robust nonlinear terms of which are smoothed to be deployed in the method. The uniqueness analysis of the solution is discussed in terms of the Banach contraction principle. An error analysis technique is non-linearly theorized and applied to improve the solutions. A programme for the method is especially developed. Thus, the outcomes of five fractional model problems constrained by terminal conditions are numerically and graphically evaluated in tables and figures. Based on the investigation of the results, one can claim that the method presents a sustainable and effective mathematical procedure for the aforementioned problems.

A Game—Theoretic Model for a Stochastic Linear Quadratic Tracking Problem

In this paper, we solve a stochastic linear quadratic tracking problem. The controlled dynamical system is modeled by a system of linear Itô differential equations subject to jump Markov perturbations. We consider the case when there are two decision-makers and each of them wants to minimize the deviation of a preferential output of the controlled dynamical system from a given reference signal. We assume that the two decision-makers do not cooperate. Under these conditions, we state the considered tracking problem as a problem of finding a Nash equilibrium strategy for a stochastic differential game. Explicit formulae of a Nash equilibrium strategy are provided. To this end, we use the solutions of two given terminal value problems (TVPs). The first TVP is associated with a hybrid system formed by two backward nonlinear differential equations coupled by two algebraic nonlinear equations. The second TVP is associated with a hybrid system formed by two backward linear differential equations coupled by two algebraic linear equations.

Terminal value problem for Riemann‐Liouville fractional differential equation in the variable exponent Lebesgue space Lp(.)$$ {L}^{p(.)} $$

In this manuscript, the existence, uniqueness, and stability of solutions to the terminal value problem of Riemann‐Liouville fractional equations are established in the variable exponent Lebesgue spaces . We convert the variable exponent Lebesgue spaces to the Lebesgue spaces using the generalized intervals and piece‐wise constant function. Further, the Banach contraction principle is used, the Ulam‐Hyers‐stability is examined, and finally, we construct an example.

TERMINAL VALUE PROBLEM FOR STOCHASTIC FRACTIONAL EQUATION WITHIN AN OPERATOR WITH EXPONENTIAL KERNEL

In this paper, we investigate a terminal value problem for stochastic fractional diffusion equations with Caputo–Fabrizio derivative. The stochastic noise we consider here is the white noise taken value in the Hilbert space [Formula: see text]. The main contribution is to investigate the well-posedness and ill-posedness of such problem in two distinct cases of the smoothness of the Hilbert scale space [Formula: see text] (see Assumption 3.1), which is a subspace of [Formula: see text]. When [Formula: see text] is smooth enough, i.e. the parameter [Formula: see text] is sufficiently large, our problem is well-posed and it has a unique solution in the space of Hölder continuous functions. In contract, in the different case when [Formula: see text] is smaller, our problem is ill-posed; therefore, we construct a regularization result.

Terminal value problem for neutral fractional functional differential equations with Hilfer-Katugampola fractional derivative

In this paper, we establish the existence of solutions for a class of nonlinear neutral fractional differential equations with terminal condition and Hilfer-Katugampola fractional derivative. The arguments are based upon the Banach contraction principle, and Krasnoselskii?s fixed point theorem. An example is included to show the applicability of our results.

Terminal value problem for nonlinear parabolic and pseudo-parabolic systems

In this paper, we study on the backward problem for parabolic system and pseudo-parabolic system with conformable derivative. Two these problems have many applications in engineering such as image processing, geophysics, biology, etc. There are two main contributions in this paper. The first major result deals with ill-posedness and regularization for terminal value problem for parabolic system. By applying a new truncation method, we construct a regularized solution. We investigate the existence and uniqueness of regularized problem. Under some suitable conditions of the terminal data, we provide the error estimate between the mild and regularized solutions. The second contribution concerns the existence of a global solution of pseudo-parabolic system. Finally, we also obtain the convergence of the mild solution as the order of derivative approaches $ 1^-. $

Remarks on the Initial and Terminal Value Problem for Time and Space Fractional Diffusion Equation

The fractional problem for partial differential equation has many applications in science and technology. The main objective of the paper is to investigate the convergence of the mild solution of the diffusion equation with time and space fractional. We consider the problem in two cases which are forward problem and inverse problem. We use new techniques to overcome some of the complex assessments.