Differential Boundary Value Problems Research Articles

Riemann–Hilbert Problems on a Cut Plane

We consider the Riemann–Hilbert-type boundary-value problems in the case of several unknown functions and obtain solvability conditions and index formulas for the linear conjugation problem, the Riemann–Hilbert problem, and the Riemann–Hilbert–Poincare problem on a plane cut along several regular arcs in various weighted functional classes. We also examine the general differential boundary-value problem for analytic functions and boundary-value problems with shifts and complex conjugation on a cut plane.

Existence and Nonexistence of Solutions for Fourth-Order Nonlinear Difference Boundary Value Problems via Variational Methods

This paper is concerned with boundary value problems for a fourth-order nonlinear difference equation. Via variational methods and critical point theory, sufficient conditions are obtained for the existence of at least two nontrivial solutions, the existence ofndistinct pairs of nontrivial solutions, and nonexistence of solutions. Some examples are provided to show the effectiveness of the main results.

Positive Solutions for a System of Semipositone Fractional Difference Boundary Value Problems

Using the fixed point index, we establish two existence theorems for positive solutions to a system of semipositone fractional difference boundary value problems. We adopt nonnegative concave functions and nonnegative matrices to characterize the coupling behavior of our nonlinear terms.

Positive Solutions to n-Order Fractional Differential Equation with Parameter

By employing the properties of the sum operators, we investigate the solutions of the n-order fractional differential equations multiple point boundary value problem (in short BVP) with the boundary conditions contains a parameter. We not only obtain the existence and uniqueness of solutions about this BVP but also construct an iterative scheme to approximate the solution which is important for practical application. An example is given to demonstrate the validity of our main results.

Existence of positive solutions for a high order fractional differential equation integral boundary value problem with changing sign nonlinearity

In this paper, we investigate the existence of positive solutions for a class of high order fractional differential equation integral boundary value problems with changing sign nonlinearity. By applying cone expansion and cone compression fixed point theorem, we have obtained and proved theorems related to the existence of positive solutions, which highlight the influences of the parameters in different ranges on the existence of positive solutions. Finally, we also give some examples to illustrate our main results.

B-SPLINE COLLOCATION SIMULATION OF NON-LINEAR TRANSIENT MAGNETIC NANOBIO-TRIBOLOGICAL SQUEEZE-FILM FLOW

A mathematical model is presented for magnetized nanofluid bio-tribological squeeze-film flow between two approaching disks. The nanofluid comprises a suspension of metal oxide nanoparticles with an electrically-conducting base fluid, making the nanosuspension responsive to applied magnetic field. The governing viscous momentum, heat and species (nanoparticle) conservation equations are normalized with appropriate transformations which renders the original coupled, non-linear partial differential equation system into a more amenable ordinary differential boundary value problem. The emerging model is shown to be controlled by a number of parameters, viz nanoparticle volume fraction, squeeze number, Hartmann magnetic body force number, disk surface transpiration parameter, Brownian motion parameter, thermophoretic parameter, Prandtl number and Lewis number. Computations are conducted with a B-spline collocation numerical method. Validation with previous homotopy solutions is included. The numerical spline algorithm is shown to achieve excellent convergence and stability in non-linear bio-tribological boundary value problems. The interaction of heat and mass transfer with nanofluid velocity characteristics is explored. In particular, smaller nanoparticle (high Brownian motion parameter) suspensions are studied. The study is relevant to enhanced lubrication performance in novel bio-sensors and intelligent knee joint (orthopaedic) systems.

Existence of solutions for [formula omitted]-order nonlinear differential boundary value problems by means of fixed point theorems

This paper is devoted to prove the existence of one or multiple solutions of a wide range of nonlinear differential boundary value problems.To this end, we obtain some new fixed point theorems for a class of integral operators. We follow the well-known Krasnoselskiĭ’s fixed point Theorem together with two fixed point results of Leggett–Williams type. After obtaining a general existence result for a one parameter family of nonlinear differential equations, are proved, as particular cases, existence results for second and fourth order nonlinear boundary value problems.

One-dimensional nonlinear boundary value problems with variable exponent

In this paper, a class of nonlinear differential boundary value problems with variable exponent is investigated. The existence of at least one non-zero solution is established, without assuming on the nonlinear term any condition either at zero or at infinity. The approach is developed within the framework of the Orlicz-Sobolev spaces with variable exponent and it is based on a local minimum theorem for differentiable functions.

Four Point Implicit Methods for the Second Derivatives of the Solution of First Type Boundary Value Problem for One Dimensional Heat Equation

We construct four-point implicit difference boundary value problem for the first derivative of the solution u(x,t) of the first type boundary value problem for one dimensional heat equation with respect to the time variable t. Also, for the second derivatives of u(x,t) special four-point implicit difference boundary value problems are proposed. It is assumed that the initial function belongs to the Hölder space C8+α,0 < α < 1, the heat source function given in the heat equation is from the Hölder space [see formula in PDF], the boundary functions are from [see formula in PDF], and between the initial and the boundary functions the conjugation conditions of orders q = 0,1,2,3,4 are satisfied. We prove that the solution of the proposed difference schemes converge uniformly on the grids of the order O(h2+τ) (second order accurate in the spatial variable x and first order accurate in time t) where, h is the step size in x and τ is the step size in time. Theoretical results are justified by numerical examples.

Differential and Difference Boundary Value Problem for Loaded Third-Order Pseudo-Parabolic Differential Equations and Difference Methods for Their Numerical Solution

Boundary value problems for loaded third-order pseudo-parabolic equations with variable coefficients are considered. A priori estimates for the solutions of the problems in the differential and difference formulations are obtained. These a priori estimates imply the uniqueness and stability of the solution with respect to the initial data and the right-hand side on a layer, as well as the convergence of the solution of each difference problem to the solution of the corresponding differential problem.

Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator

In this paper, we investigate the existence of positive solutions for a system of nonlinear fractional differential equations nonlocal boundary value problems with parameters and p-Laplacian operator. Under different combinations of superlinearity and sublinearity of the nonlinearities, various existence results for positive solutions are derived in terms of different values of parameters via the Guo-Krasnosel’skii fixed point theorem.

Lyapunov-type inequalities for fractional difference operators with discrete Mittag-Leffler kernel of order 2 < α < 5/2

Fractional difference operators with discrete-Mittag-Leffler kernels of order α > 1 are defined and their corresponding fractional sum operators are confirmed. We prove existence and uniqueness theorems for the discrete fractional initial value problems in the frame of discrete Caputo (ABC) and Riemann (ABR) operators by using Banach contraction theorem. Then, we prove Lyapunov type inequality for a Riemann type fractional difference boundary value problem of order 2 < α < 5∕2 within discrete Mittag-Leffler kernels, where the limiting case α → 2+ results in the ordinary difference Lyapunov inequality. Examples are given to clarify the applicability of our results and an application about the discrete fractional Sturm-Liouville eigenvalue problem is analyzed.

Robust point-to-point trajectory planning for nonlinear underactuated systems: Theory and experimental assessment

Abstract This paper proposes the theory and the experimental assessment of a robust model-based trajectory planning algorithm for underactuated nonlinear systems in point-to-point motion. The method has been developed to increase the insensitivity of the resulting trajectory to parametric uncertainties of the plant. The proposed method is based on an augmented model that considers an approximate dynamics of the servo-controlled axis driving the actuated degrees of freedom. Trajectory planning is accomplished by computing the motion reference for the actuated degrees of freedom to reduce the effects of the uncertainty on the dynamic model. By exploiting an indirect variational formulation method, the necessary optimality conditions deriving from the Pontryagin’s minimum principle are imposed, thus leading to a differential Two-Point Boundary Value Problem (TPBVP). Numerical solution of the latter is accomplished by means of collocation techniques to handle model nonlinearities. Robustness is achieved by including additional conditions on the sensitivity functions for the initial and final points of the trajectory. The experimental evaluation of the effectiveness of the proposed method is performed on a double-pendulum crane, by comparing the transient and residual vibration. A comparison is provided with three well-established input-shaping methods, and robustness against unmodeled parametric perturbations and tracking errors is evaluated. The experimental evidence indicates that the inclusion of the additional constrains results in an effective reduction of the residual vibration, and that the proposed method is well suited to perform high speed motion.

Inverse problems for difference equations with quadratic eigenparameter dependent boundary conditions

This paper inductively investigates an inverse problem for difference boundary value problems with boundary conditions that depend quadratically on the eigenparameter. In particular, given the eigenvalues and the weights, we provide an algorithm to uniquely reconstruct the potential.

Coupled systems of Caputo type fractional [formula omitted]-difference boundary value problems at resonance

The main research line of this paper is concerned with the existence and uniqueness of solutions for a certain class of coupled systems of Caputo type fractional Δ-difference boundary value problems at resonance. To this aim, we use coincidence degree theory to obtain existence results and impose growth controlling conditions on nonlinearities, uniqueness results will be concluded. At the end by means of an illustrative example the obtained main results will be implemented.

Existence Results for Fractional Hahn Difference and Fractional Hahn Integral Boundary Value Problems

The existence and uniqueness results of two fractional Hahn difference boundary value problems are studied. The first problem is a Riemann-Liouville fractional Hahn difference boundary value problem for fractional Hahn integrodifference equations. The second is a fractional Hahn integral boundary value problem for Caputo fractional Hahn difference equations. The Banach fixed-point theorem and the Schauder fixed-point theorem are used as tools to prove the existence and uniqueness of solution of the problems.

Weighted cubic and biharmonic splines

In this paper we discuss the design of algorithms for interpolating discrete data by using weighted cubic and biharmonic splines in such a way that the monotonicity and convexity of the data are preserved. We formulate the problem as a differential multipoint boundary value problem and consider its finite-difference approximation. Two algorithms for automatic selection of shape control parameters (weights) are presented. For weighted biharmonic splines the resulting system of linear equations can be efficiently solved by combining Gaussian elimination with successive over-relaxation method or finite-difference schemes in fractional steps. We consider basic computational aspects and illustrate main features of this original approach.

Positive solutions for a class of fractional difference boundary value problems

Positive solutions for a class of fractional difference boundary value problems

Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications

Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order0&lt;α≤1with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extension is given to both left and right fractional differences and sums. Then, existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial difference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional difference boundary value problems of order2&lt;α≤3is proved and the ordinary difference Lyapunov inequality then follows asαtends to2from right. Illustrative examples are discussed and an application about Sturm-Liouville eigenvalue problem in the sense of this new fractional difference calculus is given.

Mathematical Statement and Methods of Solution of the Vibroconsolidation Problem for Salty Soils

We formulate an initial-boundary value problem describing the process of vibroconsolidation of salty soils. We establish the existence and uniqueness of a solution and discuss methods for solving this problem. For the correspondence finite difference boundary value problem we study errors of the locally one-dimensional scheme and derive a priori estimates. Conclusions based on a numerical experiment are given.