Theorems about the Existence of Solutions to Problems with Nonlocal Initial Value

High regularity of the solutions of the telegraph system subjected to nonlinear boundary conditions

In 1908 Birkhoff defined a class of series generalizing the Fourier and Sturm-Liouville series which arise from ordinary linear homogeneous differential equations of the second order with linear homogeneous boundary conditions. He starts from a differential system of order n with boundary conditions restricted to be regular and proves that an arbitrary function with a sufficient number of derivatives can be represented in terms of the solutions of this differential system.t Subsequently, Tamarkin extended the discussion to apply to functions integrable in the sense of Lebesgue by the device of comparing the formal Fourier and Birkhoff series for the same function.t Recently the author has published further researches on the problem. ? It is now proposed to complete in some respects our knowledge of regular differential systems of the second order. In our paper cited above, we left certain necessary conditions unstudied and now desire to fill this gap in the theory. 11 In addition, the attempt to discuss the formal expansions for functions integrable in accordance with definitions of integration more extended than that of Lebesgue has led us to results quite different from those hitherto established. We shall indicate the facts for functions integrable in the sense of Denjoy. The situation may be described roughly by saying that, whereas all the regular expansions associated with an arbitrary function integrable in the sense of Lebesgue have virtually the same behavior, the series formed for an arbitrary function integrable in the sense of Denjoy or non-absolutely integrable in the sense of Riemann fall into a non-denumerable infinity of distinct types exhibiting individual behavior. The result is due essentially to the fact that the Riemann-Lebesgue theorem does not hold true for nonabsolutely integrable functions. On the other hand, if the series for an arbi-

In this thesis we investigate the existence of solutions to boundary value problems (BVPs) for nonlinear systems of first-order ordinary differential equations (ODEs) and difference equations. The thesis is divided into two parts. Part one is devoted to the study of difference equations where two types of problems are studied. First, we derive the existence of solutions to first-order systems of difference equations that arise when one applies the trapezoidal rule to approximate solutions of second-order scalar ODEs. Strict discrete lower solutions are used with maximum principle arguments in the discrete problem to obtain a priori bounds on solutions. The a priori bounds on difference quotients for solutions of the discrete problem are obtained from the Bernstein-Nagumo condition. We use homototopy to find solutions of the discrete amroximations. Second, motivated by Ma [60] we consider existence and uniqueness of solutions to difference equations which arise as discrete approximations to three-point BVPs for systems of first-order ODES after employing the Euler method. Several existence and uniqueness results for three-point BVPs are established, using the contraction mapping theorem and the Brouwer fixed point theorem in Rn. In part two, two types of problems for systems of first-order ODES are analyzed. First, we study the existence of a unique solution to a system of first-order differentia1 equations under multi-point BVPs with nonlinear boundary conditions. We prove the existence and uniqueness of solutions for a system of first-order three-point BVP under which the properties of linear three-point BVP are preserved under small nonlinear perturbations of both the differentia1 equation and the boundary conditions. The method of proof of the existence of solutions using the Banach contraction mapping principle is similar to that used by Rodriguez [75], who studied the discrete analogue of this BVP. Second, motivated by Cronin 120,211 we investigate the existence of solutions to three-point BVPs in perturbed systems of first-order ODEs at resonance where the associated homogeneous linear problem has nontrivial solutions. Based on the second part of the investigation of three-point BVPs in perturbed systems of first-order ODEs at resonance, four different results were obtained: (i) The existence of solutions via a version of the Brouwer Fixed Point Theorem which is due to Miranda. (ii) The existence of solutions to the problem through the application of the Implicit Function Theorem. These two results extend the work of Feng and Webb [34], and Gupta [39] for the problems at resonance in Euclidean 2-space. (iii) Existence results for three-point BVPs at resonance for general BVPs through the application of Brouwer degree theory; we show the degree is non zero through the application of Borsuk's Theorem for one problem and through a result of Cronin involving pairs of polynomials in two variables whose terms of highest order have no common factors. (iv) The existence of solutions to the entrainment of frequency problem in Euclidean 2-space. We will investigate conditions under which entrainment of frequency for three-point as well as two-point BVPs by adapting the work of Cronin in periodic case. We will describe such conditions and then show the degree is non zero through a result of Cronin involving pairs of polynomials in two variables whose terms of lowest order have no common factors.

We consider the one-dimensional linear free space Schrödinger equation on a bounded interval subject to homogeneous linear boundary conditions. We prove that, in the case of pseudoperiodic boundary conditions, the solution of the initial-boundary value problem exhibits the phenomenon of revival at specific (“rational”) times, meaning that it is a linear combination of a certain number of copies of the initial datum. Equivalently, the fundamental solution at these times is a finite linear combination of delta functions. At other (“irrational”) times, for suitably rough initial data, e.g., a step or more general piecewise constant function, the solution exhibits a continuous but fractal-like profile. Further, we express the solution for general homogenous linear boundary conditions in terms of numerically computable eigenfunctions. Alternative solution formulas are derived using the Unified Transform Method (UTM), that can prove useful in more general situations. We then investigate the effects of general linear boundary conditions, including Robin, and find novel “dissipative” revivals in the case of energy decreasing conditions.

In this work, boundary control problems governed by a system of semilinear parabolic PDEs with pointwise control constraints are considered. This class of problems is related to applications in the chemical catalysis. After discussing existence and uniqueness of the state equation with both linear and nonlinear boundary conditions, the existence of an optimal solution is shown. Necessary and sufficient optimality conditions are derived to deal with numerical examples, which conclude the paper.

The highly fruitful conception of adjoint pairs of linear systems, each consisting of a homogeneous linear differential equation of the nth order and of n homogeneous linear boundary conditions, was first introduced in an explicit and general manner by BIRKHOFF.t That we have here a conception of fundamental importance is hardly yet generally appreciated. It is my purpose in the present paper to make an application of this idea to an important question in the theory of linear boundary problems which has so far been treated only in very special cases.+ I will mention in passing that the results may also be obtained by the use of integral equations; though, if we wish to include all cases, not perhaps so simply or obviously as might at first sight appear. In § 1 the conception of adjoint system and a few other fundamental matters concerning linear systems are explained and a few preliminary questions disposed of. In § 2 come the main results of the paper, namely Theorems I and II, which apply to the case of complex as well as to that of real independent variNble. These theorems are then applied in some detail to the equation of the second order, whereby the relation to MASON'S results becomes apparent. In § 3 an extension to systems of equations of any order is briefly indicated; and a somewhat generalized type of differential equation is also considered. Finally, ill § 4, the conception of adjoint system is extended to the case in which the number of boundary conditions is not the same as the order of the differential equation.

The purpose of this paper is to study a Lions type regularization of the telegraph system with nonlinear boundary conditions. An asymptotic expansion of order zero for the solution of this regularization is established, including some boundary layer corrections. Specifically, under some appropriate smoothness and compatibility conditions on the data an estimate for the remainder term of the expansion is derived with respect to the \(C([0,T];L^2(0,1)^2)\) norm. Our main theorem generalizes a result regarding the particular case of homogeneous linear boundary conditions reported by Apreutesei and Djafari Rouhani (Nonlinear Anal 35:3049–3061, 2010).

In this study, nonlinear vibrations of a slightly curved Timoshenko beam with nonlinear boundary conditions are investigated. The formulation is developed based on Timoshenko beam theory, incorporating shear deformation and rotary inertia effects. Geometric nonlinearity is introduced through the von Karman strain–displacement relations, and the governing equations of motion are derived using Hamilton’s principle. The resulting partial differential equations are reduced to a set of ordinary differential equations via the Galerkin method, where the mode shapes of a linear straight Timoshenko beam with linear spring boundary conditions are employed as admissible functions. The temporal response of the system is obtained using the method of multiple scales. Validation of the proposed formulation is carried out by examining the limiting case of vanishing curvature and by comparison with the nonlinear response of straight Timoshenko beams and thick shallow arch models. The results reveal that both hardening and softening behaviors may occur due to the presence of second- and third-order nonlinear terms. The proposed semi-analytical framework provides an efficient tool for analyzing nonlinear vibrations of slightly curved Timoshenko beams, capturing the combined effects of geometric nonlinearity, shear deformation, rotary inertia, and boundary flexibility.

Numerical methods for bifurcation problems of the form ( ∗ ) L y = λ f ( y ) , B y = 0 , \begin{equation}\tag {$\ast $} Ly = \lambda f(y),\quad By = 0,\end{equation} where f ( 0 ) = 0 f(0) = 0 and f ′ ( 0 ) ≠ 0 f’(0) \ne 0 , are considered. Here y is a scalar function, λ \lambda is a real scalar, L is a linear differential operator and B y = 0 By = 0 represents some linear homogeneous two-point boundary conditions. Under certain assumptions, it is shown that if ( ∗ ) (\ast ) is replaced by an appropriate difference scheme, then there exists a unique branch of nontrivial solutions of the discrete problem in a neighborhood of a branch of nontrivial solutions of ( ∗ ) (\ast ) bifurcating from the trivial solution and that the discrete branch converges to the continuous one. Error estimates are derived and an illustrative numerical example is included.

Homotopy method of fundamental solutions for solving nonlinear heat conduction problems

Consider the initial-boundary value problem of the linear wave equation wtt-wxx=0 on an interval. The boundary condition at the left endpoint is linear homogeneous, injecting energy into the system, while the boundary condition at the right endpoint has cubic nonlinearity of a van der Pol type. We show that the interactions of these linear and nonlinear boundary conditions can cause chaos to the Riemann invariants (u,v) of the wave equation when the parameters enter a certain regime. Period-doubling routes to chaos and homoclinic orbits are established. We further show that when the initial data are smooth satisfying certain compatibility conditions at the boundary points, the space-time trajectory or the state of the wave equation, which satisfies another type of the van der Pol boundary condition, can be chaotic. Numerical simulations are also illustrated.

Many effects that generally characterise the phenomenon of turbulence, in particular, the emergence of structures (including the cascade process of birth of coherent structures of decreasing scales) and self-stochasticity can be observed in simple examples of dynamical systems which are generated by one- and two-dimensional boundary value problems (BVP) consisting of linear partial differential equations and nonlinear boundary conditions. But mathematical models using these BVP (without certain additional conditions or assumptions) do not describe adequately the behaviour of real physical objects when time is sufficiently large, at least, because the diameters of structures appearing in these BVPs can decrease up to zero, which is not in agreement with discrete nature of time and space. Thus these models can describe adequately a real process only within certain time-space limits. Most likely the models have to be adjusted beginning with certain time-space scales. At the same time, results obtained by numerical investigation of these models are probably more close (in some sense) to the reality than exact solutions not in spite but owing to incorrect calculations.

In this paper, by using the Mountain Pass Lemma, we study the existence of nontrivial solutions for a nonlocal elliptic Kirchhoff type equation together with nonlinear boundary conditions.

Effect of a thin layer on the measurement of the thermal diffusivity of a material by a flash method

The nonsingular, nonlinear Volterra integral equation of the second kind for the surface temperature of a cylinder with a continuous heat sink is solved numerically by a simple, but robust, procedure based on a composite formula. The present scheme is capable of handling stiff and nonstiff kernels expressed in terms of infinite series that contain standard functions. The numerical method is tested by imposing linear and nonlinear boundary conditions at the cylinder surface. The boundary conditions and source terms are treated explicitly without resorting to a linearization technique. To show the versatility of the numerical scheme, two classes of nonlinear boundary conditions are considered, the Stefan Boltzmann boundary condition and the time-dependent Newton's cooling law. The accuracy and robustness of the present method for solving Volterra equations of the second kind are verified considering two different forms of the source term. Two additional methods of solution are used in this investigation as benchmarks. One of them is a pseudospectral approach and the other is a standard finite difference calculation. The integral equation scheme is found to work well over the entire nondimensional time domain. Numerical results are presented for a wide range of Stark numbers and steady-state Biot numbers. The numerical scheme is computationally efficient and yields stable solutions. Changes in the geometry alter the form of the kernel in the integral equation, but the general principles used should be applicable to solve nonlinear integral equations.