Pedro Almenar

A Lyapunov inequality for a second order nonlinear differential equation

Abstract This paper presents a Lyapunov-type inequality for the second order nonlinear equation ( r ( x ) y ′ ) ′ + p ( x ) f ( y ( x ) ) = 0 , with r ( x ) , p ( x ) > 0 and f ( y ) odd and positive for y > 0 . It also compares it with similar results.

Mixed problems for separated variable coefficient wave equations: analytic-numerical solutions with a priori error bounds

This paper deals with the construction of accurate analytic-numerical solutions of mixed problems related to the separated variable dependent wave equation utt=(b(t)/a(x))uxx,0 0. Based on the study of the growth of eigenfunctions of the underlying Sturm–Liouville problems, an exact theoretical series solution is firstly obtained. Explicit bounds allow truncation of the series solution so that the error of the truncated approximation is less than e1 in a bounded domain Ω(d)={(x,t);0⩽x⩽L,0⩽t⩽d}. Since the approximation involves only a finite number of exact eigenvalues λi2,1⩽i⩽n0, the admissible error for the approximated eigenvalues λi2,1⩽i⩽n0, is determined in order to construct an analytical numerical solution of the mixed problem, involving only approximated eigenvalues λi2, so that the total error is less than e uniformly in Ω(d). Uniqueness of solutions is also treated.

The Distribution of Zeroes and Critical Points of Solutions of a Second Order Half-Linear Differential Equation

This paper reuses an idea first devised by Kwong to obtain upper bounds for the distance between a zero and an adjacent critical point of a solution of the second order half-linear differential equation , with , , and real such that . It also compares it with other methods developed by the authors.

Solvability of Nth Order Linear Boundary Value Problems

This paper presents a method that provides necessary and sufficient conditions for the existence of solutions of nth order linear boundary value problems. The method is based on the recursive application of a linear integral operator to some functions and the comparison of the result with these same functions. The recursive comparison yields sequences of bounds of extremes that converge to the exact values of the extremes of the BVP for which a solution exists.

Improving Results on Solvability of a Class of th-Order Linear Boundary Value Problems

This paper presents a modification of a recursive method described in a previous paper of the authors, which yields necessary and sufficient conditions for the existence of solutions of a class of th-order linear boundary value problems, in the form of integral inequalities. Such a modification simplifies the assessment of the conditions on restricting the inequality to be verified to a single point instead of the full interval where the boundary value problem is defined. The paper also provides an error bound that needs to be considered in the integral inequalities of the previous paper when they are calculated numerically.

The Distance between Points of a Solution of a Second Order Linear Differential Equation Satisfying General Boundary Conditions

This paper presents a method to obtain lower and upper bounds for the minimum distance between points and of the solution of the second order linear differential equation satisfying general separated boundary conditions of the type and . The method is based on the recursive application of a linear operator to certain functions, a recursive application that makes these bounds converge to the exact distance between and as the recursivity index grows. The method covers conjugacy and disfocality as particular cases.

Convergent Disfocality and Nondisfocality Criteria for Second-Order Linear Differential Equations

This paper presents a method to determine whether the second-order linear differential equation is either disfocal or nondisfocal in a fixed interval. The method is based on the recursive application of a linear operator to certain functions and yields upper and lower bounds for the distances between a zero and its adjacent critical points, which will be shown to converge to the exact values of such distances as the recursivity index grows.

On the Zeroes and the Critical Points of a Solution of a Second Order Half-Linear Differential Equation

This paper presents two methods to obtain upper bounds for the distance between a zero and an adjacent critical point of a solution of the second-order half-linear differential equation , with and piecewise continuous and , and being real such that . It also compares between them in several examples. Lower bounds (i.e., Lyapunov inequalities) for such a distance are also provided and compared with other methods.

Solvability of a Class of N -th Order Linear Focal Problems

This paper presents a recursive method which yields necessary and sufficient conditions for the existence of solutions of a class of n-th order linear focal boundary value problems in the interior of a given interval, in the form of integral inequalities. Some results on the sign of the derivatives of the Green functions of n-th order linear focal boundary value problems will also be provided.