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The paper deals with energy (weak) solutions u (t; x) of the class of equations with the model representativeup−1ut−Δpu=0,tx∈0T×Ω,Ω∈ℝn,n≥1,p>0,and with the following blow-up condition for the energy:εt≔∫Ωutxp+1dx+∫0t∫Ω∇xuτxp+1dxdτ→∞ast→T,where Ω is a smooth bounded domain. In the case of flat peaking, namely, under the conditionεt≤Fαtω0T−t−α∀t0,α>1p+1,a sharp estimate of the profile of a solution has been obtained in a neighborhood of the blow-up time t = T. [ABSTRACT FROM AUTHOR]

In this paper, we present a general methodology for solving buckling problems for inhomogeneous columns. Columns that are treated are functionally graded in axial direction. The buckling mode is postulated as the general order polynomial function that satisfies all boundary conditions. For specificity, we concentrate on the boundary conditions of simple support, and employ the second-order ordinary differential equation that governs the buckling behavior. A quadratic polynomial is adopted for the description of the column's flexural rigidity. Satisfaction of the governing differential equation leads to a set of nonlinear algebraic equations that are solved exactly. In addition to the recovery of the solutions previously found by Duncan and Elishakoff, several new solutions are arrived at. [ABSTRACT FROM AUTHOR]

In this paper, a mid-knot cubic non-polynomial spline is applied to obtain the numerical solution of a system of second-order boundary value problems. The numerical method is proved to be uniquely solvable and it is of second-order accuracy. We further include three examples to illustrate the accuracy of our method and to compare with other methods in the literature. [ABSTRACT FROM AUTHOR]