Solutions and stability for p-Laplacian differential problems with mixed type fractional derivatives

In this note, the main emphasis is to study two kinds of high-order fractional p-Laplacian differential equations with mixed derivatives, which include Caputo type and Riemann–Liouville type fractional derivative. Based on fixed point theorems on the cone, the existence-uniqueness of positive solutions for equations and two iterative schemes to uniformly approximate the unique solutions are discussed theoretically. What’s more, the sufficient conditions for stability of the equations are given. Some exact examples are further provided to verify the analytical results at the end of the article.

Synthesis of a smoothing bicubic spline for differentiating experimental data of nonparametric identification

Over the past two decades, so-called Voltaire series have been used to describe the dynamics of nonlinear systems in terms of input-output. Nonparametric identification of models using Voltaire series consists in constructing estimates for impulse transition functions (IPFs) depending on two or more arguments, which naturally makes identification algorithms much more complicated than in the one-dimensional case. So, in order to identify the two-dimensional IPF (corresponding to the quadratic term of the Voltaire series), it is necessary to calculate the second-order mixed derivatives of the output two-dimensional signal of the system, when a series of rectangular pulses of different amplitudes at different times are fed to its input. Everyone knows, the problem of differentiation is an ill-posed problem and one of the manifestations of incorrectness is poor resistance to errors in the initial data. It is proposed to use two-dimensional smoothing cubic (bicubic) spline (abbreviated SBS) to overcome this problem. The two tasks that constitute SBS synthesis: assignment and implementation of different types of boundary conditions at the border of the rectangular region where SBS is determined; optimal values estimation of two smoothing parameters due to the different “smoothness” of IPF for different two arguments. An acceptable solution to this synthesis problem is proposed in the paper. Our performed computational experiment showed the efficiency of the proposed algorithm for calculating second-order mixed derivative from noisy initial data.

Precise Numerical Differentiation of Thermodynamic Functions with Multicomplex Variables

The multicomplex finite-step method for numerical differentiation is an extension of the popular Squire–Trapp method, which uses complex arithmetics to compute first-order derivatives with almost machine precision. In contrast to this, the multicomplex method can be applied to higher-order derivatives. Furthermore, it can be applied to functions of more than one variable and obtain mixed derivatives. It is possible to compute various derivatives at the same time. This work demonstrates the numerical differentiation with multicomplex variables for some thermodynamic problems. The method can be easily implemented into existing computer programs, applied to equations of state of arbitrary complexity, and achieves almost machine precision for the derivatives. Alternative methods based on complex integration are discussed, too.

Hexagonal Grid Computation of the Derivatives of the Solution to the Heat Equation by Using Fourth-Order Accurate Two-Stage Implicit Methods

We give fourth-order accurate implicit methods for the computation of the first-order spatial derivatives and second-order mixed derivatives involving the time derivative of the solution of first type boundary value problem of two dimensional heat equation. The methods are constructed based on two stages: At the first stage of the methods, the solution and its derivative with respect to time variable are approximated by using the implicit scheme in Buranay and Arshad in 2020. Therefore, Oh4+τ of convergence on constructed hexagonal grids is obtained that the step sizes in the space variables x1, x2 and in time variable are indicated by h, 32h and τ, respectively. Special difference boundary value problems on hexagonal grids are constructed at the second stages to approximate the first order spatial derivatives and the second order mixed derivatives of the solution. Further, Oh4+τ order of uniform convergence of these schemes are shown for r=ωτh2≥116,ω>0. Additionally, the methods are applied on two sample problems.

Construction Of Three-Variable High-Order Defferential Equation Polynomial Solutions By Combinatorics Method

In this paper, we study how basic systems of polynomial solutions of a differential equation of high order with mixed derivatives of a function of three variables are constructed using combinatorial methods

Cebyšev inequalities for co-ordinated \(QC\)-convex and \((s,QC)\)-convex

In this paper, we establish some new Cebyšev type inequalities for functions whose modulus of the mixed derivatives are co-ordinated quasi-convex and \(\alpha \)-quasi-convex and \(s\)-quasi-convex functions.

Switched coupled system of nonlinear impulsive Langevin equations with mixed derivatives

<abstract><p>In this paper, we consider switched coupled system of nonlinear impulsive Langevin equations with mixed derivatives. Some sufficient conditions are constructed to observe the existence, uniqueness and generalized Ulam-Hyers-Rassias stability of our proposed model, with the help of generalized Diaz-Margolis's fixed point approach, over generalized complete metric space. We give an example which supports our main result.</p></abstract>