A Numerical Method for Computing Double Integrals with Variable Upper Limits

We propose and justify a numerical method for computing the double integral with variable upper limits that leads to the variableness of the region of integration. Imposition of simple variables as functions for upper limits provides the form of triangles of integration region and variable in the external limit of integral leads to a continuous set of similar triangles. A variable grid is overlaid on the integration region. We consider three cases of changes of the grid for the division of the integration region into elementary volumes. The first is only the size of the imposed grid changes with the change of variable of the external upper limit. The second case is the number of division elements changes with the change of the external upper limit variable. In the third case, the grid size and the number of division elements change after fixing their multiplication. In these cases, the formulas for computing double integrals are obtained based on the application of cubatures in the internal region of integration and performing triangulation division along the variable boundary. The error of the method is determined by expanding the double integral into the Taylor series using Barrow’s theorem. Test of efficiency and reliability of the obtained formulas of the numerical method for three cases of ways of the division of integration region is carried out on examples of the double integration of sufficiently simple functions. Analysis of the obtained results shows that the smallest absolute and relative errors are obtained in the case of an increase of the number of division elements changes when the increase of variable of the external upper limit and the grid size is fixed.

An Existence Result for Henstock-Kurzweil-Stieltjes-⋄-Double Integral of Interval-Valued Functions on Time Scales

We employ the concept of interval-valued functions to state and prove an existence result for the Henstock-Kurzweil-Stieltjes-⋄-double integral on time scales.

Minimality of balls in the small volume regime for a general Gamow-type functional

We consider functionals given by the sum of the perimeter and the double integral of some kernel g : ℝ N × ℝ N → ℝ + {g:\mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb{R}^{+}} , multiplied by a “mass parameter” ε. We show that, whenever g is admissible, radial and decreasing, the unique minimizer of this functional among sets of given volume is the ball as soon as ε ≪ 1 {\varepsilon\ll 1} .

An Efficient Numerical Scheme for the Solution of a Stochastic Volatility Model Including Contemporaneous Jumps in Finance

The model of stochastic volatility with contemporaneous jumps is written for pricing under a partial integro-differential equation (PIDE) having a double integral and a nonsmooth initial value. To tackle this problem, first, a new radial basis function (RBF) as a convex combination of two known RBFs is given. Second, the weighting coefficients of the RBF generated finite difference (FD) method are contributed and the associated error equations are derived. To deal with the integral part, the new idea is to apply an estimate for the unknown function for every cell and do an integration of the density function. The contributed approach is competitive and reduces both the calculational efforts and elapsed time.

Some analytical and numerical results for a fractional q-differential inclusion problem with double integral boundary conditions

In this work, we study a q-differential inclusion with doubled integral boundary conditions under the Caputo derivative. To achieve the desired result, we use the endpoint property introduced by Amini-Harandi and quantum calculus. Integral boundary conditions were considered on time scale $\mathcal{T}_{t_{0}}=\{t_{0},t_{0}q,t_{0}q^{2}, \ldots\}\cup \{0\}$ T t 0 = { t 0 , t 0 q , t 0 q 2 , … } ∪ { 0 } . To better evaluate the validity of our results, we provided an example, some graphs, and tables.

Design of Nonlinear Backstepping Double-Integral Sliding Mode Controllers to Stabilize the DC-Bus Voltage for DC–DC Converters Feeding CPLs

This paper proposes a composite nonlinear controller combining backstepping and double-integral sliding mode controllers for DC–DC boost converter (DDBC) feeding by constant power loads (CPLs) to improve the DC-bus voltage stability under large disturbances in DC distribution systems. In this regard, an exact feedback linearization approach is first used to transform the nonlinear dynamical model into a simplified linear system with canonical form so that it becomes suitable for designing the proposed controller. Another important feature of applying the exact feedback linearization approach in this work is to utilize its capability to cancel nonlinearities appearing due to the incremental negative-impedance of CPLs and the non-minimum phase problem related to the DDBC. Second, the proposed backstepping double integral-sliding mode controller (BDI-SMC) is employed on the feedback linearized system to determine the control law. Afterwards, the Lyapunov stability theory is used to analyze the closed-loop stability of the overall system. Finally, a simulation study is conducted under various operating conditions of the system to validate the theoretical analysis of the proposed controller. The simulation results are also compared with existing sliding mode controller (ESMC) and proportional-integral (PI) control schemes to demonstrate the superiority of the proposed BDI-SMC.