When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Based on general theory for positivity, with an explicit third-order Runge-Kutta method (we will refer to it as RK3 method) pos...

The framework of first-, second-, and third-order embodiment (1-3E; Metzinger, 2014) was developed for exploring how phenomenal properties are physically and computationally grounded. Metzinger’s goal for 1-3E is to show how the experience of being a self (i.e., phenomenal selfhood) is generated within an embodied system and thus to embed his self-model theory (Metzinger, 2004) in the context o...

Singularly perturbed two-point boundary-value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative with a discontinuous source term is considered. The given fourth-order BVP is transformed into a system of weakly coupled systems of two second-order ODEs, one without the parameter and the other with the paramet...

Impulsive differential equations, which arise in physics, population dynamics, economics, and so forth, are important mathematical tools for providing a better understanding of many real-world models, we refer to [1–5] and the references therein. About the applications of the theory of impulsive differential equations to different areas, for example, see [6–15]. Boundary value problems (BVPs) f...

Abstract The paper presents an accelerating of solving potential boundary value problems (BVPs) with curvilinear boundaries by modified parametric integral equations system (PIES). fast multipole method (FMM) known from the literature was included into PIES. To consider complex shapes a boundary, modification binary tree used FMM is proposed. combined PIES, called also allows significant reduct...

A theory of third order extended regular variation is developed, somewhat similar to the corresponding theories of first and second order.

In recent work, Schweber [7] introduces a framework for reverse mathematics in a third order setting and investigates several natural principles of transfinite recursion. The main result of that paper is a proof, using the method of forcing, that in the context of two-person perfect information games with moves in R, open determinacy (Σ1 -DET) is not implied by clopen determinacy (∆ R 1 -DET). ...

SUMMARY For a variable and parameter of the same dimension, the tangent exponential model (Fraser, 1988) approximates an asymptotic model to third order in a rst derivative neighbourhood of the data point and to second order otherwise. For the more usual case of a variable of larger dimension than the parameter, we obtain a unique expression for the third order ancillary distribution as project...