The Multiplicative Base Solution to the Differential Equations in Analog Circuits

This work focuses to solve any order of scalar differential equation involved in analog circuit representation. These types of mathematical representations have many applications in analysis and processing such as noise, filter, audio, RLC distributed interconnection (nodes) and transmission lines. Such systems are represented with scalar type differential equations and use numerical method to find a solution. One of the most successful methods is the fourth-order Runge–Kutta. This study introduced a multiplicative version of Runge–Kutta (MRK4) method. The performance analysis of the MRK4 is examined based on the error analysis method. The MRK4 method is applied to solve equations representing the linear and the nonlinear type systems. Results indicate the MRK4 to be superior with respect to the RK4 method.

The stability analysis of the market price using Lambert function method

In this article we are going to analyze market price stability with different market intensity coefficient and delay argument values. Market price is described as a scalar differential equation with a delay argument. In order to find solutions for the transcendental equation we will use method based on Lambert function. We will present examples of the applications of the method.

One method for investigating the solvability of boundary value problems for an implicit differential equation

The article concernes a boundary value problem with linear boundary conditions of general form for the scalar differential equation f(t,x(t),x ̇(t))=y ̂(t), not resolved with respect to the derivative x ̇ of the required function. It is assumed that the function f satisfies the Caratheodory conditions, and the function y ̂ is measurable. The method proposed for studying such a boundary value problem is based on the results about operator equation with a mapping acting from a metric space to a set with distance (this distance satisfies only one axiom of a metric: it is equal to zero if and only if the elements coincide). In terms of the covering set of the function f(t,x_1,•): R→R and the Lipschitz set of the function f(t,•,x_2): R →R, conditions for the existence of solutions and their stability to perturbations of the function f generating the differential equation, as well as to perturbations of the right-hand sides of the boundary value problem: the function y ̂ and the value of the boundary condition, are obtained.

Effects of Elliptical Ring Electrodes on Shear Vibrations of Quartz Crystal Plates

A theoretical analysis is performed on the thickness-shear vibrations of an AT-cut quartz piezoelectric crystal plate with elliptical ring electrodes. The scalar differential equation by Tiersten and Smythe is used. An analytical solution is obtained. Numerical results from the solution show that the thickness-shear mode of interest may be trapped by the ring electrodes and can have a convex, concave, or nearly flat vibration distribution near the plate center, which is fundamentally important when the plate is used as an acoustic wave mass sensor. The vibration distribution is found to be sensitive to both the geometric and physical parameters of the electrodes. Therefore, a careful design is needed to realize the desired trapped mode with suitable center convexity for sensor application

Multiple Positive Solutions for the Dirichlet Boundary Value Problems by Phase Plane Analysis

We consider boundary value problems for scalar differential equationx′′+λfx=0,x(0)=0,x(1)=0, wheref(x)is a seventh-degree polynomial andλis a parameter. We use the phase plane method combined with evaluations of time-map functions and make conclusions on the number of positive solutions. Bifurcation diagrams are constructed and examples are considered illustrating the bifurcation processes.