Oscillation of Solutions of LDE’s in Domains Conformally Equivalent to Unit Disc

Oscillation of solutions of $$f^{(k)} + a_{k-2} f^{(k-2)} + \cdots + a_1 f' +a_0 f = 0$$ f ( k ) + a k - 2 f ( k - 2 ) + ⋯ + a 1 f ′ + a 0 f = 0 is studied in domains conformally equivalent to the unit disc. The results are applied, for example, to Stolz angles, horodiscs, sectors, and strips. The method relies on a new conformal transformation of higher order linear differential equations. Information on the existence of zero-free solution bases is also obtained.

Analytical Solutions of Fuzzy Linear Differential Equations in the Conformable Setting

In this paper, we provide the generalization of two predefined concepts under the name fuzzy conformable differential equations. We solve the fuzzy conformable ordinary differential equations under the strongly generalized conformable derivative. For the order $\Psi$, we use two methods. The first technique is to resolve a fuzzy conformable differential equation into two systems of differential equations according to the two types of derivatives. The second method solves fuzzy conformable differential equations of order $\Psi$ by a variation of the constant formula. Moreover, we generalize our results to solve fuzzy conformable ordinary differential equations of a higher order. Further, we provide some examples in each section for the sake of demonstration of our results.

On the absence of logarithmic singularities in the solutions of Lamé-type equations

The object of this research is linear differential equations of the second order with regular singularities. We extend the concept of a regular singularity to linear partial differential equations. The general solution of a linear differential equation with a regular singularity is a linear combination of two linearly independent solutions, one of which in the general case contains a logarithmic singularity. The well-known Lamé equation, where the Weierstrass elliptic function is one of the coefficients, has only meromorphic solutions. We consider such linear differential equations of the second order with regular singularities, for which as a coefficient instead of the Weierstrass elliptic function we use functions that are the solutions to the first Painlevé or Korteweg – de Vries equations. These equations will be called Lamé-type equations. The question arises under what conditions the general solution of Lamé-type equations contains no logarithms. For this purpose, in the present paper, the solutions of Lamé-type equations are investigated and the conditions are found that make it possible to judge the presence or absence of logarithmic singularities in the solutions of the equations under study. An example of an equation with an irregular singularity having a solution with an logarithmic singularity is given, since the equation, defining it, has a multiple root.

Modeling and forecasting the probability of the states of technical support systems for the use of weapons and military equipment

The article discusses a probabilistic model of processes in complex systems of technical support for military vehicles. One of the methods for studying such complex systems is their representation in the form of a set of typical states in which the system can be. Transitions occur between states, the intensities and probabilities of which are assumed to be known. The system is graphically represented using a graph of states and transitions, and the subject of research is the probability of finding the technical support system in these states. The graph of states and transitions is associated with a system of first order linear differential equations with respect to the probabilities of finding the support system in its basic states. To obtain a solution, this system must be supplemented with certain conditions. These are, firstly, the initial conditions that specify the probabilities of all states at the initial moment of time. Second, this is the normalization condition, which states that at any moment in time the sum of the probabilities of all states is equal to unity. An approximate solution to the problem is described in the literature. Such approximate solution is getting more accurate when the sought probabilities depend on time weaker. We propose a method of the exact solution of the above mentioned system of differential equations based on the use of operational calculus. In this case, the system of linear differential equations is transformed into a system of linear algebraic equations for the Laplace images of unknown probabilities. The use of matrix calculus made it possible to write down the obtained results in a compact form and to use effective numerical algorithms of linear algebra for further calculations. The model is illustrated by the example of solving the problem of technical support for the march of a battalion tactical group column, including wheeled and tracked vehicles. The boundaries of the validity of the results of a simpler approximate solution are established.

Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity

We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of non-linear differential equations to several cases of equations containing non-polynomial non-linearity. The main idea of the study is to use an appropriate transformation at Step (1.) of SEsM. This transformation has to convert the non-polynomial non- linearity to polynomial non-linearity. Then, an appropriate solution is constructed. This solution is a composite function of solutions of more simple equations. The application of the solution reduces the differential equation to a system of non-linear algebraic equations. We list 10 possible appropriate transformations. Two examples for the application of the methodology are presented. In the first example, we obtain kink and anti- kink solutions of the solved equation. The second example illustrates another point of the study. The point is as follows. In some cases, the simple equations used in SEsM do not have solutions expressed by elementary functions or by the frequently used special functions. In such cases, we can use a special function, which is the solution of an appropriate ordinary differential equation, containing polynomial non-linearity. Specific cases of the use of this function are presented in the second example.

New Criteria for Oscillation of Half-Linear Differential Equations with p-Laplacian-Like Operators

In this paper, new oscillatory properties for fourth-order delay differential equations with p-Laplacian-like operators are established, using the Riccati transformation and comparison method. Moreover, our results are an extension and complement to previous results in the literature. We provide some examples to examine the applicability of our results.