Skip to main content
Top

2024 | Book

New Trends in the Applications of Differential Equations in Sciences

NTADES 2023, Saints Constantine and Helena, Bulgaria, July 17–20

insite
SEARCH

About this book

This book convenes peer-reviewed, selected papers presented at the Tenth International Conference New Trends in the Applications of Differential Equations in Sciences (NTADES) held in Saints Constantine and Helena, Bulgaria, July 17–20, 2023. Contributions are devoted to many applications of differential equations in different fields of science. A number of phenomena in nature (physics, chemistry, biology) and in society (economics) result in problems leading to the study of linear and nonlinear differential equations, stochastic equations, statistics, analysis, numerical analysis, optimization, and more. The main topics are presented in the five parts of the book - applications in mathematical physics, mathematical biology, financial mathematics, neuroscience, and fractional analysis.

In this volume, the reader will find a wide range of problems concerning recent achievements in both theoretical and applied mathematics. The main goal is to promote the exchange of new ideas and research between scientists, who develop and study differential equations, and researchers, who apply them to solve real-life problems. The book promotes basic research in mathematics leading to new methods and techniques useful for applications of differential equations.

Table of Contents

Frontmatter

Applications in Mathematical Physics

Frontmatter
2-D Minimal Surface Flow with the Oblique Condition and Translators

In this paper, we study evolved surfaces over convex planar domains following the minimal surface flow $$\begin{aligned}u_{t}= div\left( \frac{Du}{\sqrt{1+|Du|^2}}\right) -H(x,Du).\end{aligned}$$ u t = d i v Du 1 + | D u | 2 - H ( x , D u ) . Here, we specify the angle of contact of the evolved surface to the boundary cylinder. The interesting question is to find translating solitons of the form $$u(x,t)=\omega t+w(x)$$ u ( x , t ) = ω t + w ( x ) where $$\omega \in \mathbb R$$ ω ∈ R . Under an angle condition on the boundary, we can prove the a priori estimate holds true for the translating solitons (i.e., translator), which makes the solitons exist. Then, we can prove for suitable condition on the function H(x, p) that there is the global solution of the minimal surface flow. Finally, we show that once the translating soliton exists, the global solutions converge to such translator.

Li Ma, Yuxin Pan
Graphical Portraits of the Solutions of Binary First Order Nonlinear Ordinary Differential Equation Near Their Singular Point

This paper deals with the appropriate normal homeomorphic forms of a nonlinear binary first order ordinary differential equation (ODE) with smooth coefficients near the critical (singular) point (0, 0) . The normal forms (A. Davydov, then T. Fukui) depend on a real parameter $$ \lambda $$ λ and can be semicubic parabola, folded saddle point, folded node and folded focus. The corresponding graphical portraits in the plane are proposed in Theorem 2 and are illustrated geometrically by several figures.

Petar Popivanov, Angela Slavova
Physics Informed Cellular Neural Networks for Solving Partial Differential Equations

Physics-Informed Neural Networks (PINNs) are a scientific machine learning technique used to solve a broad class of problems. PINNs approximate problems’ solutions by training a neural network to minimize a loss function; it includes terms reflecting the initial and boundary conditions along the space-time domain’s boundary. PINNs are deep learning networks that, given an input point in the integration domain, produce an estimated solution in that point of a differential equation after training. The basic concept behind PINN training is that it can be thought of as an unsupervised strategy that does not require labelled data, such as results from prior simulations or experiments. In this paper we generalize the idea of PINNs for solving partial differential equations by introducing physics informed cellular neural networks (PICNNs). We shall present example of the solutions of reaction-diffusion obtained by PICNNs. The advantages of the proposed new method are in the fastest algorithms and real time solutions.

Angela Slavova, Elena Litsyn
Several Relationships Connected to a Special Function Used in the Simple Equations Method (SEsM)

The use of certain classes of simple equations in the Simple Equations Method (SEsM) favors the occurrence of a specific special function in these solutions. We discuss this special function and its specific cases and derive some relationships which connect the special functions possessing different parameters.

Nikolay K. Vitanov
On the Exact Solutions of a Sequence of Nonlinear Differential Equations Possessing Polynomial Nonlinearities

We apply the Simple Equations Method (SEsM) to a sequence of nonlinear differential equations possessing polynomial nonlinearities and connected to the SEIR model of epidemics spread. Exact solutions are obtained and several of these solutions are discussed from the point of view of application to the model of epidemic waves.

Zlatinka I. Dimitrova
Improved Hardy Inequality with Logarithmic Term

New Hardy type inequality with double singular kernel and with additional logarithmic term in a ball $$B\subset {\text {I}}\!{\text {R}}^n$$ B ⊂ I R n is proved. As an application an estimate from below of the first eigenvalue for Dirichlet problem of p-Laplacian in a bounded domain $$\varOmega \subset {\text {I}}\!{\text {R}}^n$$ Ω ⊂ I R n is obtain.

Nikolai Kutev, Tsviatko Rangelov
Robin Boundary Value Problem for Some Nonlinear Nonlocal Elliptic Partial Differential Equations

This paper deals with Robin type boundary value problems for the mean field equation in the unit circle and for a nonlocal autonomous second order ordinary differential equation with cubic nonlinearity in the unit interval. The radial solutions in the first case and the classical solutions in the second one are written into explicit form. The real- valued spectral parameter can be bounded and unbounded and participates in the formulae for the corresponding solutions.

Petar Popivanov, Angela Slavova
The Ericksen-Leslie System for Data on a Plane

The work deals with the Ericksen-Leslie system for nematic liquid crystals on the whole space with dimension greater or equal than 3. In our work we suppose the initial condition of the orientation field stays on an arc connecting two fixed orthogonal vectors on the unit sphere. Thanks to this geometric assumption, we prove through energy a priori estimates the local existence and the global existence with decay in time for small initial data of a solution in Sobolev spaces asking low regularity for the initial conditions.

Daniele Barbera, Vladimir Georgiev
High Frequency Weighted Resolvent Estimates for the Dirichlet Laplacian in the Exterior Domain

In this paper, we want to present several resolvent estimates for the Dirichlet Laplacian in exterior domain. The estimates evaluate a weighted $$L^2$$ L 2 norm with a weight measured by a negative power of the distance from the boundary. We consider an exterior domain $$\varOmega $$ Ω , that is the complementary of a compact in $$\textbf{R}^n$$ R n , and the inhomogeneous Helmotz equation on it. If the exterior domain is non-trapping, there are cut-off resolvent estimates without weights. Our main result is that we can improve the estimates putting the weights. The main idea is the polar change of coordinates, where $$r=d(x,\partial \varOmega )$$ r = d ( x , ∂ Ω ) , that allows us to use the Hardy inequality close to the boundary of the domain. Kato smoothing estimate is obtained as a consequence of the weighted cut-off resolvent estimates.

Vladimir Georgiev, Mario Rastrelli
Influence of Stimulus on the Motion of Substance in a Channel of Network

We discuss the flow of substance in a channel of network in presence of stimulus for an increase in the amount of flow in certain nodes of the channel. The influence of the stimulus on the distribution of the substance in the network node is studied. The model is used for a study of the changes in the numbers of young and experienced researches in a research organization.

Zlatinka I. Dimitrova, Yoana Chorbadzhiyska-Stamenova
Klein-Gordon Equation with Critical Initial Energy and Nonlinearities with Variable Coefficients

In this paper, we focus on the global behavior of the weak solutions to the Cauchy problem for Klein-Gordon equation with critical initial energy. We consider the polynomial-type nonlinearities with coefficients, depending on the space variables. All coefficients have a constant sign, except one of them, which may change its sign. By means of the sign-preserving properties of the Nehari functional and the concavity method, we prove nonexistence of global solutions or non-blowing up of the weak solutions.

Nikolai Kutev, Milena Dimova, Natalia Kolkovska
Definite/Indefinite Integrals Involving Non-integral Powers for Certain Trigonometric Functions Times Special Functions

Treatment of definite integrals possessing non-integral power are mainly presented. Based on definite integrals with integral power(s), non-integral cases are constructed through power series. Especially cases given by a trigonometric function are decomposed into smaller than or equal to components. Hypergeometric functions, Pochhammer functions, Gamma functions involved are easy to avoid overflow in numerical computation.

Yoshihiro Mochimaru
A Dynamic Green’s Function for the Homogeneous Viscoelastic and Isotropic Half-Space

A dynamic 3D Green’s function for the homogeneous, isotropic and viscoelastic (of the Zener type) half-space is derived in a closed form. The results obtained here can be used as either stand-alone solutions for simple problems or in conjunction with a boundary integral equation formulations to account for complex boundary conditions. In the later case, mesh-reducing boundary element formulations can be constructed as an alternative method for numerical implementation purposes.

Tsviatko V. Rangelov, Petia S. Dineva, George D. Manolis
Area and Perimeter Full Distribution Functions for Planar Poisson Line Processes and Voronoi Diagrams

The challenges of examining random partitions of space are a significant class of problems in the theory of geometric transformations. Richard Miles calculated moments of areas and perimeters of any order (including expectation) of the random division of space in 1972. In the paper we calculate whole distribution function of random divisions of plane by Poisson line process. Our idea is to interpret a random polygon as the evolution of a segment along a moving straight line. In the plane example, the issue connected with an infinite number of parameters is overcome by considering a secant line. We shall take into account the following tasks: 1. On the plane, a random set of straight lines is provided, all shifts are equally likely, and the distribution law is of the form $$F(\varphi ).$$ F ( φ ) . What is the area distribution of the partition’s components? 2. On the plane, a random set of points is marked. Each point A has an associated area of attraction, which is the collection of points in the plane to which the point A is the nearest of the designated ones. In the first problem, the density of moved sections adjacent to the line allows for the expression of the balancing ratio in kinetic form. Similarly, one can write the perimeters’ kinetic equations. We will demonstrate how to reduce these equations to the Riccati equation using the Laplace transformation in this paper. In fact, we formulate the distribution function of area and perimeter and the joint distribution of them with a Poisson line process based on differential equations. Also, for Voronoi diagrams. These are the main search results (see Theorems 1, 2, 3).

Alexei Kanel-Belov, Mehdi Golafshan, Sergey Malev, Roman Yavich
An Application of the Simplest Equations Method to Logarithmic Schrödinger Equation

In this paper we apply the Simple Equations Method (SEsM) to obtain exact solution of equations which are connected to the nonlinear logarithmic Equation of Schrödinger. The used simple equations are more simple than the solved nonlinear partial differential equation but these simple equations in fact can be quite complicated. We consider the specific case of SEsM for obtaining exact solution of one nonlinear partial differential equation. We use specific case of SEsM which is based on the use of 2 simple equations.

Ivan P. Jordanov
Explicit Solutions of the Nonlinear Schrödinger-Type Equation

In this work, we focus on the construction of explicit solutions of the nonlinear Schrödinger-type equation, which has various applications in physics. To obtain different kinds of exact solutions the Jacobi elliptic function method is applied. As a result, abundant new exact solutions are received including Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions, etc. Furthermore, for certain parameter values, the obtained solutions are presented in 2D and 3D plots. The method can be also applied to other nonlinear partial differential equations.

Arailym Syzdykova, Gaziz Kudaibergenov
A Didactic Approach to Study Mass—Inertial Characteristics of Bodies in Plane Motion

The STEM (Science, Technology, Engineering, Math) education employs a hands-on learning approach that integrates various subjects to foster a thorough grasp of concepts and their real-world applications. In the present paper, an experimental set up to study mass-inertial characteristics of different shape cylindrical bodies in plane motion is presented. The aim of the experimental set up is to show students visually the influence of mass distribution of circle shape bodies with different cross sections on their planar motion. Moreover, the meaning of each term of differential equations of motions as well as their initial conditions are explained and shown. We have designed and developed an experimental set up utilizing a 3D printer and plastic materials, as well as time measure sensors and display in order to give some examples. Also, we have designed and printed 2 cylinders with different cross sections with different diameters with the same mass. Moment of inertia of these two bodies are calculated. We plug into the differential equations the calculated mass—characteristics of these three bodies. Also, we discussed the behavior of the system with different initial conditions. The obtained result shows If two cylinders with equal mass but different diameters and shapes of its cross sections are move in the same plane the cylinder with small diameter moves faster.

Vladimir Kotev, Radoslav Rusinov, Miriam Rimeh, Georgi Ivanov, Mirjana Ivanova, Ivan Jordanov
On the Loss of Regularity in a Degenerate Vibrating Beam Equation

We study the well-posedness in Sobolev spaces $$H^s(\textbf{R})$$ H s ( R ) of the Cauchy problem for $$D_t^2u=y(1+D_y^2)^2yu$$ D t 2 u = y ( 1 + D y 2 ) 2 y u , where $$(t,y)\in [0,\infty )\times \textbf{R}$$ ( t , y ) ∈ [ 0 , ∞ ) × R . Our results show that solutions u(t, y) undergo infinite losses of derivatives $$D_y$$ D y on a subset of positive time measure $$S\subset [0,T]$$ S ⊂ [ 0 , T ] for every $$T>\pi .$$ T > π . We also find explicitly a basis of eigenfunctions for the degenerate elliptic operator which do not belong to any Sobolev space with index $$s\ge 5/2.$$ s ≥ 5 / 2 .

Petar Popivanov, Borislav Yordanov

Applications in Fractional Calculus

Frontmatter
Mittag-Leffler Stability for Non-instantaneous Impulsive Generalized Proportional Caputo Fractional Differential Equations

Fractional calculus is a powerful tool in applied mathematics and is used to study problems in mathematical physics, finance, hydrology, biophysics, thermodynamics, control theory, statistical mechanics, astrophysics, cosmology and bioengineering. In studying stability for nonlinear fractional differential equations, there are several approaches in the literature, one of which is the Lyapunov approach. There are however several difficulties encountered when one applies the Lyapunov technique to fractional differential equations and one of the main difficulties is connected with the appropriate definition of derivatives of Lyapunov functions among differential equations of fractional order. In this paper fractional differential equations with non-instantaneous impulses and generalized proportional Caputo fractional derivative are studied. The case of changeable lower limit of the fractional derivative at any and time point of the non-instantaneuos impulse is considered. The Mitatg-Leffler stability with respect to the impulses is defined. Also, its partial case of exponential stability with respect to impulses is given. Sufficient conditions by the help with Lyapunov like functions are obtained. An example is given to illustrate our results.

Snezhana Hristova
Uniqueness Functions to Conformable Differential Inclusions

We study conformable evolution inclusions with the help of uniqueness (Perron) functions. Our conditions are much weaker than commonly used Lipschitz continuity. Existence of solutions, continuous dependence on the initial conditions and relaxation theorem have been proved. Finally we study conformable differential inclusions under one sided Lipshitz condition.

Tzanko Donchev, Jamil Abbas, Iveta Nikolova, Stanislava Stoilova
Numerous Exact Solutions of the Wu-Zhang System with Conformable Time–Fractional Derivatives via Simple Equations Method (SEsM): The Case of Two Simple Equations

In this study we consider the nonlinear Wu-Zhang system with fractional derivative order. The system describes the propagation of dispersive long waves on shallow waters. By an appropriate transformation we reduce the nonlinear fractional partial differential equations to nonlinear ordinary differential equations with integer order. We apply the Simple Equations Method (SEsM) for obtaining exact analytical solutions of the reduced system. We present the general solution of the system equations by composite functions of two simpler functions which are power series of solutions of two simple equations. We choose the simple equations used for this study to be ordinary differential equations of second order. Depending on numerical values of the coefficients of the simple equations, numerous particular solutions of the studied system are obtained. Several numerical examples of the obtained solutions are presented.

Elena V. Nikolova
Impulses in Generalized Proportional Caputo Fractional Differential Equations and Equivalent Integral Presentation

In this paper we present both main approaches in the interpretation of the impulses in generalized proportional Caputo fractional differential equations. We started with both equivalent interpretations in differential equations with integer order derivatives and based on them we presented both main cases: with fixed lower limit of the fractional derivative at the initial time and with a changeable lower limit at any impulsive time. In both cases we give an integral presentation of teh solution. Several examples illustrate the concepts.

Snezhana Hristova, Radoslava Terzieva
Reconstruction of the Time-Dependent Diffusion Coefficient in a Space-Fractional Parabolic Equation

We propose two algorithms for investigation the numerical reconstruction of the time-dependent diffusion coefficient in a space-fractional parabolic problem at integral and point measured outputs. In the first one, by implicit Euler method we perform a linearization of the quadratically nonlinear initial boundary value problem on each time level. Then, we apply a decomposition to this problem solution around the unknown diffusion coefficient. Finally, using the integral or point observations we express in exact form by the solutions of the new subproblems, the required diffusion coefficient. By the second one, on each time level, an iterative algorithm based on the secant method for the numerical identification of the diffusion coefficient, is proposed. We compare the two methods on computational test examples.

Miglena N. Koleva, Lubin G. Vulkov
On the Traveling Wave Solutions of the Fractional Diffusive Predator—Prey System Incorporating an Allee Effect

We consider a system of two fractional-derivative order partial differential equations which describes the dynamics of predator-prey communications where the prey as well as the predator are subjected to an Allee effect. Using fractional traveling wave transformations, the fractional-derivative order partial differential equations are converted to ordinary differential equations with integer order. We apply the Simple Equations Method (SEsM) for obtaining traveling wave solutions of the reduced system. We present the solutions of the system equations (for the prey and for the predator) as finite series of the solutions of two different simple equations. One of obtained solutions is simulated numerically and the visualized traveling waves of the prey and the predator population densities are analyzed.

Elena V. Nikolova
Several Exact Solutions of the Fractional Predator—Prey Model via the Simple Equations Method (SEsM)

We consider a system of two fractional-derivative order partial differential equations which describes the dynamics of predator-prey communications where the prey per capita growth rate is a subject to an Allee effect. Using conformable fractional transformations, the fractional-derivative order partial differential equations are converted to ordinary differential equations with integer order. We apply the Simple Equations Method (SEsM) for obtaining exact analytical solutions of the reduced system. We present the solutions of the system equations as finite series of the solutions of two different simple equations. One of obtained solutions is simulated numerically and is analized in the context of predator—prey dynamics.

Radoslav G. Nikolov, Elena V. Nikolova, Vilislav N. Boutchaktchiev

Applications in Financial Mathematics

Frontmatter
Comparison Between the Chain Ladder Method and the Bornhuetter-Ferguson Method for Third Party Liability Insurance

The Chain-ladder method (CLM) and the Bornhuter-Ferguson method (BF method) are the most widely used methods from the insurance companies for calculating the claims reserves for covering claims that have been incurred but not reported (IBNR). The purpose of this paper is to estimate the required “pending payment” reserves using the both methods and to make comparisons between the results obtained. Some numerical experiments have been presented.

Elitsa Raeva, Velizar Pavlov, Hedie Redzheb
Iterative Calibration of Implied Volatility for European Options: A Computational Approach

Implied volatility is a crucial variable for options trading as it helps determine profitability. It reflects the future movement of the underlying asset’s price and predicts the extent of potential price fluctuation, which can be used to determine if options are likely to be profitable before expiry. The main focus of this research paper is on an iterative numerical method used to calculate the implied volatility of European options. This value is a crucial component in determining the option price. The Black–Scholes model, which assumes that the underlying asset’s volatility is constant and known, does not reflect the reality of the market. Empirical and theoretical studies have shown that the implied volatility of the underlying asset prices follows a persistent smile pattern, indicating a clear relationship between the option strike price and its implied volatility. Moreover, the volatility term structure reflects the relationship between the implied volatility and the time to option expiration, which is not constant. The research aims to provide a better understanding of the complex dynamics of implied volatility and its impact on options trading profitability.

Teodora Klimenko, Velizar Pavlov
Mixed Approach Between Capital Asset Pricing Model and ARIMA Model for Estimating the Standard and Poor’s Stocks

The Capital Asset Pricing Model (CAPM) uses a formula that calculates a stock's expected return by taking into account the risk-free rate of return, the risk premium, and the stock's beta coefficient. The data on which the assessment is made are considered on an annual basis, or for a given fixed period. At the same time, the assessment of the risk-free yield can be underestimated or overestimated for the specific moment of calculations. The Autoregressive Integrated Moving Average (ARIMA) models, in turn, offer a forecast for a particular observation. This article examines an approach to estimate the expected rate of return on a share from Standard and Poor's 500 (S&P500), which combines the assessment by CAPM and ARIMA models. Amazon's expected rate of return is estimated relative to the S&P500 using the CAPM. A forecast was made with an ARIMA model for Amazon's expected change. A forecast of the S&P500's expected rate of return is then made. This prediction is included in Amazon's CAPM as the expected rate of return in the model. In this way, three results are obtained, which were compared for different dates.

Elitsa Raeva, Iliyana Raeva, Yovana Ivanova
Measuring of Inferred Loss Rate with Application to Capital Adequacy

The loss rate of a bank’s portfolio traditionally measures what portion of the exposure is lost in case of default. To overcome the lack of private data and the difficulties involved in its computation, we introduce the notion of implied loss rate (ILR). We prove that ILR is sufficiently close to the actual loss rate in properties that facilitates capital adequacy analysis. To further demonstrate its usefulness, as an example, we estimate ILR for portfolios in the Bulgarian bank system using data on the quality of assets of bank groups reported by the Bulgarian National Bank.

Vilislav Boutchaktchiev
Comparative Analysis on Neural Networks and ARIMA for Forecasting Heterogeneous Portfolio Returns

In the domain of portfolio management theory, the principle of separation postulates that all investors will achieve an identical optimal risk portfolio given the same inputs. However, the realization of true optimality depends on the accuracy of technical analysis, conducted by portfolio managers or investors in predicting the rate of return for the financial assets, encompassed in the portfolio. This article leverages the Nonlinear AutoRegressive with Exogenous inputs Neural Network (NARXNN) for the purpose of predicting the prices of ten financial instruments, facilitating the computation of their respective rates of return. Subsequently, a multi-objective optimization problem is formulated to construct an optimal risk portfolio that concurrently maximizes return while minimizing risk. The resulting portfolio is then compared with a similar portfolio derived from the same dataset, utilizing Autoregressive Integrated Moving Average (ARIMA) models to forecast the rates of return for the assets.

Aleksandra Klimenko, Vesela Mihova, Slavi Georgiev, Ivan Georgiev, Velizar Pavlov
Some Modifications of the Kies Distribution. Applications

The Kies model can be regarded as an efficient model in terms of goodness–of–fit in the field of Software Reliability Analysis. In the books. Pavlov et al., (Some software reliability models: Approximation and modeling aspects. LAP LAMBERT Academic Publishing, Chisinau, 2018) [20], Pavlov et al., (Nontrivial Models in Debugging Theory (Part 2). LAP LAMBERT Academic Publishing, Chisinau, 2018) [21], we explore some models which correspond to debugging theory. In this article we consider the three–parameter Kies model modified as “deterministic” model. We give how a modification of the model with “polynomial variable transfer” can be applied. The usage of such new model for approximation of key data from many scientific fields which concern growth theory. CAS Mathematica gives us possibility for successful implementation of our results.

Nikolay Kyurkchiev, Tsvetelin Zaevski, Anton Iliev, Asen Rahnev
Portfolio Construction Using Neural Networks and Multiobjective Optimization

In recent times, financial markets have been increasingly affected by significant volatility and uncertainty. Given this backdrop, it is beneficial for investors to explore a broader range of asset classes when constructing their financial portfolios. This paper examines the concept of a mixed portfolio from 10 different assets. A technical analysis on the selected data has been conducted using Excel and MATLAB. Subsequent price movements of the chosen instruments were then predicted for the subsequent period employing NARXNN method. This led to the evaluation of the expected rates of return for these financial instruments. The estimations were then blended into an optimal risk portfolio, which maximizes return and minimizes risk, based on the solution to a multi-objective optimization problem. To assess risk, the standard deviations of the rates of return and the correlation matrix between the return rates of the considered financial instruments were utilized.

Tsvetelin Tsonev, Slavi Georgiev, Ivan Georgiev, Vesela Mihova, Velizar Pavlov
Advanced Stochastic Monte Carlo Optimization Methods for Two-Dimensional European Style Options

Multidimensional option pricing poses significant challenges and is a fundamental area in large-scale finance. A European call option grants the holder the right to buy a specific quantity of an underlying asset (S) at a predetermined price (E) and time (T), without the obligation to do so. Monte Carlo and quasi-Monte Carlo methods are powerful tools for solving various financial problems. This paper addresses the challenge of determining the fair value of two and higher dimensional European style options. Monte Carlo methods are particularly effective and useful, especially in higher dimensions, for option pricing problems. This paper proposes simulation optimization methods that employ both low discrepancy sequences and variance reduction techniques to enhance the accuracy of standard approaches for European style options. Improving accuracy is critical for more reliable European option pricing results. Additionally, this approach can be used in situations where other deterministic methods fail, such as high dimensions, complex contract specifications, and other challenging scenarios.

Venelin Todorov, Slavi Georgiev

Applications in Mathematical Biology

Frontmatter
Simple Equations Method (SEsM): Exact Solutions for Description of COVID-19 Epidemic Waves

We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear nonlinear differential equations connected to the SIR model of epidemic spread. We illustrate the methodology by obtaining a solution on the basis of the use of Bernoulli equation as a simple equation.

Nikolay K. Vitanov
Numerical Determination of Age-Dependent Coefficients in an Integro-Hyperbolic System of Honeybee Population Dynamics

We investigate the problem of recovering of a space-dependent reaction coefficient in a integro-hyperbolic system of equations. This is a model of honeybee population dynamics with unknown functions, susceptible population of hive bees $$H_{S}$$ H S and foraging bees $$F_{S}$$ F S and infected population $$H_{I}$$ H I and $$F_{I}$$ F I . In the present study we concentrate on the linearized infection model, proposed in [6]. The recovering model is solved at integral observation that can be considered as generalized point observations. A numerical approach based on the characteristics of the hyperbolic part of the integro-differential operator is developed. Computational examples are discussed.

Slavi Georgiev, Lubin Vulkov
Edge of Chaos in Reaction-Diffusion System with Memristor Synapses

In this paper principles of local activity theory will be presented for studying complex behavior of reaction-diffusion systems. For reaction-diffusion models, one can determine the domain of the cell parameters in order for the cells to be locally active, and thus potentially capable of exhibiting complexity. In the literature, the so called edge of chaos (EC) means a region in the parameter space of a dynamical system, where complex phenomena and information processing can emerge. In this paper edge of chaos domain will be determined for a reaction-diffusion model with memristor synapses. In our model each cell will be arranged on a two-dimensional square grid and will be connected to adjacent cells through coupling devices that mimic 2-D spatial diffusion and transmit the cell’s state to its neighboring cells. Numerical simulations will illustrate the obtained theoretical results.

Angela Slavova, Ventsislav Ignatov
Inverse Modelling of the Cellular Immune Response to SARS-CoV-2

The ongoing struggle with COVID-19 persists as most researchers endeavour to find a successful method to curb global pandemics. Consequently, there has been significant advancement in our comprehension of this disease, and we are now more informed than during the initial phases of this global health crisis. Yet, these endeavours have not fully halted the virus’ spread, leaving numerous queries still unresolved. In this context, grasping the relationship between the SARS-CoV-2 virus and the target cells it impacts is absolutely essential. In this paper, we develop a model to study the complex interaction between the innate and adaptive immune responses to SARS-CoV-2, and the temporal mismatch between them, in particular. We suggest a method to calibrate the important rates which cannot be measured directly. This would help developing a proactive approach to advance the treatment of known and future variants of concern.

Slavi Georgiev

Applications in Numerical Methods and Computer Science

Frontmatter
Reconstruction of Boundary Conditions of a Parabolic-Hyperbolic Transmission Problem

We consider a special kind of interface partial differential equations problems, which solution is defined in a few disjoint distant domains, where the effect of the intermediate region (layer) is modeled by means of nonlocal jump conditions across the layer. Our aim is the numerical identification of external boundary conditions for parabolic-hyperbolic problems on disjoint domains from given point data. We develop decomposition techniques to obtain exact formulas for the unknown boundary conditions. A number of numerical examples are discussed.

Miglena N. Koleva, Lubin G. Vulkov
Novel Monte Carlo Algorithm for Linear Algebraic Systems

A novel Monte Carlo algorithm has been introduced and analyzed for solving linear algebraic equations. This hybrid algorithm is comparable to the ”Walk on Equations”; Monte Carlo approach developed by Ivan Dimov, Sylvain Maire, and Jean Michel Sellier. A comparison with the Gauss-Siedel method has been made for matrices up to a size of 10000. The algorithms’ performance is enhanced by choosing appropriate values for the relaxation parameters, resulting in a significant reduction in computation time and lower relative errors for a given number of iterations. A theorem proving the algorithms convergence has been presented. By balancing the iteration matrix, the original algorithm can be optimized. In addition, a sequential Monte Carlo method developed by John Halton based on an iterative application of the control variate method has been employed. The most notable numerical experiment involves a large system derived from a finite element approximation of a problem that describes a beam structure in constructive mechanics.

Venelin Todorov, Slavi Georgiev, Ivan Dimov
A Note on the Hwang-Kim’s Universal Activation Function

In this talk we give some notes on Universal activation function UA(t) proposed from Seung-Yeon Hwang and Leong-Joon Kim (Computers, Materials and Continua 75:2, 2022). First of all, we will note that this new activation function is, in a certain sense, a ”superposition” of activation functions suggested by Jun (Mathematics 7:3, 2019) and Vasileva and Kyurkchiev (Int. J. Differ. Equ. and Appl. 21:1, 2022). This fact is missing in the work of Hwang and Kim. Our main goal is to investigate the smooth approximation to UA(t) using Gaussian error function. The theoretical results are important in elucidating the intrinsic properties of this new universal activation function. We present simple dynamic programming module for graphical representation of smooth approximation of UA(t) developed by CAS Wolfram Mathematica.

Maria Vasileva, Nikolay Kyurkchiev
Special Lattice and Digital Sequences for Multidimensional Air Pollution Modelling

This paper presents an advanced multidimensional sensitivity analysis using innovative stochastic approaches for air pollution modeling on a large-scale long-range transport model of air pollutants, specifically the Unified Danish Eulerian Model (UNI-DEM). This mathematical model is important for studying the harmful effects of high air pollution levels, and in this paper, we aim to use it to address critical environmental protection questions. We develop advanced Monte Carlo and quasi-Monte Carlo methods using special lattice and digital sequences to improve the computational efficiency of multidimensional numerical integration. We also enhance the existing stochastic approaches for digital ecosystem modeling. The study focuses on analyzing the sensitivity of UNI-DEM model output to variations in input emissions of anthropogenic pollutants and rates of several chemical reactions. The algorithms are applied to compute global Sobol sensitivity measures for several input parameters’ influence on important air pollutant concentrations in various European cities with different geographical locations. The research aims to improve understanding of the factors affecting air pollution and inform effective strategies for mitigating its harmful effects on the environment.

Venelin Todorov, Slavi Georgiev, Ivan Dimov
Some Applications of the Dickson Polynomials of Higher Kind for Modeling Radiation Diagrams

In this paper we consider the Dickson polynomials $$D_{m+1,n}(x,a)$$ D m + 1 , n ( x , a ) of the sixth and seven kind (i.e. $$m=5$$ m = 5 and $$m=6$$ m = 6 , respectively). First, a model with Dickson polynomials as corrections in the Lienard system is presented and “level curves” are studied. In the second place, we will note that some specifics of the amplitudes of these high-degree polynomials open up the possibility of modeling signals from the field of antenna-feeder technology. So, for example, changing the variable t by $$t=b\cos \theta +c$$ t = b cos θ + c ( $$\theta $$ θ is the azimuthal angle and c is the phase difference) in the y(t)-component of the solution of Lienard differential system results in the generation of radiation diagrams. Numerical examples, illustrating our results using CAS MATHEMATICA are given.

Maria Vasileva, Vesselin Kyurkchiev, Anton Iliev, Asen Rahnev, Nikolay Kyurkchiev
Novel Stochastic Method for Multidimensional Fredholm Integral Equations

Integral equations have broad applications in various fields, such as applied mathematics, physics, engineering, geophysics, electricity and magnetism, kinetic theory of gases, quantum mechanics, mathematical economics, and queuing theory. Therefore, it is essential to develop and explore efficient and reliable approaches for solving integral equations. For multidimensional problems, existing biased stochastic algorithms based on a finite number of integrals suffer from the high dimensionality effect since they are based on quadrature points. Thus, advanced unbiased algorithms are required to solve multidimensional problems, which we propose in this paper. We introduce a new unbiased stochastic method for solving multidimensional Fredholm integral equations of the second kind, which we analyze and compare to the old unbiased stochastic algorithm for both one-dimensional and multidimensional problems. This research aims to enhance the understanding of unbiased stochastic algorithms and improve their effectiveness and reliability for solving multidimensional integral equations.

Venelin Todorov, Slavi Georgiev, Yuri Dimitrov
Metadata
Title
New Trends in the Applications of Differential Equations in Sciences
Editor
Angela Slavova
Copyright Year
2024
Electronic ISBN
978-3-031-53212-2
Print ISBN
978-3-031-53211-5
DOI
https://doi.org/10.1007/978-3-031-53212-2

Premium Partner