Computational Methods and Mathematical Modeling in Cyberphysics and Engineering Applications 2
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This book develops a coherent framework for modeling coupled processes in porous media, heat and mass transfer, biodegradation-driven consolidation, and micro-irrigation networks, alongside modern operator-theoretic and variational methods in Banach spaces. It integrates finite element and finite difference techniques, quasiconformal mappings for electrical impedance tomography, fractional and biparabolic evolution equations, and homogenization in composite media. This book addresses high-performance and grid computing, geometric integration in Hamiltonian magnetic levitation systems, and hierarchical predictors for lossless image compression. It also explores AI-oriented transformer architectures for bioinformatics and optimization algorithms for variational inequalities.
By combining rigorous mathematical foundations with scalable computational strategies, this book offers researchers and advanced practitioners a unified perspective on simulation, identification, and optimization in complex engineering and physical systems.
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Content
Chapter 1. Solution of Differential Equations Systems that Arise During the Analysis of Complex Multicomponent Environments 1
V. BOHAIENKO, O. MARCHENKO and T. SAMOILENKO
1.1. Construction of an approximate solution of the axisymmetric parabolic problem 1
1.2. The analysis of numerical modeling of soil mass dynamics in the presence of unsteady pressure filtration 14
1.3. The analysis of numerical simulation of non-isothermal processes in soil 26
1.4. The study of soil massif state in the foundations of hydraulic structures 34
1.5. Parallel algorithm for AMLI preconditioner and its application to model soil massif state 46
1.6. References 68
Chapter 2. Computer Simulation of Transdermal Drug Delivery Using Soluble Microneedles 71
D.A. KLYUSHIN, S. LYASHKO, V.V. ONOTSKYI and O.S. BONDAR
2.1. Introduction 71
2.2. Mathematical model 72
2.3. Mathematical methods 78
2.4. Numerical experiments 80
2.5. Results and discussion 81
2.6. Conclusion 91
2.7. References 91
Chapter 3. Homogenization and Modeling of Processes in Composites Similar to Photonic Crystals 95
G.V. SANDRAKOV
3.1. Introduction 95
3.2. Composite media with periodic structures and contrast properties 98
3.3. Regular homogenized asymptotic expansions of solutions 100
3.4. Singular homogenized asymptotic expansions of solutions 107
3.5. Computational aspects of modeling by homogenization 117
3.6. Spectral aspects of modeling by homogenization 119
3.7. Conclusion 129
3.8. References 129
Chapter 4. Polynomial Operator Interpolation and its Applications 133
V.L. MAKAROV and O.F. KASHPUR
4.1. Introduction 133
4.2. Formulation of Lagrange's operator interpolation problem 135
4.3. Solution of the Lagrange's operator interpolation problem 136
4.4. Solution operator equations by the interpolation method 139
4.5. Interpolation in Euclidean spaces 143
4.6. Construction of surfaces 146
4.7. Conclusion 151
4.8. References 151
Chapter 5. New Fractional Differential Analogues of the Biparabolic Evolution Equation and Some Boundary Value Problems 155
V.M. BULAVATSKY and S. LYASHKO
5.1. Introduction 155
5.2. Some boundary value problems for the fractional-differential analogue of the biparabolic equation with non-locality in time and space 159
5.3. Generalization of the model equation based on Hilfer-type fractional derivatives 169
5.4. Fractional-differential analogue of the biparabolic evolution equation with Caputo and Caputo-Fabrizio derivatives 173
5.5. Conclusions 178
5.6. References 178
Chapter 6. Optimal Control for Integro-differential Systems of Hyperbolic Type 181
A.V. ANIKUSHYN, Kh.M. HRANISHAK, V.S. LYASHKO and O.S. BONDAR
6.1. Introduction 181
6.2. Main notations and spaces 189
6.3. Generalized control problem 191
6.4. A priori inequalities for the differential part of the operator 197
6.5. A priori inequalities for the integro-differential operator 205
6.6. Example of an optimal control problem 212
6.7. Conclusion 219
6.8. References 219
Chapter 7. Self-adaptive Operator Extrapolation Method for Operator Inclusions in Banach Space 223
V. SEMENOV and S. DENYSOV
7.1. Introduction 223
7.2. Preliminaries 227
7.3. Algorithm 231
7.4. Convergence 233
7.5. Variants 239
7.6. Application to variational inequalities 241
7.7. Conclusion 243
7.8. Acknowledgments 243
7.9. References 243
Chapter 8. Forecasting Algorithms Based on Intellectual Analysis of Polynomial Extrapolation and Divided Differences 247
Y. TURBAL, M. TURBAL and A. BOMBA
8.1. Introduction - Problem of the time series forecasting 247
8.2. Method of finding the predictive value based on a polynomial of any degree without finding the coefficients of the polynomial 249
8.3. The optimal polynomial extrapolation problem 254
8.4. Condition of forecast efficiency based on the arithmetic mean of polynomial forecasts 260
8.5. Improved algorithm for optimal polynomial forecasting 262
8.6. Numerical results of the polynomial forecast 266
8.7. Pyramidal method of extrapolation 271
8.8. Numerical results for the pyramidal methods 283
8.9. Conclusions 285
8.10. References 286
Chapter 9. Transformer with BPE Tokenization for Analysis of Interactions of Chemical Substances and Proteins 289
M. ZOZIUK, P. KRYSENKO, S. DOVGIY, V. MAKAROV, Y. YAKIMENKO and D. KOROLIOUK
9.1. Introduction 290
9.2. Methods and data 291
9.3. The model's architecture 293
9.4. The model's training 295
9.5. Conclusion 298
9.6. References 298
List of Authors 301
Index 305
1
Solution of Differential Equations Systems that Arise During the Analysis of Complex Multicomponent Environments
V. BOHAIENKO1, O. MARCHENKO1 and T. SAMOILENKO2
1V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, Kyiv, Ukraine
2National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine
1.1. Construction of an approximate solution of the axisymmetric parabolic problem
In this section, we propose an algorithm for calculating the dynamics of the non-isothermal process of moisture transfer in an axisymmetric setting, which is essential when studying the state of soil moisture around, for example, vertical drains, wells, piles, etc. We formulate the initial-boundary value problem for a system of non-stationary moisture and heat transfer equations for an isotropic environment in a cylindrical coordinate system with heterogeneous mixed boundary conditions, including the conditions of heat exchange with the external environment (Marchenko and Samoilenko 2020).
The obtained results are also important for further research in cylindrical coordinates of problems simulating the migration of moisture in the process of seasonal freezing of the soil, as a continuation of the approach presented in Pavlov and Permyakov (1983), Marchenko and Lezhnina (2002) and Marchenko and Samoilenko (2012), in which phase transitions from unfrozen water to ice are taken into account in the entire volume of soil massif without highlighting the crystallization front. In this case, moisture exchange and heat exchange characteristics are functions of the total moisture content, and the moisture transfer equation is written relative to the "fictitious" moisture content. Taking into account the directions of moisture migration relative to the freezing/thawing front, we consider the convective heat transfer along the vertical axis to be essential, which allows us to achieve sufficient agreement with experimental data (Kislitsyn et al. 2018).
The system of equations has the following form:
[1.1]where , , 0 < r0 = r = r1 < 8, 0 = z = z1 < 8, W(r, z, t) is the volumetric moisture content, T(r, z, t) is the temperature, k(W, T) is the hydraulic conductivity, is the thermal conductivity coefficient, cT, cw are the volume heat capacities of soil and liquid in pores and u(W) is the moisture transfer-filtration rate along the z axis.
Let us set the boundary conditions in the form
[1.2] [1.3] [1.4] [1.5]Ta is the ambient temperature.
The initial conditions are as follows:
[1.6]We assume that a = const > 0 and the given functions k(W, T), u(W), W0(r, z), T0(r, z) are sufficiently smooth on , are bounded and satisfy the Lipschitz condition, i.e.:
[1.7]The classical solution of the initial-boundary value problem [1.1]-[1.6] is a vector function , whose components together with their partial derivatives , , i = 1, 2, are continuous in the domain , have bounded continuous partial derivatives , , , i = 1, 2 in , satisfy the equation [1.1] and the conditions [1.2]-[1.6].
We denote by Z the set of vector functions , whose components belong to the Sobolev space and , hi(r, z, 0), i = 1, 2 belong to L2(O). Accordingly, we denote by Z0 the set of vector functions , whose components belong to the space .
The generalized Galerkin solution of the initial-boundary value problem for the system [1.1]-[1.6] is the vector function h(r, z, t) ? Z, which, for an arbitrary vector function v(r, z) ? Z0, satisfies the integral relations
[1.8] [1.9]where
[1.10]( , ) is the scalar product in L2(O), h0 = (W0, T0)T.
We will search the approximate generalized solution of this Cauchy problem in a finite-dimensional subspace ZN ? Z using the finite element method (FEM) in the following form:
[1.11]where ai(t), are the functions integrable together with their second derivatives on
[1.12]is the basis of the space obtained from ZN by fixing ; , is the set of linearly independent functions corresponding to the nodal points of the FEM, which are built on complete polynomials of degree k(k = 1, 2, 3) and have a limited support in .
The basis of the space similarly consists of 2N vector functions Fi(r, z) so that an arbitrary function can be presented in the form:
[1.13]where ßi are constants.
The approximate generalized solution hN(r, z, t) ? ZN of the problem [1.8], [1.9] satisfies the integral relations
[1.14] [1.15]Let us use the following notations:
[1.16]The following theorem is true.
THEOREM 1.1.-
Let the classical solution of the initial-boundary value problem [1.1]-[1.6] has bounded continuous partial derivatives on .
Then, for the approximate generalized solution , of the problem [1.8], [1.9], there exists a constant C > 0 such that the following estimate holds:
where is the maximum length of the sides of the triangles, k =1, 2, 3 is the degree polynomials of FEM.
PROOF.- Let h and hN be the classical and the approximate generalized solution of the problem [1.1]-[1.6]; then,
[1.17]Using the relation [1.17], consider the expression (Deineka et al. 1995)
[1.18]Let us estimate the left and right parts of the relation [1.18] taking into account [1.10], [1.7] as
[1.19]where ei, are some positive constants of e-inequalities,
[1.20]From [1.18], [1.19], we obtain the inequality in the form
[1.21]where constants , , are positive due to appropriately selected ei, .
Multiplying both parts of the inequality [1.21] by e-ct and integrating the result over the variable from 0 to s, , we obtain the inequality
[1.22]Let us obtain an upper estimate to the last term of the relation [1.22]. Integrating by parts and taking into account the Cauchy-Buniakovsky inequality, e-inequality and notation [1.10], we have
[1.23]where .
Taking into account the following estimate of the norm on the boundary ?O of the given domain:
Friedrichs' inequality (Rectoris 1977), written in the general case, and e-inequality, we have
In the last inequality, we choose e such that 1 - C3C1 e > 0. So, for the considered domain O ? C4 = const > 0 such that
In view of the last estimate and the inequality [1.23], we rewrite the estimate [1.22] as follows:
[1.24]From the equalities [1.9], [1.15], we have
Taking into account the last relation, from the inequality [1.24] and the notations [1.16], we obtain
The statement of the theorem follows from the last inequality and well-known estimates of interpolation by FEM polynomials (Deineka et al. 1995).
The problem [1.14], [1.15] taking into account [1.11]-[1.13] can be rewritten in the matrix form
[1.25] [1.26]where
elements of matrices , and vectors , , s = 1,2, are calculated according to the formulas
To estimate the time-discrete approximate generalized solution, we use the Crank-Nicolson scheme (Deineka et al. 1995).
Let for some integer J = 1. We are going to find a sequence , Hj(r, z) = H(r, z, jt) such that Hj approximates hN ? ZN optimally in . Let us denote
The Crank-Nicolson scheme for the Cauchy problem [1.25],...
