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This monograph is a valuable contribution to the highly topical and extremely productive field of regularization methods for inverse and ill-posed problems. The author is an internationally outstanding and accepted mathematician in this field. In his book he offers a well-balanced mixture of basic and innovative aspects. He demonstrates new, differentiated viewpoints, and important examples for applications. The book demonstrates the current developments in the field of regularization theory, such as multi parameter regularization and regularization in learning theory. The book is written for graduate and PhDs

Numerical analysis --- Numerical differentiation. --- Graphic differentiation --- Functions --- Improperly posed problems in numerical analysis --- Improperly posed problems. --- Ill-posed problems --- Balancing Principle. --- Blood Glucose Prediction. --- Convergence Rate. --- Discrepancy Principle. --- Error Bound Estimation. --- Ill-posed Problem. --- Learning Theory, Meta-learning. --- Multi-parameter Regularization. --- Regularization Method.

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Mathematical models of various natural processes are described by differential equations, systems of partial differential equations and integral equations. In most cases, the exact solution to such problems cannot be determined; therefore, one has to use grid methods to calculate an approximate solution using high-performance computing systems. These methods include the finite element method, the finite difference method, the finite volume method and combined methods. In this Special Issue, we bring to your attention works on theoretical studies of grid methods for approximation, stability and convergence, as well as the results of numerical experiments confirming the effectiveness of the developed methods. Of particular interest are new methods for solving boundary value problems with singularities, the complex geometry of the domain boundary and nonlinear equations. A part of the articles is devoted to the analysis of numerical methods developed for calculating mathematical models in various fields of applied science and engineering applications. As a rule, the ideas of symmetry are present in the design schemes and make the process harmonious and efficient.

Information technology industries --- high-order methods --- Brinkman penalization --- discontinuous Galerkin methods --- embedded geometry --- high-order boundary --- IMEX Runge–Kutta methods --- boundary value problems with degeneration of the solution on entire boundary of the domain --- the method of finite elements --- special graded mesh --- multigrid methods --- Hermitian/skew-Hermitian splitting method --- skew-Hermitian triangular splitting method --- strongly non-Hermitian matrix --- lie symmetries --- invariantized difference scheme --- numerical solutions --- finite integration method --- shifted Chebyshev polynomial --- direct and inverse problems --- Volterra integro-differential equation --- Tikhonov regularization method --- quartic spline --- triangulation --- scattered data --- continuity --- surface reconstruction --- positivity-preserving --- interpolation --- jaw crusher --- symmetrical laser cladding path --- FEPG --- wear --- high-order methods --- Brinkman penalization --- discontinuous Galerkin methods --- embedded geometry --- high-order boundary --- IMEX Runge–Kutta methods --- boundary value problems with degeneration of the solution on entire boundary of the domain --- the method of finite elements --- special graded mesh --- multigrid methods --- Hermitian/skew-Hermitian splitting method --- skew-Hermitian triangular splitting method --- strongly non-Hermitian matrix --- lie symmetries --- invariantized difference scheme --- numerical solutions --- finite integration method --- shifted Chebyshev polynomial --- direct and inverse problems --- Volterra integro-differential equation --- Tikhonov regularization method --- quartic spline --- triangulation --- scattered data --- continuity --- surface reconstruction --- positivity-preserving --- interpolation --- jaw crusher --- symmetrical laser cladding path --- FEPG --- wear

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It is very well known that differential equations are related with the rise of physical science in the last several decades and they are used successfully for models of real-world problems in a variety of fields from several disciplines. Additionally, difference equations represent the discrete analogues of differential equations. These types of equations started to be used intensively during the last several years for their multiple applications, particularly in complex chaotic behavior. A certain class of differential and related difference equations is represented by their respective fractional forms, which have been utilized to better describe non-local phenomena appearing in all branches of science and engineering. The purpose of this book is to present some common results given by mathematicians together with physicists, engineers, as well as other scientists, for whom differential and difference equations are valuable research tools. The reported results can be used by researchers and academics working in both pure and applied differential equations.

dynamic equations --- time scales --- classification --- existence --- necessary and sufficient conditions --- fractional calculus --- triangular fuzzy number --- double-parametric form --- FRDTM --- fractional dynamical model of marriage --- approximate controllability --- degenerate evolution equation --- fractional Caputo derivative --- sectorial operator --- fractional symmetric Hahn integral --- fractional symmetric Hahn difference operator --- Arrhenius activation energy --- rotating disk --- Darcy–Forchheimer flow --- binary chemical reaction --- nanoparticles --- numerical solution --- fractional differential equations --- two-dimensional wavelets --- finite differences --- fractional diffusion-wave equation --- fractional derivative --- ill-posed problem --- Tikhonov regularization method --- non-linear differential equation --- cubic B-spline --- central finite difference approximations --- absolute errors --- second order differential equations --- mild solution --- non-instantaneous impulses --- Kuratowski measure of noncompactness --- Darbo fixed point --- multi-stage method --- multi-step method --- Runge–Kutta method --- backward difference formula --- stiff system --- numerical solutions --- Riemann-Liouville fractional integral --- Caputo fractional derivative --- fractional Taylor vector --- kerosene oil-based fluid --- stagnation point --- carbon nanotubes --- variable thicker surface --- thermal radiation --- differential equations --- symmetric identities --- degenerate Hermite polynomials --- complex zeros --- oscillation --- third order --- mixed neutral differential equations --- powers of stochastic Gompertz diffusion models --- powers of stochastic lognormal diffusion models --- estimation in diffusion process --- stationary distribution and ergodicity --- trend function --- application to simulated data --- n-th order linear differential equation --- two-point boundary value problem --- Green function --- linear differential equation --- exponential stability --- linear output feedback --- stabilization --- uncertain system --- nonlocal effects --- linear control system --- Hilbert space --- state feedback control --- exact controllability --- upper Bohl exponent

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This Special Issue deals with the theory and applications of differential and difference equations, and includes papers for different branches of differential equations, such as - Boundary Value Problems for Fractional Differential Equations and Inclusions - Spectral Theory for Fractional Differential Equations - Generalized Abel's Integral Equations - Oscillation Results for Higher Order Differential Equations - Stability of Equilibria under Stochastic Perturbations - Harmonic Functions - Coincidence Continuation Theory for Multivalued Maps - Generalized Briot–Bouquet Differential Equation - Nonlocal Inverse Problem - Lyapunov Type Theorems for Exponential Stability - Fuzzy Functions on Time Scales - Modified Helmholtz Equation on a Regular Hexagon

Research & information: general --- Mathematics & science --- generating functions --- functional equations --- partial differential equations --- special numbers and polynomials --- Bernoulli numbers --- Euler numbers --- Stirling numbers --- Bell polynomials --- Cauchy numbers --- Poisson-Charlier polynomials --- Bernstein basis functions --- Daehee numbers and polynomials --- combinatorial sums --- binomial coefficients --- p-adic integral --- probability distribution --- Mittag-Leffler function --- spectrum --- eigenvalue --- fractional derivative --- q-Homotopy analysis transform method --- Natural decomposition method --- Whitham–Broer–Kaup equations --- Caputo derivative --- liner recursions --- convolution formulas --- Gegenbauer polynomials --- Humbert polynomials --- classical polynomials in several variables --- classical number sequences --- Riemann–Liouville fractional integral --- Mittag–Leffler function --- Babenko’s approach --- generalized Abel’s integral equation --- harmonic functions --- janowski functions --- starlike functions --- extreme points --- subordination --- ocillation --- higher-order --- differential equations --- p-Laplacian equations --- rumor spreading model --- white noise --- stochastic differential equations --- asymptotic mean square stability --- stability in probability --- linear matrix inequality --- Co-infection of HIV-TB --- equilibrium point --- reproduction number --- stability analysis --- backward bifurcation --- harmonic univalent functions --- generalized linear operator --- differential operator --- Salagean operator --- coefficient bounds --- essential maps --- coincidence points --- topological principles --- selections --- univalent function --- analytic function --- unit disk --- integro-differential equation --- mixed type equation --- spectral parameters --- integral conditions --- solvability --- exponential stability --- linear skew-product semiflows --- Lyapunov functions --- fractional differential equations --- fractional differential inclusions --- existence --- fixed point theorems --- fuzzy functions time scales --- Hukuhara difference --- generalized nabla Hukuhara derivative --- fuzzy nabla integral --- caputo fractional derivative --- multi-term fractional differential equations --- fixed point --- difference equations --- periodicity character --- nonexistence cases of periodic solutions --- hypersingular integral equations --- iterative projection method --- Lyapunov stability theory --- MADE --- eigenfunction --- convergence --- Fourier transform --- singular Cauchy problem --- asymptotic series --- regularization method --- turning point --- unified transform --- modified Helmholtz equation --- global relation --- triple q-hypergeometric function --- convergence region --- Ward q-addition --- q-integral representation --- generating functions --- functional equations --- partial differential equations --- special numbers and polynomials --- Bernoulli numbers --- Euler numbers --- Stirling numbers --- Bell polynomials --- Cauchy numbers --- Poisson-Charlier polynomials --- Bernstein basis functions --- Daehee numbers and polynomials --- combinatorial sums --- binomial coefficients --- p-adic integral --- probability distribution --- Mittag-Leffler function --- spectrum --- eigenvalue --- fractional derivative --- q-Homotopy analysis transform method --- Natural decomposition method --- Whitham–Broer–Kaup equations --- Caputo derivative --- liner recursions --- convolution formulas --- Gegenbauer polynomials --- Humbert polynomials --- classical polynomials in several variables --- classical number sequences --- Riemann–Liouville fractional integral --- Mittag–Leffler function --- Babenko’s approach --- generalized Abel’s integral equation --- harmonic functions --- janowski functions --- starlike functions --- extreme points --- subordination --- ocillation --- higher-order --- differential equations --- p-Laplacian equations --- rumor spreading model --- white noise --- stochastic differential equations --- asymptotic mean square stability --- stability in probability --- linear matrix inequality --- Co-infection of HIV-TB --- equilibrium point --- reproduction number --- stability analysis --- backward bifurcation --- harmonic univalent functions --- generalized linear operator --- differential operator --- Salagean operator --- coefficient bounds --- essential maps --- coincidence points --- topological principles --- selections --- univalent function --- analytic function --- unit disk --- integro-differential equation --- mixed type equation --- spectral parameters --- integral conditions --- solvability --- exponential stability --- linear skew-product semiflows --- Lyapunov functions --- fractional differential equations --- fractional differential inclusions --- existence --- fixed point theorems --- fuzzy functions time scales --- Hukuhara difference --- generalized nabla Hukuhara derivative --- fuzzy nabla integral --- caputo fractional derivative --- multi-term fractional differential equations --- fixed point --- difference equations --- periodicity character --- nonexistence cases of periodic solutions --- hypersingular integral equations --- iterative projection method --- Lyapunov stability theory --- MADE --- eigenfunction --- convergence --- Fourier transform --- singular Cauchy problem --- asymptotic series --- regularization method --- turning point --- unified transform --- modified Helmholtz equation --- global relation --- triple q-hypergeometric function --- convergence region --- Ward q-addition --- q-integral representation

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This Special Issue deals with the theory and applications of differential and difference equations, and includes papers for different branches of differential equations, such as - Boundary Value Problems for Fractional Differential Equations and Inclusions - Spectral Theory for Fractional Differential Equations - Generalized Abel's Integral Equations - Oscillation Results for Higher Order Differential Equations - Stability of Equilibria under Stochastic Perturbations - Harmonic Functions - Coincidence Continuation Theory for Multivalued Maps - Generalized Briot–Bouquet Differential Equation - Nonlocal Inverse Problem - Lyapunov Type Theorems for Exponential Stability - Fuzzy Functions on Time Scales - Modified Helmholtz Equation on a Regular Hexagon

generating functions --- functional equations --- partial differential equations --- special numbers and polynomials --- Bernoulli numbers --- Euler numbers --- Stirling numbers --- Bell polynomials --- Cauchy numbers --- Poisson-Charlier polynomials --- Bernstein basis functions --- Daehee numbers and polynomials --- combinatorial sums --- binomial coefficients --- p-adic integral --- probability distribution --- Mittag-Leffler function --- spectrum --- eigenvalue --- fractional derivative --- q-Homotopy analysis transform method --- Natural decomposition method --- Whitham–Broer–Kaup equations --- Caputo derivative --- liner recursions --- convolution formulas --- Gegenbauer polynomials --- Humbert polynomials --- classical polynomials in several variables --- classical number sequences --- Riemann–Liouville fractional integral --- Mittag–Leffler function --- Babenko’s approach --- generalized Abel’s integral equation --- harmonic functions --- janowski functions --- starlike functions --- extreme points --- subordination --- ocillation --- higher-order --- differential equations --- p-Laplacian equations --- rumor spreading model --- white noise --- stochastic differential equations --- asymptotic mean square stability --- stability in probability --- linear matrix inequality --- Co-infection of HIV-TB --- equilibrium point --- reproduction number --- stability analysis --- backward bifurcation --- harmonic univalent functions --- generalized linear operator --- differential operator --- Salagean operator --- coefficient bounds --- essential maps --- coincidence points --- topological principles --- selections --- univalent function --- analytic function --- unit disk --- integro-differential equation --- mixed type equation --- spectral parameters --- integral conditions --- solvability --- exponential stability --- linear skew-product semiflows --- Lyapunov functions --- fractional differential equations --- fractional differential inclusions --- existence --- fixed point theorems --- fuzzy functions time scales --- Hukuhara difference --- generalized nabla Hukuhara derivative --- fuzzy nabla integral --- caputo fractional derivative --- multi-term fractional differential equations --- fixed point --- difference equations --- periodicity character --- nonexistence cases of periodic solutions --- hypersingular integral equations --- iterative projection method --- Lyapunov stability theory --- MADE --- eigenfunction --- convergence --- Fourier transform --- singular Cauchy problem --- asymptotic series --- regularization method --- turning point --- unified transform --- modified Helmholtz equation --- global relation --- triple q-hypergeometric function --- convergence region --- Ward q-addition --- q-integral representation