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This Special Issue focuses mainly on techniques and the relative formalism typical of numerical methods and therefore of numerical analysis, more generally. These fields of study of mathematics represent an important field of investigation both in the field of applied mathematics and even more exquisitely in the pure research of the theory of approximation and the study of polynomial relations as well as in the analysis of the solutions of the differential equations both ordinary and partial derivatives. Therefore, a substantial part of research on the topic of numerical analysis cannot exclude the fundamental role played by approximation theory and some of the tools used to develop this research. In this Special Issue, we want to draw attention to the mathematical methods used in numerical analysis, such as special functions, orthogonal polynomials, and their theoretical tools, such as Lie algebra, to study the concepts and properties of some special and advanced methods, which are useful in the description of solutions of linear and nonlinear differential equations. A further field of investigation is dedicated to the theory and related properties of fractional calculus with its adequate application to numerical methods.
risk assessment  numerical analysis  ignition hazard  effective field strength  offshore plant  Hamiltonian system  complex Lagrangian  Noether symmetries  first integrals  symplectic Runge–Kutta methods  effective order  partitioned rungekutta methods  symplecticity  hamiltonian systems  RungeKutta type methods  fourthorder ODEs  order conditions  Bseries  quadcolored trees  khypergeometric differential equations  nonhomogeneous  khypergeometric series  special function  general solution  Frobenius method  Chebyshev polynomials  pseudoChebyshev polynomials  recurrence relations  differential equations  composition properties  orthogonality properties  numerical analysis  heat generation  chemical reaction  thin needle  nanofluid  fourthorder  nonoscillatory solutions  oscillatory solutions  delay differential equations  particle accelerator  coupling impedance  dual integral equations  ClenshawCurtis quadrature  steepest descent method  logarithmic singularities  Cauchy singularity  highly oscillatory integrals  secondorder  nonoscillatory solutions  oscillatory solutions  delay differential equations  Fredholm integral equations  multiresolution analysis  unitary extension principle  oblique extension principle  Bsplines  wavelets  tight framelets  Swift–Hohenberg type of equation  surfaces  narrow band domain  closest point method  operator splitting method
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The use of scientific computing tools is currently customary for solving problems at several complexity levels in Applied Sciences. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and better performing numerical methods that are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application.
time fractional differential equations  mixedindex problems  analytical solution  asymptotic stability  conservative problems  Hamiltonian problems  energyconserving methods  Poisson problems  Hamiltonian Boundary Value Methods  HBVMs  line integral methods  constrained Hamiltonian problems  Hamiltonian PDEs  highly oscillatory problems  boundary element method  finite difference method  floating strike Asian options  continuous geometric average  barrier options  isogeometric analysis  adaptive methods  hierarchical splines  THBsplines  local refinement  linear systems  preconditioners  Cholesky factorization  limited memory  Volterra integral equations  Volterra integro–differential equations  collocation methods  multistep methods  convergence  Bspline  optimal basis  fractional derivative  Galerkin method  collocation method  spectral (eigenvalue) and singular value distributions  generalized locally Toeplitz sequences  discretization of systems of differential equations  higherorder finite element methods  discontinuous Galerkin methods  finite difference methods  isogeometric analysis  Bsplines  curl–curl operator  time harmonic Maxwell’s equations and magnetostatic problems  low rank completion  matrix ODEs  gradient system  ordinary differential equations  Runge–Kutta  tree  stump  order  elementary differential  edgehistogram  edgepreserving smoothing  histogram specification  initial value problems  onestep methods  Hermite–Obreshkov methods  symplecticity  Bsplines  BS methods  hyperbolic partial differential equations  high order discontinuous Galerkin finite element schemes  shock waves and discontinuities  vectorization and parallelization  high performance computing  generalized Schur algorithm  nullspace  displacement rank  structured matrices  stochastic differential equations  stochastic multistep methods  stochastic Volterra integral equations  meansquare stability  asymptotic stability  numerical analysis  numerical methods  scientific computing  initial value problems  onestep methods  Hermite–Obreshkov methods  symplecticity  Bsplines  BS methods
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The many technical and computational problems that appear to be constantly emerging in various branches of physics and engineering beg for a more detailed understanding of the fundamental mathematics that serves as the cornerstone of our way of understanding natural phenomena. The purpose of this Special Issue was to establish a brief collection of carefully selected articles authored by promising young scientists and the world's leading experts in pure and applied mathematics, highlighting the stateoftheart of the various research lines focusing on the study of analytical and numerical mathematical methods for pure and applied sciences.
ultraparabolic equation  ultradiffusion process  probabilistic representation  mathematical finance  linear elastostatics  layer potentials  fredholmian operators  fractional differential equations  fractional derivative  Abeltype integral  time delay  distributed lag  gamma distribution  macroeconomics  Keynesian model  integral transforms  Laplace integral transform  transmutation operator  generating operator  integral equations  differential equations  operational calculus of Mikusinski type  Mellin integral transform  fractional derivative  fractional integral  Mittag–Leffler function  Riemann–Liouville derivative  Caputo derivative  Grünwald–Letnikov derivative  spacetime fractional diffusion equation  fractional Laplacian  subordination principle  MittagLeffler function  Bessel function  exterior calculus  exterior algebra  electromagnetism  Maxwell equations  differential forms  tensor calculus  Fourier Theory  DFT in polar coordinates  polar coordinates  multidimensional DFT  discrete Hankel Transform  discrete Fourier Transform  Orthogonality  multispecies biofilm  biosorption  free boundary value problem  heavy metals toxicity  method of characteristics  relativistic diffusion equation  Caputo fractional derivatives of a function with respect to another function  BesselRiesz motion  Mittag–Leffler function  matrix function  Schur decomposition  Laplace transform  fractional calculus  central limit theorem  anomalous diffusion  stable distribution  fractional calculus  power law  n/a
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Researches and investigations involving the theory and applications of integral transforms and operational calculus are remarkably widespread in many diverse areas of the mathematical, physical, chemical, engineering and statistical sciences.
highly oscillatory  convolution quadrature rule  volterra integral equation  Bessel kernel  convergence  higher order Schwarzian derivatives  Janowski starlike function  Janowski convex function  bound on derivatives  tangent numbers  tangent polynomials  Carlitztype qtangent numbers  Carlitztype qtangent polynomials  (p,q)analogue of tangent numbers and polynomials  (p,q)analogue of tangent zeta function  symmetric identities  zeros  Lommel functions  univalent functions  starlike functions  convex functions  inclusion relationships  analytic function  Hankel determinant  exponential function  upper bound  nonlinear boundary value problems  fractionalorder differential equations  RiemannStieltjes functional integral  LiouvilleCaputo fractional derivative  infinitepoint boundary conditions  advanced and deviated arguments  existence of at least one solution  Fredholm integral equation  Schauder fixed point theorem  Hölder condition  generalized Kuramoto–Sivashinsky equation  modified Kudryashov method  exact solutions  Maple graphs  analytic function  Hadamard product (convolution)  partial sum  Srivastava–Tomovski generalization of Mittag–Leffler function  subordination  differential equation  differential inclusion  Liouville–Caputotype fractional derivative  fractional integral  existence  fixed point  Bernoulli spiral  Grandi curves  Chebyshev polynomials  pseudoChebyshev polynomials  orthogonality property  symmetric  encryption  password  hash  cryptography  PBKDF  q–Bleimann–Butzer–Hahn operators  (p,q)integers  (p,q)Bernstein operators  (p,q)Bleimann–Butzer–Hahn operators  modulus of continuity  rate of approximation  Kfunctional  HurwitzLerch zeta function  generalized functions  analytic number theory  ?generalized HurwitzLerch zeta functions  derivative properties  series representation  basic hypergeometric functions  generating functions  qpolynomials  analytic functions  Mittag–Leffler functions  starlike functions  convex functions  Hardy space  vibrating string equation  initial conditions  spectral decomposition  regular solution  the uniqueness of the solution  the existence of a solution  analytic  ?convex function  starlike function  stronglystarlike function  subordination  q Sheffer–Appell polynomials  generating relations  determinant definition  recurrence relation  q Hermite–Bernoulli polynomials  q Hermite–Euler polynomials  q Hermite–Genocchi polynomials  Volterra integral equations  highly oscillatory Bessel kernel  Hermite interpolation  direct Hermite collocation method  piecewise Hermite collocation method  differential operator  qhypergeometric functions  meromorphic function  Mittag–Leffler function  Hadamard product  differential subordination  starlike functions  Bell numbers  radius estimate  (p, q)integers  Dunkl analogue  generating functions  generalization of exponential function  Szász operator  modulus of continuity  function spaces and their duals  distributions  tempered distributions  Schwartz testing function space  generalized functions  distribution space  wavelet transform of generalized functions  Fourier transform  analytic function  subordination  Dziok–Srivastava operator  nonlinear boundary value problem  nonlocal  multipoint  multistrip  existence  Ulam stability  functions of bounded boundary and bounded radius rotations  subordination  functions with positive real part  uniformly starlike and convex functions  analytic functions  univalent functions  starlike and qstarlike functions  qderivative (or qdifference) operator  sufficient conditions  distortion theorems  Janowski functions  analytic number theory  ?generalized Hurwitz–Lerch zeta functions  derivative properties  recurrence relations  integral representations  Mellin transform  natural transform  Adomian decomposition method  Caputo fractional derivative  generalized mittagleffler function  analytic functions  Hadamard product  starlike functions  qderivative (or qdifference) operator  Hankel determinant  qstarlike functions  fuzzy volterra integrodifferential equations  fuzzy general linear method  fuzzy differential equations  generalized Hukuhara differentiability  spectrum symmetry  DCT  MFCC  audio features  anuran calls  analytic functions  convex functions  starlike functions  strongly convex functions  strongly starlike functions  uniformly convex functions  Struve functions  truncatedexponential polynomials  monomiality principle  generating functions  Apostoltype polynomials and Apostoltype numbers  Bernoulli, Euler and Genocchi polynomials  Bernoulli, Euler, and Genocchi numbers  operational methods  summation formulas  symmetric identities  Euler numbers and polynomials  qEuler numbers and polynomials  HurwitzEuler eta function  multiple HurwitzEuler eta function  higher order qEuler numbers and polynomials  (p, q)Euler numbers and polynomials of higher order  symmetric identities  symmetry of the zero
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