Article
Mathematics
Pedro Gonzalez-Rodelas, Miguel Pasadas, Abdelouahed Kouibia, Basim Mustafa
Summary: In this paper, we propose an approximation method for solving second kind Volterra integral equation systems using radial basis functions. The method minimizes a suitable functional in a discrete space generated by compactly supported radial basis functions of Wendland type. Two convergence results are proven, which is a highlight as most recent published papers in the literature do not include any. Numerical examples are presented to demonstrate and justify the validity of the proposed method. The proposed technique achieves acceptable accuracy with minimal data usage and low computational cost.
Article
Mathematics, Applied
M. Ahmadinia, H. Afshariarjmand, M. Salehi
Summary: This paper proposes a numerical method based on the least squares method and the second kind Chebyshev wavelets to solve the multi-dimensional Ito Volterra integral equations. The equations are converted to a non-square linear system of equations using Clenshaw-Curtis quadrature rules, and the least squares solution is obtained through QR factorization method. Parallel computing process is used to reduce computation time for the computationally intensive equations. The convergence rate of the method is proven, and numerical results demonstrate its reliability and efficiency.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
D. Rostamy, F. Mirzaei
Summary: This paper proposes a method to solve parabolic Volterra integro-differential equations with bounded and unbounded domains by transforming them into well-posed linear and nonlinear dynamical systems and solving them using a new class of algorithms. The error bounds in both bounded and unbounded domains demonstrate the efficiency and accuracy of the proposed methods, and stability and convergence are also established. Additionally, numerical simulations are conducted to test the performance of the proposed methods on data with measurement noise.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Multidisciplinary Sciences
Sanda Micula
Summary: This paper presents an iterative numerical method for approximating solutions of two-dimensional Fredholm-Volterra integral equations, using Mann-type successive approximations and suitable numerical integration formulas. The existence and uniqueness of the solution are derived under certain conditions, along with error estimates and numerical examples showing the feasibility and effectiveness of the method.
Article
Mathematics, Applied
Reza Zolfaghari, Jacob Taylor, Raymond J. Spiteri
Summary: The method described in the text is used for analyzing the structure of a system of nonlinear integro-differential-algebraic equations (IDAEs) by focusing on the sparsity pattern of the IDAE and the nu-smoothing property of a Volterra integral operator. It aims to determine the index by analyzing which equations need to be differentiated and how many times, while also revealing hidden constraints and compatibility conditions to prove the existence of a solution.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Chuanli Wang, Biyun Chen
Summary: We propose a multi-step spectral collocation method for solving Caputo-type fractional integro-differential equations with weakly singular kernels. By reformulating the problem as a second type Volterra integral equation with two different weakly singular kernels, we construct a multi-step Legendre-Gauss spectral collocation scheme and rigorously establish the hp-version convergence. Numerical experiments are conducted to validate the effectiveness of the suggested method and the theoretical results.
Article
Mathematics, Applied
Walid Remili, Azedine Rahmoune, Chenkuan Li
Summary: This paper investigates the Galerkin spectral method for solving linear second-kind Volterra integral equations with weakly singular kernels on large intervals. By using variable substitutions, the equation is transformed into an equivalent semi-infinite integral equation with nonsingular kernel, allowing the inner products from the Galerkin procedure to be evaluated using Gaussian quadrature. The error analysis based on the Gamma function demonstrates the spectral rate of convergence, and several numerical experiments are conducted to validate the theoretical results.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics
Samad Noeiaghdam, Sanda Micula
Summary: This study discusses mathematical modeling of load leveling problems and energy storage systems, introduces the Lagrange-collocation method and control methods, and proposes a novel condition instead of traditional conditions.
Article
Mathematics
Svetlana Solodusha, Mikhail Bulatov
Summary: This paper examines two types of Volterra integral equations of the first kind related to the inverse problems of controlled heat power systems dynamics. The focus is on studying the specifics of the classes of Volterra equations of the first kind that arise from describing nonlinear dynamics using the apparatus of Volterra integro-power series. The research area is represented by a simulation model of a heat exchange unit element, showing numerical results of solving the identification problem of transient characteristics and the importance of practical recommendations for linear multidimensional Volterra equations of the first kind.
Article
Mathematics, Applied
Xiaohua Ma, Chengming Huang
Summary: This work analyzes a Legendre collocation approximation for third-kind Volterra integro-differential equations, providing a rigorous error analysis. To avoid the low-order accuracy caused by singularity at the initial point, the idea of smooth transformation is adopted. The validity and applicability of the method are verified through several numerical experiments.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Yongtao Zhou, Martin Stynes
Summary: A class of block boundary value methods is proposed for linear neutral Volterra integro-differential equations with weakly singular kernels, showing optimal convergence rates on special graded meshes. The methods can also be extended to solve 2nd kind linear Volterra integral equations with weakly singular kernels, as confirmed by numerical experiments and comparison with piecewise polynomial collocation methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Farshid Mirzaee, Erfan Solhi, Shiva Naserifar
Summary: In this paper, a technique utilizing moving least squares (MLS) and spectral collocation method is extended to solve nonlinear stochastic Volterra integro-differential equations without the need for preprocessing like mesh refinement. By converting the problem into a system of algebraic equations, the proposed method reduces the computational burden while ensuring convergence and reliability with an error bound. The efficiency and applicability of this technique are demonstrated through illustrative examples presented in the paper.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Salwa A. Mohamed, Norhan A. Mohamed, Sarah I. Abo-Hashem
Summary: An integration matrix operator is introduced in this work to discretize integro-differential equations, along with a generic differential-integral quadrature method (DIQM). Stability analysis and numerical results demonstrate the exponential convergence and applicability of the proposed method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Hui Liang
Summary: This paper explores the convergence of solving second-kind Volterra integral equations using DC, CC, DG, and CG methods, demonstrating their equivalence and validating the theoretical results through numerical examples.
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2022)
Article
Mathematics, Interdisciplinary Applications
Anqi Zhang, Roghayeh Moallem Ganji, Hossein Jafari, Mahluli Naisbitt Ncube, Latifa Agamalieva
Summary: This paper presents a numerical algorithm to solve distributed order integro-differential equations, using matrices derived from shifted Legendre polynomials to approximate the solution and compute numerical values of the polynomial coefficients. Theoretical aspects of error bounds are discussed, and illustrative examples are included to demonstrate the validity and applicability of the method.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2022)
Article
Chemistry, Physical
Luojia Zhang, Evgeny Zhuravlev, Jun Yi, Qijie Zhai, Christoph Schick, Yulai Gao, Bingge Zhao
Summary: Crystal nucleation is a crucial step in the crystallization of metallic glasses. However, the nucleation kinetics can deviate from the classical nucleation theory as the undercooling increases. This study examines the nucleation kinetics in glass and undercooled melt using nanocalorimetry. By varying the cooling rate, the processes of crystallization, homogeneous nucleation, and heterogeneous nucleation are distinguished. The critical cooling rates for suppressing crystallization and nucleation are estimated, and a well-identified amorphous phase is produced for nucleation studies. The underlying kinetic mechanism is revealed through the analysis of crystallization heat and overall heat using the Johnson-Mehl-Avrami method.
JOURNAL OF ALLOYS AND COMPOUNDS
(2023)
Article
Chemistry, Physical
Ruslan A. Andrianov, Juern W. P. Schmelzer, Rene Androsch, Timur A. Mukhametzyanov, Christoph Schick
Summary: The specific features of crystal nucleation have a significant impact on the morphology of the crystalline material. Due to their small size and stochastic nature, it is usually difficult to observe the development of nuclei directly. However, an experimental approach using fast scanning calorimetry has been developed to determine the specific features of the cluster size distribution. This approach allows for estimating the time evolution of the largest detectable clusters in the distribution and determining their radial growth rate.
JOURNAL OF CHEMICAL PHYSICS
(2023)
Article
Materials Science, Ceramics
Semen E. Lapuk, Marat A. Ziganshin, Radik A. Larionov, Timur A. Mukhametzyanov, Christoph Schick, Alexander V. Gerasimov
Summary: Robust determination of parameters governing the stability of amorphous drugs is crucial in modern pharmaceutics. The kinetic stability of these systems greatly impacts their practical applications. This study determined the critical cooling rates and kinetic parameters of cold crystallization for four slowly crystallizing sulfonamides. The Nakamura crystallization model demonstrated good prognostic ability by determining the stability time profile of drug systems prone to crystallization from non-isothermal thermokinetic data.
JOURNAL OF NON-CRYSTALLINE SOLIDS
(2023)
Article
Thermodynamics
Aleksey V. Buzyurov, Ruslan N. Nagrimanov, Timur A. Mukhametzyanov, Marat A. Ziganshin, Boris N. Solomonov, Christoph Schick
Summary: The vapor pressures and enthalpies of sublimation/vaporization/fusion were measured for acetanilide and eight derivatives. The obtained vapor pressures resolved contradictions in the literature and confirmed previous values for certain compounds. Sublimation and vaporization enthalpies were determined and mostly agreed with literature values. Fusion enthalpies of certain compounds were determined using different methods and showed consistency with literature values.
JOURNAL OF CHEMICAL AND ENGINEERING DATA
(2023)
Article
Polymer Science
Akihiko Toda, Yoshitomo Furushima, Christoph Schick
Summary: In this study, the isothermal crystallization kinetics of poly(butylene terephthalate) at low temperatures near the glass transition was investigated using chip-sensor fast scanning calorimetry. The Avrami analysis showed that the Avrami index decreased from 4 to less than 2 in the low-temperature peak of the crystallization rate, resulting in significantly low crystallinity near the glass-transition temperature. This phenomenon was attributed to the inhibition of crystal growth by the rigid amorphous fraction (RAF) that is constrained by crystals, as proposed by Schawe for other crystalline polymers. The transformation of the mobile amorphous fraction (MAF) into RAF with the progress of crystallization was confirmed near the glass-transition temperature, leading to the rigidification of all amorphous fractions.
Article
Medicine, Research & Experimental
Semen E. Lapuk, Timur A. Mukhametzyanov, Christoph Schick, Alexander Gerasimov
Summary: The application of drugs in the amorphous state is being actively researched in pharmaceutical science to enhance their bioavailability. In this study, the kinetic stability and glass-forming ability of thermally labile quinolone antibiotics were investigated using fast scanning calorimetry. The critical cooling rates for preventing crystallization of oxolinic acid, pipemidic acid, and sparfloxacin were determined to be 10,000, 40, and 80 K·s-1 respectively, indicating their strong glass-forming ability. The Nakamura model was found suitable for describing the crystallization process of the amorphous forms of the quinolone antibiotics using a combination of nonisothermal and isothermal kinetic approaches.
MOLECULAR PHARMACEUTICS
(2023)
Article
Chemistry, Multidisciplinary
Alexander Minakov, Christoph Schick
Summary: This article studies the influence of various mesoscopic effects on interfacial thermal conductance (ITC) during fast melting processes and the contributions to ITC in pre-melting and melting processes of metal microparticles. The gained knowledge is useful for understanding and optimizing technological processes involving fast melting, such as laser additive manufacturing.
APPLIED SCIENCES-BASEL
(2023)
Article
Chemistry, Multidisciplinary
Karina V. Gataullina, Aleksey V. Buzyurov, Alexander V. Gerasimov, Askar K. Gatiatulin, Marat A. Ziganshin, Christoph Schick, Valery V. Gorbatchuk
Summary: Solid-state guest exchange is an efficient method for preparing highly metastable polymorphs of indomethacin. This method allows the preparation of metastable forms with low first melting points, and the characterization of indomethacin polymorphs was performed using fast scanning calorimetry.
CRYSTAL GROWTH & DESIGN
(2023)
Article
Physics, Multidisciplinary
Juern W. P. Schmelzer, Timur V. Tropin, Christoph Schick
Summary: In the theoretical treatment of crystallization, it is commonly assumed that liquid relaxation processes are faster than crystal nucleation and growth. However, near and below the glass transition temperature, this assumption is often not valid. Deviations from the metastable equilibrium state of the liquid must be considered when determining the kinetic coefficients, thermodynamic driving force, and surface tension of the crystal phase. These factors can greatly influence the overall crystallization process.
Article
Polymer Science
Yoshitomo Furushima, Nobuhiro Hirota, Masaru Nakada, Akihiro Masuda, Kazuma Okada, Masatoshi Ohkura, Christoph Schick
Summary: Fast scanning calorimetry (FSC) allows rapid cooling of samples from the melt at rates of several thousand Kelvin per second. By fast cooling at rates of 5000 K s(-1), a perfect amorphous state for polypropylene (PP) can be achieved. In this study, a novel method combining FSC and differential scanning calorimetry was proposed to determine the PP/PE mixing ratio with a standard deviation of 1.4 mass%. The proposed method takes into account the thermal behavior of polyolefin blend components. It is expected to provide a convenient way to analyze the composition of recycled polyolefins without the need for high-temperature solvents or calibration curves.
JOURNAL OF APPLIED POLYMER SCIENCE
(2023)
Article
Polymer Science
Akihiko Toda, Yoshitomo Furushima, Christoph Schick
Summary: The relationship between the changes in the crystallization kinetics and the crystal domains of poly(butylene terephthalate) was investigated under isothermal conditions. The Avrami exponent n characterizing the nucleation and growth kinetics of the crystal domains exhibited a continuous change within the target temperature range, indicating a change in the nucleation mode for spherical domains. The morphology of the crystal domains responsible for this change was identified, with a continuous change in the size of spherulites observed. The formation of 10 nm scale granular nodules with high nuclei density at low temperatures was also observed.
Article
Polymer Science
Rene Androsch, Katalee Jariyavidyanont, Andreas Janke, Christoph Schick
Summary: In this study, the melt-crystallization process of PBS under different melt-supercooling conditions was analyzed using X-ray scattering techniques, microscopy, and fast scanning chip calorimetry. It was found that lamellar thickening is not the main mechanism of isothermal secondary crystallization, and low-temperature crystallization leads to the presence of crystal defects.
Article
Chemistry, Physical
Semen Lapuk, Marina Ponomareva, Marat Ziganshin, Radik Larionov, Timur Mukhametzyanov, Christoph Schick, Ivan Lounev, Alexander Gerasimov
Summary: In this study, the glass transition of the biocompatible polymer polyvinylpyrrolidone was investigated using differential scanning calorimetry, fast scanning calorimetry, and broadband dielectric spectroscopy. The dependence of the difference in isobaric specific heat capacities between the liquid and glass on the dynamic glass transition temperature, volume, and number of particles in the cooperatively rearranging regions was determined. A linear relationship between the shift factor from the Frenkel-Kobeko-Reiner equation and the molecular mass of polyvinylpyrrolidone was established. These findings contribute to the selection of optimal excipients for the development of solid dispersions based on amorphous polymers.
PHYSICAL CHEMISTRY CHEMICAL PHYSICS
(2023)
Article
Thermodynamics
Timur A. Mukhametzyanov, Ruslan A. Andrianov, Dmitrii N. Bolmatenkov, Mikhail I. Yagofarov, Boris N. Solomonov, Christoph Schick
Summary: The nucleation and crystallization of the rapidly crystallizing organic compound benzocaine show different behaviors under different supercooling conditions, and the temperature transition of nucleation and crystallization is sharp and sample-dependent.
THERMOCHIMICA ACTA
(2023)
Article
Polymer Science
Akihiko Toda, Yoshitomo Furushima, Christoph Schick
Summary: The relationship between changes in crystallization kinetics and crystal domains of poly(butylene terephthalate) was investigated under isothermal conditions. The study found that the nucleation and growth kinetics of the crystal domains exhibited a continuous change within a certain temperature range. This change corresponded to the nucleation mode for spherical domains, such as spherulites and nodules, which depended on whether nucleation occurred from foreign heterogeneities or from the homogeneous melt. The morphology of the crystal domains, specifically the size and formation of spherulites and granular nodules, was identified as a factor contributing to the changes in crystallization kinetics.
