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The paper addresses a computational method to simulate more accurate models for population growth with immigration, focusing on integral equations (IEs) featuring a delay parameter in the time variable. The proposed method utilizes Legendre wavelets within the Galerkin scheme as a orthonormal basis. Legendre wavelets are known for their localized functions, offering suitable precision and stability in simulating time-delay biological models. This approach employs the composite Gauss-Legendre (CGL) quadrature rule to compute integrals appeared in the scheme. An error bound analysis demonstrates a convergence rate of order 2 − M k. Additionally, various numerical examples are presented to show the efficiency, accuracy and validate the theoretical error estimate of the novel technique. [ABSTRACT FROM AUTHOR]

• Exact solution of the problem of Eshelby's inhomogeneity in strain gradient elasticity theory. • Green's function technique and integral equation developed for arbitrary boundary conditions. • Analytical expressions of mean strain inside inclusion. • Analytical expressions of effective elastic properties with strain gradient. • Effect of the scale parameter, the contrast of elastic behavior and the volume fraction of inhomogeneity. A new micromechanical approach to deal with the problem of Eshelby's inhomogeneity is developed for the prediction of the effective properties of composite materials according to the strain gradient elasticity theory. The method is based on the Green's function technique leading to an integral equation of the heterogeneous elastic problem. Within the simplified strain gradient elasticity theory, the integral equation for an infinite heterogeneous medium subjected to non-homogeneous boundary conditions is acquired. Thanks to this integral equation, the exact solution of Eshelby's inhomogeneity problem is detailed for spherical inhomogeneity and isotropic elastic behavior. From the expression of strain localization relations, the effective elastic properties of a two-phase composite material are then predicted through Mori Tanaka's homogenization scheme. To test the relevance of the suggested approach, its predictions are compared with results issued from some reference models and experimental data. [ABSTRACT FROM AUTHOR]

A visco-acoustic medium is an approximate representation of the absorption and attenuation characteristics of the actual earth medium, which will cause energy attenuation and velocity dispersion of seismic waves. The propagation law of scattered waves in viscoelastic media can be revealed through seismic wave forward simulation. However, the numerical simulation of scattered waves in a visco-acoustic medium using the De Wolf approximation method based on the Lippmann-Schwingr (L-S) equation has not been studied. Therefore, based on the Futterman dispersion relation, the mathematical expression of the integral equation of acoustic wave propagation and scattering in the visco-acoustic medium is derived. We combine the De Wolf approximation with the quality factor Q. The thin slab approximation and screen approximation methods are two realizations of the De Wolf approximation. The thin slab approximation and the screen approximation continuation operator in the viscous-acoustic medium are constructed and applied to the numerical simulation of the scattered wave field. The research results of the concave model and complex model indicate that due to the influence of medium viscosity, the energy of the deep reflected waves is weaker in the forward modeling records of the thin slab approximation method and screen approximation method, and the dominant frequency of seismic waves moves towards the low-frequency direction; Compared with the finite difference results, they only calculate one reflection, and there are no multiple wave signals in the seismic records. Finally, We discuss the thin slab approximation and the screen approximation method for detail around the division of the thickness of the thin slab, the selection of the reference frequency, and the non-uniformity of the medium. • Based on the De Wolf approximation, we proposed a numerical simulation method for scattered waves in visco-acoustic media. • The proposed method can reveal the propagation law of scattered waves in viscous acoustic media. • We combined the De Wolf approximation with the quality factor Q , and used two methods to achieve it. • We analyzed the influence of different thin slab thicknesses on the forward modeling of scattered waves. [ABSTRACT FROM AUTHOR]

In this paper, integral equation formulations and field-dependent micromechanical modeling of global nonlinear magneto-mechanical behavior of magneto-elastic composites under high strain and magnetic field are elaborated. The modeling is obtained based on an extension of micro–macro transition inclusion problem to nonlinear behavior using field-dependent and highly nonlinear localization tensors. A methodological procedure is elaborated based on newly introduced strain and magnetic field-dependent Green tensors, strain and magnetic field-dependent integral equations linked to tensors of Eshelby and micromechanical approaches. The field-dependent global moduli are predicted based on the Mori–Tanaka approach and self-consistent method. Iterative incremental schemes based on the Newton–Raphson and fixed-point algorithms are examined and established precise semi-analytic equations of the effective magneto-elastic properties of composites that are dependent on the magnetic-strain field for different types of inclusions. A numerical code is elaborated for numerical predictions and the obtained field-dependent effective magneto-elastic coefficients are obtained and presented for various volume fractions of inclusion, types and shapes of the reinforced nonlinear composites. • Nonlinear micromechanical modeling of fully coupled magneto-elastic under large deformation and high magnetic field. • Magnetic-Strain dependent Green tensors and associated localization tensors. • Iterative incremental schemes for semi-analytical magnetic-strain field dependent effective magneto-elastic properties. • Magneto-mechanical field dependent effective properties for several types of matrix and inclusions phases. [ABSTRACT FROM AUTHOR]

We present a computational method for reconstructing the refractive index of an unknown complex-shaped two-dimensional medium with embedded metal inclusions from its transverse magnetic electromagnetic scattering properties. We present a novel hybrid surface-volume integral equation that generalises the Lippmann–Schwinger equation to media containing embedded scatterers. Using this hybrid equation, we show that the Frechet derivative of the forward-mapping satisfies a particular inhomogeneous wave scattering problem. We solve the inhomogeneous wave scattering problem using a novel coupled FEM-BEM formulation. Our numerical scheme is based on a thin plate spline ansatz for the refractive index, which can be constructed using a relatively small number of control points, and can be efficiently constructed and evaluated even for the complex-shaped multiply-connected domains of interest. Numerical experiments demonstrate the effectiveness of our method by reconstructing several challenging media. [ABSTRACT FROM AUTHOR]

Induced currents in the bulk superconducting cylinder of finite length placed in a transverse magnetic field are calculated by the partial and complete separation of variables. The Fourier transform over the angular and axial coordinates reduces the problem to the solution of one-dimensional integral equations. The harmonic balance method is applied to account for the highly nonlinear constitutive relation between the voltage and current. The current and field distribution is compared with the results of FEA. The approximation of the current distribution by a set of cylindrical and disc layers is presented. The magnetic moment of induced currents is analyzed in the increasing applied field when the cylinder is magnetized up to the state of saturation. [ABSTRACT FROM AUTHOR]

[Display omitted] • A new general integral transform which is covered all class of integral transform in the class of Laplace transform. • We investigated the application of this new transform for solving ODE with constant and variable coefficient. • This new transform can handle easily for fractional order integral equations and fractional order differential equations. • We have discussed the advantage and disadvantage of other integral transformed which is defined during last 2 decades. • We proved the related theorems for this new transform. Integral transforms are important to solve real problems. Appropriate choice of integral transforms helps to convert differential equations as well as integral equations into terms of an algebraic equation that can be solved easily. During last two decades many integral transforms in the class of Laplace transform are introduced such as Sumudu, Elzaki, Natural, Aboodh, Pourreza, Mohand, G_transform, Sawi and Kamal transforms. In this paper, we introduce a general integral transform in the class of Laplace transform. We study the properties of this transform. Then we compare it with few exiting integral transforms in the Laplace family such as Laplace, Sumudu, Elzaki and G\_transforms, Pourreza, Aboodh and etc. A new integral transform is introduced. Then some properties of this integral transform are discussed. This integral transform is used to solve this new transform is used for solving higher order initial value problems, integral equations and fractional order integral equation. It is proved that those new transforms in the class of Laplace transform which are introduced during last few decades are a special case of this general transform. It is shown that there is no advantage between theses transforms unless for special problems. It has shown that this new integral transform covers those exiting transforms such as Laplace, Elzaki and Sumudu transforms for different value of p (s) and q (s). We used this new transform for solving ODE, integral equations and fractional integral equations. Also, we can introduce new integral transforms by using this new general integral transform. [ABSTRACT FROM AUTHOR]

This work is concerned with the construction and the hp non-conforming a priori error analysis of a Discontinuous Galerkin DG numerical scheme applied to the hypersingular integral equation related to the Helmholtz problem in 3D. The main results of this article are an error bound in a norm suited to the problem and in the L 2 -norm. Those bounds are quasi-optimal for the h -convergence and the p -convergence. Various formulation choices and penalty functions are theoretically discussed. In particular we show that a penalty function of the shape h 2 p leads to a quasi-optimal convergence of the scheme. Some numerical experiments confirm the expected rates of convergence and the effect of the penalty function. [ABSTRACT FROM AUTHOR]

The paper is concerned with the wavelet-Galerkin method for the numerical solution of Fredholm linear integral equations and second-order integro-differential equations. We propose a construction of a quadratic spline-wavelet basis on the unit interval, such that the wavelets have three vanishing moments and the shortest support among such wavelets. We prove that this basis is a Riesz basis in the space L 2 0 , 1. We adapt the basis to homogeneous Dirichlet boundary conditions, and using a tensor product we construct a wavelet basis on the hyperrectangle. We use the wavelet-Galerkin method with the constructed bases for solving integral and integro-differential equations, and we show that the matrices arising from discretization have uniformly bounded condition numbers and that they can be approximated by sparse matrices. We present numerical examples and compare the results with the Galerkin method using other quadratic spline wavelet bases and other methods. [ABSTRACT FROM AUTHOR]

The present paper proposes an efficient computational method for solving the time-harmonic multiple scattering problem by many small dielectric circular cylinders. The strategy proposed is based on a reduced integral equation formulation associated to diagonalization properties of the integral operators for the interior problem. In addition, we use an asymptotic approximation of the separation matrices to simplify the formulation. Using a GMRES Krylov iterative solver, accelerated by a preconditioner, we solve in a robust and efficient way the generated linear system. Numerical results support the efficiency and accuracy of the new strategy. [ABSTRACT FROM AUTHOR]

Abstract In this study, we derive an asymptotic solution of the integral equation satisfied by the second moment function M 2 t , a. We first find the Laplace transform M 2 L s , a and then obtain M 2 t , a asymptotically by inversion. Further, we have derived the asymptotic expressions of M 2 t , a for some special lifetime distributions such as exponential, gamma, Weibull, lognormal and truncated normal. Finally, the asymptotic solution is compared with the numerical solution to evaluate its performance. [ABSTRACT FROM AUTHOR]

In this paper, we propose a center Lipschitz condition for the second Fréchet derivative together with the use of restricted domains in order to improve the domain of starting points for Newton’s method. In addition, we compare the new result with an older one and see that the former improves the latter. [ABSTRACT FROM AUTHOR]

An error in the demonstration process in literature Alvarez and Lizama (2016) is pointed out and a theorem is proposed to correct the corresponding conclusion. [ABSTRACT FROM AUTHOR]

We analyze the semilocal convergence of Newton’s method under a center Lipschitz condition for the second derivative of the operator involved different from that used by other authors until now. In particular, we propose to center the Lipschitz condition for the second derivative in a different point from that where Newton’s method starts. This allows us to obtain different starting points for Newton’s method and modify the domain of starting points. [ABSTRACT FROM AUTHOR]

In this work, we have used the integral equations method to compute the surface tension of several refrigerant mixtures. For this goal, we have employed the Stockmayer potential and examined theoretically the surface tension of several pure refrigerants like R134a, R32, R125, R152a, R1234yf, R1234ze and also refrigerant mixtures such as R32 + R125, R32 + R134a, R125 + R134a, R125 + R143a, and R143a + R134a, R125 + R152a, R134a + R152a, R32 + R1234ze, R32 + R1234yf and R290 + R600a. We have solved numerically the Ornstein−Zernike (OZ) equation using the hypernetted chain approximation (HNC) for binary refrigerant mixtures and obtained the radial distribution functions. Finally, the variation of surface tension as a function of temperature is studied using a new molecular thermodynamic model and the obtained results have been compared with available results. Our theoretical results show a fairly agreement with available data except for R290 + R600a. [ABSTRACT FROM AUTHOR]

This paper presents a parallel algorithm for frequency-domain acoustic modeling by the method of integral equations (IE). The algorithm is applied to seismic simulation. The IE method reduces the size of the problem but leads to a dense system matrix. A tolerable memory consumption and numerical complexity were achieved by applying an iterative solver, accompanied by an effective matrix-vector multiplication operation, based on the fast Fourier transform (FFT). We demonstrate that, the IE system matrix is better conditioned than that of the finite-difference (FD) method, and discuss its relation to a specially preconditioned FD matrix. We considered several methods of matrix-vector multiplication for the free-space and layered host models. The developed algorithm and computer code were benchmarked against the FD time-domain solution. It was demonstrated that, the method could accurately calculate the seismic field for the models with sharp material boundaries and a point source and receiver located close to the free surface. We used OpenMP to speed up the matrix-vector multiplication, while MPI was used to speed up the solution of the system equations, and also for parallelizing across multiple sources. The practical examples and efficiency tests are presented as well. [ABSTRACT FROM AUTHOR]

We have developed a multi-level parallel magnetotelluric (MT) integral equation based inversion program which uses variable sensitivity domain. The limited sensitivity of the data, which decreases with increasing frequency, is exploited by a receiver sensitivity domain, which also varies with frequency. We assess the effect of inverting principal impedances, full impedance tensor, and full tensor jointly with magnetovariational data (tipper). We first apply this method to several models and then invert the EarthScope MT data. We recover well the prominent features in the area including resistive structure associated with the Juan de Fuca slab subducting beneath the northwestern United States, the conductive zone of partially melted material above the subducting slab at the Cascade volcanic arc, conductive features in the Great Basin and in the area of Yellowstone associated with the hot spot, and resistive areas to the east corresponding to the older and more stable cratons. [ABSTRACT FROM AUTHOR]

The optimal value of sampling parameter that guarantees the global equivalence of staircase and linear scan voltammograms plotted for electrochemical reactions involving soluble species is re-examined in this second article taking the electron transfer kinetics, the uncompensated resistance and the double-layer charging current into account. The discussion is based on the numerical solution of linear/nonlinear integral-equation-based models. The comparison of voltammetric and staircase voltammetric current data is carried out in terms of the mean squared deviation between voltammograms. Main results concern the dependence of the optimal sampling parameter on the potential step amplitude of staircase signal, and the dimensionless parameters pertaining to kinetic, Ohmic and capacitive effects. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new systematic procedure of estimating elastic properties of composites constituted of two phases, matrix and inclusions. A class of integral equations based on eigenstrain (or eigenstress) with the matrix as reference material is constructed with an explicit form in Fourier space. Each integral equation belonging to this class can yield estimates of the overall elastic tensor via Neumann series expansion. The best estimates and series are selected based on the convergence rate criteria of the series, i.e the spectral radius must be minimized. The optimized series is convergent for any finite contrast between inclusions and matrix. Applying the optimized series and the associated estimates to different microstructures yields very satisfying results when compared with the related full solution. For the case of a random distribution of spherical inclusions, exact relations between the elastic tensor and n th order structure factors are demonstrated. [ABSTRACT FROM AUTHOR]

Since the pioneering work by Osteryoung et al., J. Electroanal. Chem. 222 (1987) 69, the discussion about the equivalence of staircase and linear scan voltammograms is based on the ‘Abelian’ or ‘non-Abelian’ feature of the integral equation used for modelling the voltammetric response of redox systems. In practice, further investigation is required for prediction of the optimal value of current sampling parameter ( γ opt ) that guarantees the equivalence of voltammetric and staircase voltammetric data. This first article focuses on reversible electron transfer reactions involving soluble species. The discussion is based on integral-equation-based models and the use of local/global criteria for the comparison of staircase and linear scan voltammograms. Main results in the article and the Supporting information joined should concern the dependence of γ opt on the electrode potential, the potential step amplitude of staircase signal, the reaction scheme, and the mass transport process involved. [ABSTRACT FROM AUTHOR]

In this paper, a numerical method is proposed to approximate the solution of a two-dimensional scattering problem of time-harmonic elastic wave from a rigid obstacle. By Helmholtz decomposition, the scattering problem is reduced to a system of Helmholtz equations with coupled boundary conditions. Then, we prove that the system of Helmholtz equations has only one solution under certain conditions, and propose an integral equation method to solve it numerically based on Tikhonov regularization method. Finally, numerical examples are presented to show the feasibility and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this article boundary-value problems of potential theory, that are required for transforming and continuing selected parameters of a potential field, are discussed. By the potential field parameters we understand the potential and its gradients (potential gradients) up to the third order. In particular, integral equations defined for a spherical boundary transforming potential gradients of different orders and continuing potential gradients of the same order through the 3-D space are reviewed and classified. This mathematical apparatus can be used for any harmonic potential, such as electric, magnetic or gravitational, under the assumption of its conservativeness, i.e., neglecting possible temporal variations. Integral transforms are discussed in context of geoscience applications, namely in terms of the Earth's gravitational field; however, the article can serve as a general reference for integral transforms of potential field parameters in any scientific or engineering area where potential fields are used. [ABSTRACT FROM AUTHOR]

We build a numerical approximation method, for Fredholm integral equation solution of the second type. This method is based on the regularization by convolution and Fourier series expansion. It provides a better convergence order. [ABSTRACT FROM AUTHOR]

This work is to propose high asymptotic order methods for highly oscillatory Volterra integral equations. According to the error of the Filon quadrature method, the piecewise Hermite collocation method is considered. For the computation, the highly oscillatory integrals in the collocation equations are treated efficiently. Then, both the classical order and asymptotic order are analyzed in detail with the help of some estimates on highly oscillatory integrals. At last, some numerical examples are provided to verify the sharpness of the estimate in the final part. [ABSTRACT FROM AUTHOR]

We show that, by correctly selecting the probability distribution function p ( s ) for a particle’s distance-to-collision, the nonclassical diffusion equation can be represented exactly by the nonclassical linear Boltzmann equation for an infinite homogeneous medium. This choice of p ( s ) preserves the true mean-squared free path of the system, which sheds new light on the results obtained in previous work. [ABSTRACT FROM AUTHOR]

Absorption spectroscopy is an advanced tool for flow diagnostics in measuring multiple parameters of species. Such kinds of problems can be modeled by some integral equations with known kernel, aiming to the determination of the integrands from their integration values along all possible paths of injected lasers. This paper considers the parameters detection problems in combustion process, with the purpose of recovering the gas temperature and the concentration of burned gas simultaneously using injected lasers along two directions with multiple frequencies. After establishing the nonlinear integral equations describing the energy absorption process, this ill-posed model is transformed into a nonlinear optimization problem with some penalty terms. Then we present an alternative iteration scheme (AIS) to solve this problem. The convergence of the iterative sequence for AIS algorithm together with the estimate on the value of cost functional is established, ensuring that AIS can indeed generate a satisfactory approximate solution to the original optimization problem. Numerical implementations using simulant data are presented to show the validity of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper we obtain some equivalences between cyclic contractions and non-cyclic contractions in the framework of b -metric spaces. Our results improve and complement several recent fixed point results for cyclic contractions in b -metric spaces established by George et al. (2015) and Nashine et al. (2014). Moreover, all the results are with much shorter proofs. In addition, an application to integral equations is given to illustrate the usability of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, an integral equation approach is adopted to price American-style down-and-out calls. Instead of using the probability theory as used in the literature, we use the continuous Fourier sine transform to solve the partial differential equation system governing the option prices. As a way of validating our approach, we show that the “early exercise premium representation” for American-style down-and-out calls without rebate can be re-derived by using our approach. We then examine the case that time-dependent rebates are included in the contract of American-style down-and-out calls. As a result, a more general integral representation for the price of an American-style down-and-out call, with the presence of an extra term associated with the rebate, is obtained. Our numerical method based on the newly-derived integral representation appears to be efficient in computing the price and the hedging parameters for American-style down-and-out calls with rebates. In addition, significant effects of rebates on the option prices and the optimal exercise boundaries are illustrated through selected numerical results. [ABSTRACT FROM AUTHOR]

We discuss the accurate and efficient implementation of hp -BEM for the Laplace operator in two dimensions. Using Legendre polynomials and their antiderivatives as local bases for the discrete ansatz spaces, we are able to reduce both the evaluation of potentials and the computation of Galerkin entries to the evaluation of basic integrals. For the computation of these integrals we derive recurrence relations and discuss their accurate evaluation. Our implementation of p - and hp -BEM produces accurate results even for large polynomial degrees ( p > 1000 ) while still being efficient. While this work only treats Symm's integral equation for the Laplace operator in 2D, our approach can be used to solve Symm's, hypersingular and mixed integral equations for Laplace, Lamé and Stokes problems in two dimensions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish the existence and uniqueness of the common solution for the system of nonlinear Fredholm integral equations, nonlinear Volterra integral equations and nonlinear fractional differential equations using the common fixed point results equipped with illustrative examples. Some common fixed point results satisfying the generalized contraction condition involving w -distance and weak altering distance functions are proved. Then, an example is provided to support the usability of our result along with numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, based on the principle of virtual work, we formulate the equivalent eigenstrain approach for inhomogeneous inclusions. It allows calculating the elastic deformation of an arbitrarily connected and shaped inhomogeneous inclusion, by replacing it with an equivalent homogeneous inclusion problem, whose eigenstrain distribution is determined by an integral equation. The equivalent homogeneous inclusion problem has an explicit solution in terms of a definite integral. The approach allows solving the problems about inclusions of arbitrary shape, multiple inclusion problems, and lends itself to residual stress analysis in non-uniform, heterogeneous media. The fundamental formulation introduced here will find application in the mechanics of composites, inclusions, phase transformation analysis, plasticity, fracture mechanics, etc. [ABSTRACT FROM AUTHOR]

Abstract: In the anti-plane problem about a high-frequency diffraction by an interface crack located between two different elastic materials we propose a new asymptotic approach, which reduces the problem to the Wiener–Hopf integral equations. The key point of the method is a factorization of the symbolic function which is performed in an efficient way. As a result, the leading asymptotic term is written out in an explicit analytical form. [Copyright &y& Elsevier]

Abstract: In order to compute an approximate solution of a weakly singular integral equation, we first regularize the kernel and then truncate the associated Fourier series. Applications to Green and Abel operators are given. [Copyright &y& Elsevier]

Abstract: In recent years, different fields of engineering have been increasingly incorporating functionally graded materials with variable physical properties that significantly improve a quality of elements of designs. The efficiency of practical application of thermoelastic inhomogeneous materials depends on knowledge of exact laws of heterogeneity, and to define them it is necessary to solve coefficient inverse problems of thermoelasticity. In the present research a scheme of solving the inverse problem for an inhomogeneous thermoelastic rod is presented. Two statements of the inverse problem are considered: in the Laplace transform space and in the actual space. The direct problem solving is reduced to a system of the Fredholm integral equations of the 2nd kind in the Laplace transform space and an inversion of the solutions obtained on the basis of the theory of residues. The inverse problem solving is reduced to an iterative procedure, at its each step it is necessary to solve the Fredholm integral equation of the 1st kind; to solve it the Tikhonov method is used. Specific examples of a reconstruction of variable characteristics required are given. [Copyright &y& Elsevier]

Abstract: We propose a new algorithm to compute the teleseismic P and S receiver function synthetics for a multilayered Cartesian structure with anisotropic flat layers. The algorithm is based on the first-order perturbation theory in which the layered background structure is assumed one-dimensional with isotropic variations in vertical direction. Anisotropic velocity perturbations acting as secondary sources constitute the heterogeneities in the medium. The total wavefield is solved using a convolutional type integral equation along with the Green's function of the one-dimensional reference medium extracted using the reflectivity method. The integral equation involves a five-fold integration in space and wavenumber domains. Four of these integrals are achieved analytically and the fifth integral, which is spatial integral in the vertical direction, is performed numerically for which the Born single scattering approximation greatly suffices. The proposed algorithm is demonstrated on some selected numerical examples adapted from published work in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, an investigation of radiative transfer in a rectangular medium with one-dimensional or two-dimensional graded index is presented. The integral equations of intensity moments are derived and then the cases of a cold medium exposed to diffuse irradiation at the left boundary are solved by the Nyström method. The results obtained by solving integral equations are in excellent agreement with those obtained by the Monte Carlo method and the discrete ordinates method. For the case with an increasing refractive index in the direction to the right boundary, the distribution of refractive index enhances the radiation in the direction to the right, and so the half-range flux toward the right boundary increases with the increase of the gradient of refractive index. Besides, the half-range flux toward the right boundary decreases as the position considered approaches the top or bottom boundary. [Copyright &y& Elsevier]

Abstract: Drug dissolution process of solid dosages is theoretically described by Noyes–Whitney–Nernst equation. However, the analysis of the process is demonstrated assuming some models. Normally, the model-dependent methods are idealized and require some limitations. In this study, Noyes–Whitney integral equation was proposed and applied to represent the drug dissolution profiles of a solid formulation via the non-linear least squares (NLLS) method. The integral equation is a model-free formula involving the dissolution rate constant as a parameter. In the present study, several solid formulations were prepared via changing the blending time of magnesium stearate (MgSt) with theophylline monohydrate, α-lactose monohydrate, and crystalline cellulose. The formula could excellently represent the dissolution profile, and thereby the rate constant and specific surface area could be obtained by NLLS method. Since the long time blending coated the particle surface with MgSt, it was found that the water permeation was disturbed by its layer dissociating into disintegrant particles. In the end, the solid formulations were not disintegrated; however, the specific surface area gradually increased during the process of dissolution. The X-ray CT observation supported this result and demonstrated that the rough surface was dominant as compared to dissolution, and thus, specific surface area of the solid formulation gradually increased. [Copyright &y& Elsevier]

Abstract: We introduce a cyclic generalized -weakly contractive mappings for a map in a metric space and originate existence and uniqueness results of fixed points for such mappings. Examples are given to support the usability of our results. At the end of the results, an application to the study of existence and uniqueness of solutions for a class of nonlinear integral equations is presented. [Copyright &y& Elsevier]

Abstract: The statements on solvability, solution estimates, and well-posedness of equations with conditionally covering mappings are proved. The results obtained are applied to the investigation of Volterra equations (including integral equations) unsolved for the unknown function. [Copyright &y& Elsevier]