Integration Of Differential Equations Research Articles

Методика розрахунку основних характеристик підшипників ковзання систем і агрегатів автомобілів та аналізу їх трибологічної ефективності

A methodology for calculating the main characteristics of sliding bearings of car systems and units is proposed. The specified methodical approach is important when evaluating indicators of tribological efficiency of couplings of parts, nodes, systems and aggregates and operational reliability of cars as a whole. The bearing activation model makes it possible to formulate measures to increase the tribological efficiency of the couplings of parts and the level of operational reliability of nodes, systems and aggregates, and cars as a whole. In addition, the parameters of the actuation model also respond to friction modes. It is shown that it is possible to increase the tribological efficiency of the sliding bearing by means of design, technological and operational methods. The main formulas that make it possible to develop algorithms for calculating a set of bearing characteristics are presented: contact pressure, stress, strength, sliding resistance, wear, resource. For this purpose, the following values are used: compliance coefficients and relative load, contact angle, contact pressures, contact displacements, and the stress state in the contact zone. The geometric and power parameters of the bearing are involved in the developed methodology and algorithm. Modes of friction and the nature and magnitude of the load on the bearing are also taken into account. An algorithm for calculating auxiliary and main characteristics of a sliding bearing has been developed. The conditions of continuity and equilibrium in contact are clarified. The differential-integral equation of the operation of the bearing is obtained and the dependence of the current contact angle on the friction path and other parameters of the sliding bearing operation model is obtained. It is shown that the tribological efficiency of sliding bearings is established by the method of comparative analysis of tribotechnical characteristics determined by the results of tests on friction machines and a stand.

RUIN PROBABILITY IN THE CLASSICAL RISK PROCESS WITH WEIBULL CLAIMS DISTRIBUTION

In the classical risk process, ruin is the situation when the surplus falls below zero. Ruin probability is a tool used to predict bankruptcy in the insurance company. The ruin probability can be determined by solving the Integral-Differential equation that arises from the classical risk process. In this paper, we are interested in calculating the ruin probability when the claim distribution follows the Weibull distribution. Based on the Weibull parameter, the calculation is divided into two cases: when alpha equals 1 and when . The Laplace transform gives the analytical solution of the Integral-Differential equation. However, when the analytical solution cannot be determined since the Laplace transform is no longer applicable due to the presence of an improper integral that is not possible to solve analytically. Therefore, for the case alpha greater than 1, Euler’s method is applied to determine its numerical solution. The accuracy of the numerical solution is validated by comparing it with the analytical solution for the case Then, using the accuracy determined from the first case, we apply the Euler method to determine the numerical solution for the case . The numerical method gives good accuracy to the analytical solution with the order of calculated from until

Regularization of the Solution of a Nonlinear Integral Equation of the First Kind

Integral equations, the main branch of mathematics, are widely used in physics, engineering, mechanics, control theory and other fields. Related to the application of integral equations, new fields are developing, such as economics, some sections of biology, etc. The theory of integral equations mainly developed in the late nineteenth-early twentieth century, starting with Vito Volterra (1982, 1986), Eric Ivar Fredholm (2010), David Hilbert, Erhard Schmidt, etc. scientists began to study it. Nevertheless, within the framework of mathematical concepts that existed before the first half of the twentieth century, such problems were considered incorrect because a small change in the given functions led to a greater change in the desired functions. The Volterra equation of the first kind is an integral equation that has an exact solution only in some cases. The limit of integration has been carried out in very small quantities on non-classical linear and nonlinear integral equations with variable limits and the construction of solutions in these works is based on numerical methods. Therefore, for the so-called non-classical Volterra integral equations, it is relevant to determine the conditions that ensure the uniqueness and regularization of their solutions. In this paper, the uniqueness of the solution of the non-classical nonlinear integral Volterra equation of the first kind is resolved. The aim of the study is to solve the non-classical Volterra integral equation of the first kind, that is, to determine the conditions that ensure the uniqueness of the solution of the nonlinear non-classical Volterra integral equation of the first kind. The proposed methods can be used for the study of integral, integral-differential equations such as the Volterra integral equation of the first kind, as well as for the qualitative study of some applied processes in physics, ecology, medicine, geophysics, and the theory of control of complex systems.

New investigation of the analytical behaviors for some nonlinear PDEs in mathematical physics and modern engineering

Numerous complex situations arising from science and engineering are modeled by integral-differential equations (IDEs). Examples of these situations include mathematical modeling of infectious diseases, epidemics, circuit analysis, voltage drop equations, computational neuroscience, intergalactic modeling, and many more. To explain the mathematical structure and the behavior of various phenomena in nature, two equations with simple mathematical structure namely: the (1 + 1)-dimensional integro-differential Ito equation (IDIE) and the (2 + 1)-dimensional integro-differential Sawada-Kotera equation (IDSKE) are considered. In this analytical investigation, we have conducted an in-depth enquiry to trace the distinctive closed form progressive wave solutions of these proposed models using the (G′G′+G+A)-expansion method. The graphical forms of the acquired solutions have been displayed in the bell-shaped soliton, singular periodic, and anti-kink shape soliton solutions after the values for the free parameters were specified. The two non-algebraic solutions, named, exponential and trigonometric function solutions, have emerged by applying our proposed method. Based on the general findings of our investigation for different parametric values, the attained wave solutions revealed can keep a vital role for natural balances and can be engaged in modern physical applications. The parameters have a visible effect on the wave amplitude and the swiftness of the traveling wave. The executed outcomes are innovative and have monumental applications in the present research, especially in the fields of theoretical physics, astrophysics, plasma physics, particle physics, theory of relativity, advance quantum mechanics, string theory, and geotechnical engineering.

Stochastic Modeling in Immunology Based On a Stage-Dependent Framework with Non-Markov Constraints for Individual Cell and Pathogen Dynamics

We present a systematic approach to modelling the responses of the immune system to virus infections. Two continuous-discrete stochastic models arising in mathematical immunology are developed and computationally implemented. The variables of the models are integer random variables that denote the quantity of individuals (cells and viral particles), and sets of unique types of individuals that take into account the current state and history of stay of individuals in some stages of their development. The distribution laws of the durations of the mentioned stages are different from exponential or geometric. A probabilistic description of a one-stage stochastic model of population dynamics is presented. A stochastic model of the development of HIV-1 infection in the lymph node in the initial period after infection of a healthy person is formulated. A computational algorithm based on the Monte Carlo method is given. Each of the stochastic models is complemented by a deterministic analogue in the form of integral and delay differential equations. The results of numerical simulation are presented.

On measure of noncompactness in variable exponent Lebesgue spaces and applications to integral equations

A novel measure of noncompactness is defined in variable exponent Lebesgue spaces Lp(⋅)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{p(\\cdot )}$\\end{document} on an unbounded domain R+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{+}$\\end{document} and its properties are examined. Using the fixed point method, we apply that measure to study the existence theorem for nonlinear integral equations. Our results can be handily applied in studying various types of (differential, integral, functional, and partial differential) equations in Lp(⋅)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{p(\\cdot )}$\\end{document}-spaces. The Lp(⋅)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{p(\\cdot )} $\\end{document}-spaces are natural extensions of classical constant exponent Lebesgue spaces Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L_{p}$\\end{document}, which allows us to bypass several restrictions that were previously discussed in the literature.

Application of Adomian Decomposition Method for Solving Helmholtz Equation

The Adomian decomposition method (ADM) is effective for solving linear and nonlinear ordinary and partial differential equations, integral equations, algebraic equations and integro-differential equations and it leads to an analytical solution in the form of an infinite power series. We will solve the Helmholtz equation in two dimensional by the Adomian decomposition method.
 Key words: Application of Adomian Decomposition Method, Helmholtz Equation, Adomian Decomposition Method. 

Liouville integrable binomial Hamiltonian system

In this study, we work on a novel Hamiltonian system which is Liouville integrable. In the integrable Hamiltonian model, conserved currents can be represented as Binomial polynomials in which each order corresponds to the integral of motion of the system. From a mathematical point of view, the equations of motion can be written as integrable second-order nonlinear partial differential equations in 1 + 1 dimensions.

Existence results for some stochastic functional integrodifferential systems driven by Rosenblatt process

Abstract This work investigates the existence and uniqueness of mild solutions to a class of stochastic integral differential equations with various time delay driven by the Rosenblatt process. We can obtain alternative conditions that guarantee mild solutions by using the resolvent operator in the Grimmer sense, stochastic analysis, fixed-point methods, and noncompact measures. We give an example to illustrate the theory.

A peridynamics approach to flexible multibody dynamics for fracture analysis of mechanical systems

The classical theory of continuum mechanics is formulated using partial differential equations (PDEs) that fail to describe structural discontinuities, such as cracks. This limitation motivated the development of peridynamics, reformulating the classical PDEs into integral-differential equations. In this theory, each material point interacts with its neighbours inside a characteristic length-scale through bond-interaction forces. However, while peridynamics can simulate complex multi-physics phenomena, its integration in the study of mechanical systems is still limited. This work presents a methodology that incorporates a peridynamics formulation into a planar multibody dynamics (MBD) formulation to allow the integration of flexible structures described by peridynamics into mechanical systems. A flexible body is described by a collection of point masses, in analogy with the meshless collocation scheme commonly used for peridynamics discretisations. Each point mass interacts with other point masses through nonlinear forces governed by a bond-based peridynamics (BBPD) formulation. The virtual bodies methodology enables the definition of kinematic joints connecting the flexible body with the neighbouring bodies. The implementation of the methodology proposed is illustrated using various mechanisms with different levels of complexity. Notched plates subjected to different loading conditions are compared with the results presented in the literature of the peridynamics field. The deformations of a flexible slider-crank mechanism compare well with the results obtained using a classical flexible MBD formulation. Additionally, three scenarios involving a rotating pendulum illustrate how the methodology proposed allows simulating impact scenarios. The results demonstrate how this methodology is capable to successfully simulate highly nonlinear phenomena, including crack propagation, in a multibody framework.

Distributed model for the drill-string system with multiple regenerative effects in the bit-rock interaction

In this paper, an integrated model is presented to study the axial–torsional dynamics of a distributed drill string. The coupled axial and torsional drill string dynamics typically lead to severe phenomena such as bit-bounce and stick–slip, which are the main causes of failure and efficiency reduction. The top-drive dynamics and the Bottom Hole Assembly (BHA) dynamics are interconnected through the drill pipes. In order to preserve the infinite-dimensional feature of the drill pipes in the mathematical modeling, a distributed model in terms of Delay Differential Algebraic Equations (DDAE) is proposed to represent the dynamics of the drill string. The bit-rock interaction, which consists of the cutting and the frictional components, is responsible for the coupling between the axial and torsional dynamics. A rate-independent bit-rock interaction law is employed for both cutting and frictional components. In order to capture the global bit motion, including multiple regenerative effects, the well-bottom depth function is used to determine the depth of cut. The well-bottom pattern evolution is formulated through algebraic equations rather than Partial Differential Equations (PDEs). A new formulation for the linearized drill string system in terms of Neutral-type Delay Differential Integral Equations (NDDIEs) is proposed to investigate the initiation of the oscillations around the nominal operation. By this formulation, the local stability of the distributed system is studied by determining the right-most eigenvalue of the linearized dynamics. A simulation-based case study reflecting real-life scenarios is presented to investigate the dynamical behavior of the system under different operating conditions.

Validity of Galerkin Method at Beam’s Nonlinear Vibrations of the Single Mode with the Initial Curvature

A common strategy for studying the nonlinear vibrations of beams is to discretize the nonlinear partial differential equation into a nonlinear ordinary differential equation or equations through the Galerkin method. Then, the oscillations of beams are explored by solving the ordinary differential equation or equations. However, recent studies have shown that this strategy may lead to erroneous results in some cases. The present paper carried out the following three research studies: (1) We performed Galerkin first-order and second-order truncations to discrete the nonlinear partial differential integral equation that describes the vibrations of a Bernoulli-Euler beam with initial curvatures. (2) The approximate analytical solutions of the discretized ordinary differential equations were obtained through the multiple scales method for the primary resonance. (3) We compared the analytical solutions with those of the finite element method. Based on the results obtained by the two methods, we found that the Galerkin method can accurately estimate the dynamic behaviors of beams without initial curvatures. On the contrary, the Galerkin method underestimates the softening effect of the quadratic nonlinear term that is induced by the initial curvature. This may cause erroneous results when the Galerkin method is used to study the dynamic behaviors of beams with the initial curvatures.

Fluctuation analysis for particle-based stochastic reaction–diffusion models

Recent works have derived and proven the large-population mean-field limit for several classes of particle-based stochastic reaction–diffusion (PBSRD) models. These limits correspond to systems of partial integral–differential equations (PIDEs) that generalize standard mass-action reaction–diffusion PDE models. In this work we derive and prove the next order fluctuation corrections to such limits, which we show satisfy systems of stochastic PIDEs with Gaussian noise. Numerical examples are presented to illustrate how including the fluctuation corrections can enable the accurate estimation of higher order statistics of the underlying PBSRD model.

Higher-Order Benjamin–Ono Model for Ocean Internal Solitary Waves and Its Related Properties

In this study, the propagation of internal solitary waves in oceans at great depths was analyzed. Using multi-scale analysis and perturbation expansion, the basic equation is simplified to the classical Benjamin–Ono equation with variable coefficients. To better describe the propagation characteristics of solitary waves, we derived a higher-order variable-coefficient integral differential (Benjamin–Ono) equation. Subsequently, the bilinear form of the model was derived using Hirota’s bilinear method, and a multi-soliton solution was obtained. Based on the multi-soliton solution of the model, we further studied the interaction of the soliton, which led to the discovery of Mach reflection. Some conclusions were drawn, which are of potential value for further study of solitary waves in the ocean.

Mathematical model of thermal phenomena of closure electrical contact with Joule heat source and nonlinear thermal coefficients

The mathematical model describing the dynamics of closed contact heating which involves vaporization of the metal when instantaneous explosion of micro-asperity occurs is presented through a Stefan type problem. The temperature field for metallic vaporization zone is introduced as heat resistance that decreases linearity. Temperature fields for liquid and solid phases of the metal described by spherical heat equations with nonlinear thermal coefficients and Joule heat source have to be determined as well as the free boundaries. Joule heating component depends on space and time variable when alternating current is considered. Solution method of the problem based on similarity variable transformation is applied which enables us to reduce the problem to an ordinary differential equations and nonlinear integral equations. Existence and uniqueness of the integral equations are proved by using fixed point theorem in Banach space.

A class of improved fractional physics informed neural networks

Physics informed neural networks (PINNs) are a type of deep learning framework which can use the knowledge of physical laws to facilitate the learning of various forward and inverse problems. PINNs have displayed significant efficiency in numerical computation, but there are still some limitations such as their inability in tackling differential equations with oscillations or singular perturbation. To alleviate this difficulty, a new deep learning framework called improved fractional physics informed neural networks (IFPINNs) is proposed in this paper. Compared to PINNs, the calculation in the weight propagation between layers is replaced with nonlinear functions rather than linear ones. This new architecture allows for a better usage of physical information of differential equations, thus leading to an improved fitting quality for the cases with high oscillations or singular perturbation. Under certain conditions, IFPINNs can be degenerated into PINNs. Moreover, this paper investigates the performance of IFPINNs in dealing with fractional differential equations with time-fractional orders ranging from 1 to 2. The numerical results demonstrate the proposed architecture can achieve a higher accuracy for various integral and fractional differential equations.

On some hypergeometric Sobolev orthogonal polynomials with several continuous parameters

In this paper we study the following hypergeometric polynomials: $$ \mathcal{P}_n(x) = \mathcal{P}_n(x;\alpha,\beta,\delta_1, \dots,\delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$ $$ = {}_{\rho+2} F_{\rho+1} (-n,n+\alpha+\beta+1,\delta_1+1, \dots,\delta_\rho+1;\alpha+1,\kappa_1+\delta_1+1, \dots,\kappa_\rho+\delta_\rho+1;x), $$ and $$ \mathcal{L}_n(x) = \mathcal{L}_n(x;\alpha,\delta_1,\dots, \delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$ $$ = {}_{\rho+1} F_{\rho+1} (-n,\delta_1+1,\dots,\delta_\rho+1; \alpha+1,\kappa_1+\delta_1+1,\dots,\kappa_\rho+\delta_\rho+1;x), \qquad n\in\mathbb{Z}_+, $$ where $\alpha,\beta,\delta_1,\dots,\delta_\rho\in(-1,+\infty)$, and $\kappa_1,\dots,\kappa_\rho\in\mathbb{Z}_+$, are some parameters. The natural number $\rho$ of the continuous parameters $\delta_1,\dots,\delta_\rho$ can be chosen arbitrarily large. It is seen that the special case $\kappa_1=\dots=\kappa_\rho=0$ leads to Jacobi and Laguerre orthogonal polynomials. Of course, such polynomials and more general ones appeared in the literature earlier. Our aim here is to show that polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$ are Sobolev orthogonal polynomials on the real line with some explicit matrices of measures. The importance of the orthogonality property was our main reason to concentrate our attention on polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$. Here we shall use some our tools developed earlier. In particular, it was shown recently that Sobolev orthogonal polynomials are related by a differential equation with orthogonal systems $\mathcal{A}$ of functions acting in the direct sums of usual $L^2_\mu$ spaces of square-summable (classes of the equivalence of) functions with respect to a positive measure $\mu$. The case of a unique $L^2_\mu$ is of a special interest, since it allows to use OPRL to obtain explicit systems of Sobolev orthogonal polynomials. The main problem here is \textit{to choose a suitable linear differential operator in order to get explicit representations for Sobolev orthogonal polynomials}. The proof of the orthogonality relations is then a verification of such a choice and it goes in another direction: we start from the already known polynomials to their properties. We also study briefly such properties of the above polynomials: integral representations, differential equations and location of zeros. A system of such polynomials with a kind of the bispectrality property is constructed.

Stochastic maximum principle for moving average control system

AbstractIn this paper, we consider the stochastic optimal control problem for moving average control system. The corresponding moving average stochastic differential equation is a kind of integral differential equations. We prove the existence and uniqueness of the solution of the moving average stochastic differential equations. We obtain the stochastic maximum principle of the moving average optimal control system by introducing a kind of generalized anticipated backward stochastic differential equations. We prove the existence and uniqueness of the solution of this adjoint equation, which is singular at 0. As an application, the linear quadratic moving average control problem is investigated to illustrate the main results.

Analysis of soil behavior under freezing and thawing conditions

Understanding of the thermal, hydraulic and mechanical behavior of soils under freezing and thawing is important in solving problems of permafrost engineering and artificial ground freezing. Accurate mathematical descriptions of heat and moisture transport and the associated mechanical deformations is critical for developing such understanding. Phase change and transition during freezing and thawing are characterized by a strong non-linearity due to the rapid variations of thermo-physical properties of the material and the release of the latent heat of solidification. Such processes cannot be easily and accurately modelled via the use of local (differential) formulations [1,2]. A non-local formulation, developed within the framework of peridynamics [3–5] is proposed here for ground freezing and thawing. It expands the domain of applicability of the first work on non-local modelling of heat transfer with water flow in saturated porous media [6] into modelling the thermo-hydro-mechanical behavior of unsaturated soils.
 The mathematical description of unsaturated soil behavior is based on three governing equations of conservations of water mass, energy and linear momentum, which are complemented by constitutive relationships that represent the change of soil properties.
 Using the local mathematical model of unsaturated soils [7,8], a peridynamics mathematical framework is formulated. Following [6], the bond-based peridynamics considers a soil body as a collection of particles, see Fig. 1 for the 1D case. The term `bond' refers to the interactions between two points, located at depths and . The point interacts with all other points within a certain finite region , called the horizon of z.
 The peridynamic form of the water mass conservation is:
 where is the water pressure head; is time; is the volumetric content of a soil component; is the density of a soil component; and is the vertical coordinate. The micro-diffusivity function can be defined following [9]. Here, the subscripts i, l, a and s refer to ice, liquid water, air and soil particles respectively.
 The peridynamic form of the energy conservation is:
 where is temperature; is the latent heat of water solidification; is the average thermal conductivity of soil; is the vector of water flux; is the average heat capacity of unsaturated soils and is the specific heat capacity of liquid water. Here, is a micro-conductivity function and is a micro-velocity function, for more details see [6,10].
 The peridynamic form of the linear momentum conservation is:
 where is the acceleration of the peridynamics particle ; is a pairwise force density vector between particles and that depends on its initial and deformed positions Z and Z’; is the horizon volume associated with the particle ; is the body force on the particle . The elastic stress-strain relation for unsaturated soils can be stated as:
 where is the stress increment tensor; is soil compressive modulus for unsaturated conditions, and is the strain increment tensor due to the elastic deformation of the soil matrix, that can be find from the relation for the total strain increment tensor of unsaturated soils . Here is the strain increment tensor corresponding to the change of soil volume due to the frost heave; and is the strain increment tensor corresponding to the plastic deformations of soils.
 The numerical implementation of the peridynamic formulation is verified by comparing the results of simulations with analytical solutions and the formulation is validated by comparing these results with experimental data under the condition of one-dimensional heat and water transfer obtained by laboratory experiments. In the experiments, soil cylinders were cooled from one surface by a source of negative temperature. As a result, the freezing front propagated along the cylinder. The temperature profile in the column was changing, and water migration was observed. The developed model is shown to predict the results of these experiments well.
 The proposed peridynamic model is the first to combine the flow of liquid water with heat transfer and mechanical deformations in the presence of a change of the water phase state in unsaturated soils. It is demonstrated that for the soils with relatively high initial water content, the developed approach describes effectively the change of liquid water content and predict soil deformation. The set of integral-differential equations are solved by a simple numerical scheme.

ელექტროგამტარ სითხეში წრიული ფოროვანი ფირფიტის ბრუნვის სტაციონარული ამოცანა გაჟონვის სიჩქარის დიდი მნიშვნელობებისათვის თბოგადაცემით, მაგნიტური ველისა და ჯოულის სითბოს გათვალისწინებით

The stationary problem of rotation of a circular porous plate in an electrically conductive fluid is studied by the sequential approximation method (Green's function and small parameter method) for large values of the suction velocity with heat transfer, taking into account a weak magnetic field. The problem considers the case when the influence of dissipative members on heat transfer is very small, the energy equation takes into account the influence of Joule's heat on heat transfer, and it is assumed that the temperature along with the radius of the plate changes according to the square law. For solving the problem, the Navier-Stokes and energy partial differential equations of fluid motion in a magnetic field are reduced to ordinary nonlinear differential equations using generalized Karman embeddings, the solutions of which are sought in the form of infinite series for small values of the suction parameter. The solution of the problem by the means of Green's function is reduced to the solution of integral-differential equations. Recurrence formulas have been obtained, by the means of which it is possible to calculate the solution with any approximation. The first two approximations are clearly calculated. All kinematic characteristics of fluid flow are calculated. Images for heat transfer and pressure distribution on a circular plate are obtained. The moment of resistance to rotation of the plate and the heat transfer coefficient are also calculated.