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This paper presents the method of effectiveness NTU for bayonet tube heat exchangers thermal analysis, with uniform conditions along the outer tube wall surface. Using bayonet tube control volume's energy balance, linear fluid temperatures differential equations and boundary conditions are derived at steady state conditions. The governing equations and boundary conditions are expressed in non-dimensional form. The temperature is found to be dependent on four parameters Hurd number (Hu), ratio of outer tube convective heat transfer coefficient(ξ,), number of transfer unit (NTU), and flow path. The method of fourth order Runge- Kutta was used for numerical integration of the coupled linear temperature differential equations and the results are presented graphically for selected values of Hu,ξ, NTU and flow path. The effectiveness is obtained to be dependent on NTU and fluid temperature at the shell side. It was observed from the tubes temperature distribution at a low Hu value, that little heat is exchanged within the tubes. Heat transfer to shell side fluid increases with an increase in ξ value. Maximum heat transfer is achieved by reverse flow path B. [ABSTRACT FROM AUTHOR]

In this paper, the temperature distribution of porous fins is approached by the Akbari-Ganji method using a simple analytical technique. The Darcy model and energy balance were also utilised to create the heat transfer equation. An insulated tip and a limited-length fin serve as a case for exploring thermal performance. The outcomes of the Akbari-Ganji method are compared with previous results. The boundary value is also solved numerically using the MATLAB programme for validation. The outcome indicates that the Akbari-Ganji is more reliable, appropriate, and accurate than other approaches like the perturbation method, the homotopy perturbation method, and the variational iteration method. Additionally, it has been discovered that this approach is a potent mathematical instrument that may solve a wide range of non-linear and linear issues that arise in various branches of technology and science, particularly some thermal transfer equations. It is demonstrated that for solving boundary value problems in engineering applications, the AGM approach is a superior substitute to semi-analytical and conventional numerical methods, mainly when the result's efficiency is crucial. [ABSTRACT FROM AUTHOR]

Integral transforms have been used to solve certain initial and boundary value problems. Fractional Hilbert transform, a generalization of Hilbert transform has relationship with the other classical transforms and fractional transforms. In this paper, we have derived relationship between fractional Hilbert transform with Fourier transform and Fractional Fourier Transform. [ABSTRACT FROM AUTHOR]

This work proposes multistep technique for solving the nonlinear differential equations along with boundary conditions and deals with solution of the problem and stability region analysis of such methods. Some non linear differential equations are illustrated to show the efficacy of the method discussed in this paper. Stability Regions have been analyzed. The errors obtained by the methods discussed are compared to the exact solutions. Thus, the numerical solutions obtained by these methods have high accuracy. [ABSTRACT FROM AUTHOR]

This paper proposes a non-classical theory of stressed plates. Resolving equations are obtained in terms of longitudinal forces of shear forces and internal moments, and boundary conditions are formulated. The obtained solutions simultaneously satisfy the equilibrium equations and the Beltrami-Mitchell stress compatibility equations. A boundary value problem is formulated for a plate under the action of mutually balanced loads. Some applied problems have been solved. [ABSTRACT FROM AUTHOR]

Usually, the complete system of thermoelasticity equations consists of the equation of motion, the Dugamel-Neumann relations, the Cauchy relation, and the heat influx equation. In this case, usually the boundary value problem is reduced to a system of differential equations for displacements and temperature with the corresponding initial and boundary conditions. Coupled thermoelasticity problems can be formulated in both stress and temperature, but this requires the second order of smoothness of the stress tensor. In this regard, there is an additional condition regarding stress. In this paper, the coupled problems of thermoelasticity are proposed to be considered on the basis of the equations of motion, the continuity equation, the Dugamel-Neumann relation, and the heat influx equation. In this case, a system of three interrelated equations for stress, velocity and temperature was obtained. Based on this system, the problem of the stress-strain state of a homogeneous semi-infinite thermoelastic rod under the action of a dynamic thermomechanical load applied to the end of the rod is considered. The exact analytical solution was obtained by the continuation method followed by the application of the Laplace and Fourier integral transformations. The asymptotic behavior of the stress is studied for small and large values of time. The propagation velocity of a thermoelastic wave has also been studied. Numerical calculations are carried out, the qualitative and quantitative nature of the mutual influence of temperature on the distribution of stresses and velocities of the rod sections is analyzed. [ABSTRACT FROM AUTHOR]

During the past decade, our society has become dependent on advanced mathematics for many of our daily needs. Mathematics is at the heart of the 21st century technologies and more specifically the emerging imaging technologies from thermoacoustic tomography and ultrasound computed tomography to nondestructive testing. All of these applications reconstruct the internal structure of an object from external measurements without damaging the entity under investigation. Very often the basic mathematical idea common to such reconstruction problems is based upon integral geometry. In this paper considers the problem of recovering a function from families of spheres in space. The uniqueness of the solution of the problem is proved by reducing it to the Volterra integral equation of the first and then the second kind. The methods of the theory of partial differential equations are applied. The proof of the uniqueness theorem is based on the researching of boundary value problems for auxiliary functions. Fourier transform methods are also used. Uniqueness theorems are proved for some new classes of operator equations of Volterra type in three-dimensional space. [ABSTRACT FROM AUTHOR]

In this research paper developed a three-dimensional mathematical model of the process of gas filtration in porous media in the presence of a mass transfer through the boundaries and a numerical algorithm for solving the stated initial-boundary value problem. The developed model is described using a non-linear partial differential equation with appropriate initial and boundary conditions. The proposed mathematical apparatus makes it possible to carry out hydrodynamic calculations, taking into account changes in the main factors affecting the process under consideration: permeability, porosity and thickness of formations, gas recovery coefficient, viscosity, etc. A numerical algorithm has been developed to solve the problem based on difference schemes of the second order of approximation in time and coordinates, which provide the ability to solve practical problems of analyzing and predicting the gas production process under various conditions of impact on the productive reservoir, as well as making decisions on the development of existing and design of new gas fields. Meanwhile, the use of three-dimensional models makes it possible to more accurately describe the processes occurring in reservoir porous media. Therefore, the purpose of this study is to develop a three-dimensional mathematical model of gas filtration in porous media in the presence of a mass transfer through the boundaries, as well as a numerical algorithm for solving the problem. [ABSTRACT FROM AUTHOR]

The paper is devoted to solving the Neumann boundary value problem for the Poisson equation in an elliptic body. In this case, heat transfer occurs under boundary conditions of the second kind. Based on the methods of differentiation and integration, a solution was obtained to the problem of the distribution of the temperature field of the body under study. The resulting solution has an analytical form containing hypergeometric and hyperbolic functions. The reliability of the obtained result is confirmed by the fact that the general solution of the problem coincides with the solution obtained in one of the author's works for the case of a temperature field distribution in a body with an elliptical cross section of infinite length under boundary conditions of the third kind. [ABSTRACT FROM AUTHOR]

The paper considers a model problem of the thermal diffusion process in a silicon wafer. The mathematical model of this process is an initial boundary value problem for a system of linear partial differential equations of parabolic type. In this system, one equation describes the process of heat propagation in silicon in the presence of internal heat sources, and the other describes the process of diffusion of impurities in it. Moreover, these equations are related in the same way that the diffusion coefficient of an impurity depends on temperature. A special form of the volumetric heat release function was considered, which made it possible in this particular case to write down an explicit solution in the form of a Fourier series. To find an approximate solution to the boundary value problem, an implicit difference scheme and the classical sweep method are used. A computational experiment is presented. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to introduce interval two-points fuzzy boundary value problems, in which the fuzziness course when the coefficients of the governing ordinary differential equation and/or the boundary conditions includes fuzzy numbers of either triangular or trapezoidal types. Such equations will be solved by introducing the concept of α – level sets, α ∈ [0,1] to treat the fuzzy ordinary differential equation into two nonfuzzy ordinary differential equations, which are corresponds to the lower and upper solutions of the interval fuzzy solution. The shooting method is applied to solve the resultant equations in both cases, linear and nonlinear. [ABSTRACT FROM AUTHOR]

In this paper, we consider the construction of an optimal lattice quadrature formula in the space W 2 (m) (0, 1). Further, finding the extremal function of the error functional of quadrature formulas, by virtue of the Riesz theorem on the general form of a linear continuous functional, is reduced to solving a boundary value problem for ordinary differential equations. Solving this boundary value problem, we obtain an explicit expression for the extremal function of the error functional of quadrature formulas. In addition, by virtue of the Riesz theorem, the square of the norm of the error functional is explicitly calculated and an analytical expression is obtained for the optimal coefficients of the quadrature formulas at m = 2. [ABSTRACT FROM AUTHOR]

This paper uses the Ritz method to find approximate solutions to boundary-value problems for ordinary differential equations. We present the approximate finite element solution by a linear combination of the basis functions. Here we get co-efficients of the optimal interpolation formula in W 2 (1 , 0) space as basis functions. The difference obtained in numerical results and exact solutions illustrates graphically. [ABSTRACT FROM AUTHOR]

In this paper, we study nonlocal boundary value problems for a third-order equation with an elliptic-hyperbolic operator in the principal part. Existence and uniqueness theorems for the classical solution of the stated problems are proved. The proofs of the theorem are based on energy identities, as well as on the theory of singular and Fredholm integral equations. [ABSTRACT FROM AUTHOR]

In this paper we will discuss the theoretical analysis for Bickley's method and numerical procedure for getting the approximate solution. If (sm, ck) denote the class of spline functions consisting of piecewise polynomial of degree m, joined at "knots" with at-least k continuous derivatives (k < m) then spline functions in (Sm, cm-1) are used as approximating function for two point boundary value problem in the second order differential equations. The result shows that the presented cubic spline function which interpolates the lacunary data was efficient and effective for solving such problems. [ABSTRACT FROM AUTHOR]