scholarly journals Extended Stokes' Problems for Relatively Moving Porous Half-Planes

2009 ◽  
Vol 2009 ◽  
pp. 1-10 ◽  
Author(s):  
Chi-Min Liu
Keyword(s):  
Exact Solutions ◽  
Integral Transform ◽  
Initial Conditions ◽  
Integral Transforms ◽  
Momentum Equation ◽  
Transform Method ◽  
Stokes Problems ◽  
Flow Phenomena ◽  
Mass Influx ◽  
Mathematical Techniques

A shear flow motivated by relatively moving half-planes is theoretically studied in this paper. Either the mass influx or the mass efflux is allowed on the boundary. This flow is called the extended Stokes' problems. Traditionally, exact solutions to the Stokes' problems can be readily obtained by directly applying the integral transforms to the momentum equation and the associated boundary and initial conditions. However, it fails to solve the extended Stokes' problems by using the integral-transform method only. The reason for this difficulty is that the inverse transform cannot be reduced to a simpler form. To this end, several crucial mathematical techniques have to be involved together with the integral transforms to acquire the exact solutions. Moreover, new dimensionless parameters are defined to describe the flow phenomena more clearly. On the basis of the exact solutions derived in this paper, it is found that the mass influx on the boundary hastens the development of the flow, and the mass efflux retards the energy transferred from the plate to the far-field fluid.

Exact Solutions on Mixed Convection Flow of Accelerated Non-Coaxial Rotation of MHD Viscous Fluid with Porosity Effect

2020 ◽  
Vol 399 ◽  
pp. 26-37
Author(s):  
Ahmad Qushairi Mohamad ◽  
Zulkhibri Ismail ◽  
Nur Fatihah Mod Omar ◽  
Muhammad Qasim ◽  
Muhamad Najib Zakaria ◽  
...  
Keyword(s):  
Viscous Fluid ◽  
Mixed Convection ◽  
Exact Solutions ◽  
Initial Conditions ◽  
Convection Flow ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Porosity Effect ◽  
Prandtl Numbers ◽  
Physical Boundary

Mixed convection of unsteady non-coaxial rotation flow of viscous fluid over an accelerated vertical disk is investigated. The motion in the fluid is induced due to the rotating and buoyancy force effects. The problem is formulated and extended in terms of coupled partial differential equations with some physical boundary and initial conditions. The non-dimensional equations of the problem are obtained by using the suitable non-dimensional variables. The exact solutions of non-coaxial velocity and temperature profiles are obtained by using Laplace transform method which are satisfying all the initial and boundary conditions. The physical significance of the mathematical results is shown in various plots and is discussed for Grashof and Prandtl numbers as well as magnetic, porosity and accelerated parameters. It is found that, the velocity with the effect of acceleration is higher compared to constant velocity. In limiting sense, the present solutions are found identical with published results.

New Straightforward Benchmark Solutions for Bending and Free Vibration of Clamped Anisotropic Rectangular Thin Plates

2021 ◽  
pp. 1-24
Author(s):  
Dongqi An ◽  
Zhuofan Ni ◽  
Dian Xu ◽  
Rui Li
Keyword(s):  
Free Vibration ◽  
Numerical Solutions ◽  
Integral Transform ◽  
Integral Transforms ◽  
Thin Plates ◽  
Orthotropic Plates ◽  
Transform Method ◽  
Finite Integral ◽  
Benchmark Solutions ◽  
Similar Complex

Abstract This study presents new straightforward benchmark solutions for bending and free vibration of clamped anisotropic rectangular thin plates by a double finite integral transform method. Being different from the previous studies that took pure trigonometric functions as the integral kernels, the exponential functions are adopted, and the unknowns to be determined are constituted after the integral transform, which overcomes the difficulty in solving the governing higher-order partial differential equations with odd derivatives with respect to both the in-plane coordinate variables, thus goes beyond the limit of conventional finite integral transforms that are only applicable to isotropic or orthotropic plates. The present study provides an easy-to-implement approach for similar complex problems, extending the scope of finite integral transforms with applications to plate problems. The validity of the method and accuracy of the new solutions that can serve as benchmarks are well confirmed by satisfactory comparison with the numerical solutions.

scholarly journals Complete Solutions to Extended Stokes' Problems

2008 ◽  
Vol 2008 ◽  
pp. 1-18 ◽  
Author(s):  
Chi-Min Liu
Keyword(s):  
Viscous Flow ◽  
Fluid Motion ◽  
Flow Characteristics ◽  
Integral Transforms ◽  
Transient Solutions ◽  
Stokes Problems ◽  
Moving Plane ◽  
Mathematical Techniques ◽  
Oscillating Motion ◽  
Complete Solutions

The main object of the present study is to theoretically solve the viscous flow of either a finite or infinite depth, which is driven by moving plane(s). Such a viscous flow is usually named as Stokes' first or second problems, which indicates the fluid motion driven by the impulsive or oscillating motion of the boundary, respectively. Traditional Stokes' problems are firstly revisited, and three extended problems are subsequently examined. Using some mathematical techniques and integral transforms, complete solutions which can exactly capture the flow characteristics at any time are derived. The corresponding steady-state and transient solutions are readily determined on the basis of complete solutions. Current results have wide applications in academic researches and are of significance for future studies taking more boundary conditions and non-Newtonian fluids into account.

An integral-transform method for shock–shock interaction studies

1968 ◽  
Vol 34 (2) ◽  
pp. 209-228 ◽  
Author(s):  
N. L. Arora
Keyword(s):  
Analytic Solution ◽  
Integral Transform ◽  
Strong Shock ◽  
Integral Transforms ◽  
Shock Interaction ◽  
Transform Method ◽  
Pressure Distributions ◽  
Small Incidence ◽  
Slender Bodies ◽  
Strong Shock Waves

An analytic solution is obtained for the head-on collision of strong shock waves with supersonically moving, axisymmetric, slender bodies. A new method of the solution is developed using integral transforms. Pressure distributions on the surface of a cone illustrate the effects of shock strength and body speed. The results are compared with those of Blankenship (1965), which were obtained by numerical methods.It is also shown that the same analysis can, with certain modifications, be adapted for treating the shock-shock interaction on thin two-dimensional aerofoils of arbitrary shape at small incidence.

scholarly journals Numerical evaluation of inverse integral transforms: Dynamic response of elastic materials

2020 ◽  
Vol 12 (2) ◽  
pp. 29-34
Author(s):  
M.D. Sharma ◽  
S. Nain
Keyword(s):  
Numerical Integration ◽  
Elastic Waves ◽  
Fourier Transforms ◽  
Integral Transform ◽  
Integral Transforms ◽  
Inverse Laplace Transform ◽  
Branch Points ◽  
Transform Method ◽  
Improper Integral ◽  
Romberg Integration

This study discusses the use of numerical integration in evaluating the improper integrals appearing as inverse integral transforms of non-analytic functions. These transforms appear while studying the response of various sources in an elastic medium through integral transform method. In these studies, the inverse Fourier transforms are solved numerically without bothering about the singularities and branch points in the corresponding integrands. References on numerical integration cited in relevant papers do not support such an evaluation but suggest contrary. Approximation of inverse Laplace transform integral into a series is used without following the essential restrictions and assumptions. Volume of the published papers using these dubious procedures has reached to an alarming level. The discussion presented aims to draw the attention of researchers as well as journals so as to stop this menace at the earliest possible.Keywords: Inverse Fourier transforms, inverse Laplace transforms, Romberg integration, improper integral, elastic waves

scholarly journals Hybrid integral transforms for flow development in ducts partially filled with porous media

2018 ◽  
Vol 474 (2209) ◽  
pp. 20170637 ◽  
Author(s):  
Kleber Marques Lisboa ◽  
Renato Machado Cotta
Keyword(s):  
Porous Media ◽  
Error Control ◽  
Heterogeneous Media ◽  
Convergence Rates ◽  
Integral Transform ◽  
Integral Transforms ◽  
Numerical Code ◽  
Physical Analysis ◽  
Transform Method ◽  
Flow Development

A hybrid numerical–analytical solution is developed for laminar flow development in a parallel plate duct partially filled with porous media. The integral transform method is employed in combination with a single domain reformulation strategy for representing the heterogeneous media within the channel. A novel eigenfunction expansion basis is proposed, including abrupt spatial variations of physical properties due to the domain transitions. The introduction of the new basis allows for a solution with similar convergence rates as in previous applications with simpler formulations, as demonstrated through a careful convergence analysis of the expansions. The inherent automatic error control characteristic of the integral transforms approach then provides benchmark results for the developing velocity profile. Moreover, a physical analysis further verifies the consistency of both the proposed expansion and the mixed symbolic–numerical code developed. A detailed verification with a finite-element commercial code is also performed.

scholarly journals Fractional Order Airy’s Type Differential Equations of Its Models Using RDTM

2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Daba Meshesha Gusu ◽  
Dechasa Wegi ◽  
Girma Gemechu ◽  
Diriba Gemechu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Transform Method ◽  
3 Dimensional

In this paper, we propose a novel reduced differential transform method (RDTM) to compute analytical and semianalytical approximate solutions of fractional order Airy’s ordinary differential equations and fractional order Airy’s and Airy’s type partial differential equations subjected to certain initial conditions. The performance of the proposed method was analyzed and compared with a convergent series solution form with easily computable coefficients. The behavior of approximated series solutions at different values of fractional order α and its modeling in 2-dimensional and 3-dimensional spaces are compared with exact solutions using MATLAB graphical method analysis. Moreover, the physical and geometrical interpretations of the computed graphs are given in detail within 2- and 3-dimensional spaces. Accordingly, the obtained approximate solutions of fractional order Airy’s ordinary differential equations and fractional order Airy’s and Airy’s type partial differential equations subjected to certain initial conditions exactly fit with exact solutions. Hence, the proposed method reveals reliability, effectiveness, efficiency, and strengthening of computed mathematical results in order to easily solve fractional order Airy’s type differential equations.

NUMERICAL AND STABILITY ANALYSIS OF THE TRANSMISSION DYNAMICS OF SVIR EPIDEMIC MODEL WITH STANDARD INCIDENCE RATE

2019 ◽  
Vol 4 (2) ◽  
pp. 349 ◽  
Author(s):  
Oluwatayo Michael Ogunmiloro ◽  
Fatima Ohunene Abedo ◽  
Hammed Kareem
Keyword(s):  
Reproduction Number ◽  
Disease Spread ◽  
Asymptotically Stable ◽  
Equilibrium Solutions ◽  
Transform Method ◽  
Globally Asymptotically Stable ◽  
Differential Transform ◽  
Mathematical Techniques ◽  
Fourth Order Method ◽  
Theoretical Results

In this article, a Susceptible – Vaccinated – Infected – Recovered (SVIR) model is formulated and analysed using comprehensive mathematical techniques. The vaccination class is primarily considered as means of controlling the disease spread. The basic reproduction number (Ro) of the model is obtained, where it was shown that if Ro<1, at the model equilibrium solutions when infection is present and absent, the infection- free equilibrium is both locally and globally asymptotically stable. Also, if Ro>1, the endemic equilibrium solution is locally asymptotically stable. Furthermore, the analytical solution of the model was carried out using the Differential Transform Method (DTM) and Runge - Kutta fourth-order method. Numerical simulations were carried out to validate the theoretical results. 

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