Solution of the problem of the theoretical profile of non-dimensional speed on the thickness of the boundary layer at the turbulent flow in the boundary layer based on the solution of the differential equation of Abel of the second generation with the application of the Lambert function

2018 ◽  
pp. 43-51
Author(s):  
И.Е. ЛОБАНОВ
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Turbulent Flow ◽  
Exact Analytical Solution ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Lambert Function ◽  
Theoretical Profile ◽  
Special Cases ◽  
Abel Differential Equation

В статье было найдено точное аналитическое решение дифференциального уравнения для касательных напряжений в турбулентном пограничном слое, являющихся частным случаем т.н. дифференциального уравнения Абеля второго рода, полученное с помощью специальной функции Ламберта, в то время как ранее считалось, что оно не разрешимо в квадратурах. Кроме этого, были получены ещё несколько важных решённых частных случаев этого уравнения. Полученные в статье аналитические решения преимущественно отличаются от имеющихся ранее либо численных, либо приближённых решений задачи. Полученное решение в безразмерном виде представляет собой теоретический профиль безразмерной скорости по толщине пограничного слоя при турбулентном течении в пограничном слое. An exact analytical solution of the differential equation for tangential stresses in a turbulent boundary layer, which is a special case of the so-called " of the Abel differential equation of the second kind, obtained with the help of the special Lambert function, whereas previously it was assumed that it is not solvable in quadratures. In addition, several more important solved special cases of this equation were obtained. The analytic solutions obtained in the paper are predominantly different from the previously available either numerical or approximate solutions of the problem. The solution obtained in dimensionless form is the theoretical profile of the dimensionless velocity along the thickness of the boundary layer for turbulent flow in the boundary layer.

scholarly journals Linear global instability of non-orthogonal incompressible swept attachment-line boundary-layer flow

2012 ◽  
Vol 710 ◽  
pp. 131-153 ◽  
Author(s):  
José Miguel Pérez ◽  
Daniel Rodríguez ◽  
Vassilis Theofilis
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Flow Instability ◽  
Angle Of Attack ◽  
Basic Flow ◽  
Critical Conditions ◽  
Simple Extension ◽  
Global Instability ◽  
Special Cases ◽  
Attachment Line

AbstractFlow instability in the non-orthogonal swept attachment-line boundary layer is addressed in a linear analysis framework via solution of the pertinent global (BiGlobal) partial differential equation (PDE)-based eigenvalue problem. Subsequently, a simple extension of the extended Görtler–Hämmerlin ordinary differential equation (ODE)-based polynomial model proposed by Theofilis et al. (2003) for orthogonal flow, which includes previous models as special cases and recovers global instability analysis results, is presented for non-orthogonal flow. Direct numerical simulations have been used to verify the analysis results and unravel the limits of validity of the basic flow model analysed. The effect of the angle of attack, $\mathit{AoA}$, on the critical conditions of the non-orthogonal problem has been documented; an increase of the angle of attack, from $\mathit{AoA}= 0$ (orthogonal flow) up to values close to $\lrm{\pi} / 2$ which make the assumptions under which the basic flow is derived questionable, is found to systematically destabilize the flow. The critical conditions of non-orthogonal flows at $0\leq \mathit{AoA}\leq \lrm{\pi} / 2$ are shown to be recoverable from those of orthogonal flow, via a simple algebraic transformation involving $\mathit{AoA}$.

scholarly journals Analytic Approximate Solution for Falkner-Skan Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-22 ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene ◽  
Bogdan Marinca
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Rapid Convergence ◽  
Boundary Layer Problem ◽  
Small Parameters ◽  
Optimal Homotopy Asymptotic Method ◽  
Good Agreement ◽  
Layer Problem

This paper deals with the Falkner-Skan nonlinear differential equation. An analytic approximate technique, namely, optimal homotopy asymptotic method (OHAM), is employed to propose a procedure to solve a boundary-layer problem. Our method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. The obtained results reveal that this procedure is very effective, simple, and accurate. A very good agreement was found between our approximate results and numerical solutions, which prove that OHAM is very efficient in practice, ensuring a very rapid convergence after only one iteration.

scholarly journals Convergence of approximate solutions of a quasilinear partial differential equation

1994 ◽  
Vol 50 (3) ◽  
pp. 425-433 ◽  
Author(s):  
T.R. Cranny
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Original Problem ◽  
A Priori ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Special Cases ◽  
Convergence Of Approximate Solutions ◽  
Hölder Estimates ◽  
Quasilinear Partial Differential Equation

This article is a sequel to a paper in which a quasilinear partial differential equation with nonlinear boundary condition was approximated using mollifiers, and the existence of solutions to the approximating problem shown under quite general conditions. In this paper we show that standard a priori Hölder estimates ensure the convergence of these solutions to a classical solution of the original problem. Some partial results giving such estimates for special cases are described.

scholarly journals Approximate Representations of Shaped Pulses Using the Homotopy Analysis Method

2021 ◽  
Author(s):  
Timothy Crawley ◽  
Arthur G. Palmer III
Keyword(s):  
Differential Equation ◽  
Homotopy Analysis Method ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Euler Angles ◽  
Analysis Method ◽  
Riccati Differential Equation ◽  
Spin Magnetization ◽  
Homotopy Analysis ◽  
Radiofrequency Pulse

Abstract. The evolution of nuclear spin magnetization during a radiofrequency pulse in the absence of relaxation or coupling interactions can be described by three Euler angles. The Euler angles in turn can be obtained from the solution of a Riccati differential equation; however, analytic solutions exist only for rectangular and chirp pulses. The Homotopy Analysis Method is used to obtain new approximate solutions to the Riccati equation for shaped radiofrequency pulses in NMR spectroscopy. The results of even relatively low orders of approximation are highly accurate and can be calculated very efficiently. The Homotopy Analysis Method is powerful and flexible and is likely to have other applications in theoretical magnetic resonance.

Algebraic Solution of the Horton-Izzard Turbulent Overland Flow Model of the Rising Hydrograph

1977 ◽  
Vol 8 (4) ◽  
pp. 249-256 ◽  
Author(s):  
Mohammad Akram Gill
Keyword(s):  
Differential Equation ◽  
Turbulent Flow ◽  
Flow Model ◽  
Overland Flow ◽  
Flow Equation ◽  
Algebraic Solution ◽  
Mixed Flow ◽  
Overland Flow Model

In the differential equation of the overland turbulent flow which was first postulated by Horton, Eq.(6), the value of c equals 5/3. For this value of c, the flow equation could not be integrated algebraically. Horton solved the equation for c = 2 and believed that his solution was valid for mixed flow. The flow equation with c = 5/3 is solved algebraically herein. It is shown elsewhere (Gill 1976) that the flow equation can indeed be integrated for any rational value of c.

scholarly journals An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed Al-Smadi ◽  
Nadir Djeddi ◽  
Shaher Momani ◽  
Shrideh Al-Omari ◽  
Serkan Araci
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Reproducing Kernel ◽  
Numerical Approach ◽  
Accurate Solution ◽  
Stability And Convergence ◽  
Type Model ◽  
Schmidt Orthogonalization ◽  
And Performance ◽  
Abel Differential Equation

AbstractOur aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a Caputo–Fabrizio fractional derivative. By means of such an approach, we utilize the Gram–Schmidt orthogonalization process to create an orthonormal set of bases that leads to an appropriate solution in the Hilbert space $\mathcal{H}^{2}[a,b]$ H 2 [ a , b ] . We investigate and discuss stability and convergence of the proposed method. The n-term series solution converges uniformly to the analytic solution. We present several numerical examples of potential interests to illustrate the reliability, efficacy, and performance of the method under the influence of the Caputo–Fabrizio derivative. The gained results have shown superiority of the reproducing kernel algorithm and its infinite accuracy with a least time and efforts in solving the fractional Abel-type model. Therefore, in this direction, the proposed algorithm is an alternative and systematic tool for analyzing the behavior of many nonlinear temporal fractional differential equations emerging in the fields of engineering, physics, and sciences.

scholarly journals Global small analytic solutions of MHD boundary layer equations

2021 ◽  
Vol 281 ◽  
pp. 199-257
Author(s):  
Ning Liu ◽  
Ping Zhang
Keyword(s):  
Boundary Layer ◽  
Analytic Solutions ◽  
Boundary Layer Equations ◽  
Mhd Boundary Layer

scholarly journals A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 979
Author(s):  
Sandeep Kumar ◽  
Rajesh K. Pandey ◽  
H. M. Srivastava ◽  
G. N. Singh
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Special Cases ◽  
Collocation Approach ◽  
Linear And Nonlinear ◽  
Simulation Results ◽  
Theoretical Results

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.

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