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1 – 10 of over 3000Allaberen Ashyralyev and Betul Hicdurmaz
The purpose of this paper is to introduce stability analysis for the initial value problem for the fractional Schrödinger differential equation: Equation 1 in a Hilbert space H…
Abstract
Purpose
The purpose of this paper is to introduce stability analysis for the initial value problem for the fractional Schrödinger differential equation: Equation 1 in a Hilbert space H with a self‐adjoint positive definite operator A under the condition |α(s)|<M1/s1/2 and the first order of accuracy difference scheme for the approximate solution of this initial value problem.
Design/methodology/approach
The paper considers the stability estimates for the solution of this problem and the stability estimate for the approximate solution of first order of accuracy difference scheme of this problem.
Findings
The paper finds the stability for the fractional Schrödinger differential equation and the first order of accuracy difference scheme of that equation by applications to one‐dimensional fractional Schrödinger differential equation with nonlocal boundary conditions and multidimensional fractional Schrödinger differential equation with Dirichlet conditions.
Originality/value
The paper is a significant work on stability of fractional Schrödinger differential equation and its difference scheme presenting some numerical experiments which resulted from applying obtained theorems on several mixed fractional Schrödinger differential equations.
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The purpose of this paper is to investigate the Lp‐maximal regularity for the abstract incomplete second order problem.
Abstract
Purpose
The purpose of this paper is to investigate the Lp‐maximal regularity for the abstract incomplete second order problem.
Design/methodology/approach
First, the paper gives the definition of the Lp‐maximal regularity for incomplete second‐order Cauchy problems and lists their basic properties based on Chill and Srivastava's recent work for completing second order problem. Second, the paper establishes its characterization by means of Fourier multiplier and the operator‐sum theorem. Finally, it considers an application to quasilinear systems by the regularity and linearization techniques.
Findings
Two criteria of Lp‐maximal regularity are obtained, and the existence of the local solution for the second order quasilinear problem is given. In addition, the connection on maximal regularity between second order problems with initial values and that with periodic problems is investigated. A perturbation result is given.
Originality/value
The maximal regularity is an important tool in the theory of non‐linear differential equations. The results obtained in this paper are universal because the operator is not necessarily the generator of a cosine operator function. Using this unifying approach it is possible to clarify the Lp‐maximal regularity and the existence of the solution for some systems described by partial differential equations, such as wave equations.
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A. Kaveh, H. Rahami and Iman Shojaei
The purpose of this paper is to present an efficient method for dynamic analysis of structures utilizing a modal analysis with the main purpose of decreasing the computational…
Abstract
Purpose
The purpose of this paper is to present an efficient method for dynamic analysis of structures utilizing a modal analysis with the main purpose of decreasing the computational complexity of the problem. In traditional methods, the solution of initial-value problems (IVPs) using numerical methods like finite difference method leads to step by step and time-consuming recursive solutions.
Design/methodology/approach
The present method is based on converting the IVP into boundary-value problems (BVPs) and utilizing the features of the latter problems in efficient solution of the former ones. Finite difference formulation of BVPs leads to matrices with repetitive tri-diagonal and block tri-diagonal patterns wherein the eigensolution and matrix inversion are obtained using graph products rules. To get advantage of these efficient solutions for IVPs like the dynamic analysis of single DOF systems, IVPs are converted to boundary-value ones using mathematical manipulations. The obtained formulation is then generalized to the multi DOF systems by utilizing modal analysis.
Findings
Applying the method to the modal analysis leads to a simple and efficient formulation. The laborious matrix inversion and eigensolution operations, of computational complexities of O(n2.373) and O(n3), respectively, are converted to a closed-form formulation with summation operations.
Research limitations/implications
No limitation.
Practical implications
Swift analysis has become possible.
Originality/value
Suitability of solving IVPs and modal analysis using conversion and graph product rules is presented and applied to efficient seismic optimal analysis and preliminary design.
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Ajit Kumar Parwani, Prabal Talukdar and P.M.V. Subbarao
The purpose of this paper is to develop a numerical model for estimating the unknown boundary heat flux in a parallel plate channel for the case of a hydrodynamically and…
Abstract
Purpose
The purpose of this paper is to develop a numerical model for estimating the unknown boundary heat flux in a parallel plate channel for the case of a hydrodynamically and thermally developing laminar flow.
Design/methodology/approach
The conjugate gradient method (CGM) is used to solve the inverse problem. The momentum equations are solved using an in-house computational fluid dynamics (CFD) source code. The energy equations along with the adjoint and sensitivity equations are solved using the finite volume method.
Findings
The effects of number of measurements, distribution of measurements and functional form of unknown flux on the accuracy of estimations are investigated in this work. The prediction of boundary flux by the present algorithm is found to be quite reasonable.
Originality/value
It is noticed from the literature review that study of inverse problem with hydrodynamically developing flow has not received sufficient attention despite its practical importance. In the present work, a hydrodynamically and thermally developing flow between two parallel plates is considered and unknown transient boundary heat flux at the upper plate of a parallel plate channel is estimated using CGM.
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The paper deals with a spatial discretization of transient semiconductor device equations. The method can be regarded as a combination of FDM‐ and FEM‐ideas. In the first part of…
Abstract
The paper deals with a spatial discretization of transient semiconductor device equations. The method can be regarded as a combination of FDM‐ and FEM‐ideas. In the first part of the paper the method is described and—for a weakly acute triangulation—existence, uniqueness, non‐negativity, stability and conservativity of the semidiscrete solution are proved. The second part contains an error estimation under stronger assumptions on the regularity of the analytical solution and on the uniformity of the triangulation respectively. A linear convergence rate is obtained.
David W. Lloyd, F. Mete and K. Hussain
Outlines an approach to the problem of simulating the drape of fabrics. The approach is based on representing fabrics as two‐dimensional continua and using the differential…
Abstract
Outlines an approach to the problem of simulating the drape of fabrics. The approach is based on representing fabrics as two‐dimensional continua and using the differential geometry of surfaces to describe the shape of the draped fabric and to deduce deformation measures. The approach follows the philosophical approach previously used to establish computational models of elastica theory. Summarizes the mathematical model using a compact notation, with more detail being given in the context of a particular example. Uses a simple numerical solution procedure for the example, but this has limitations that indicate that more sophisticated techniques are needed. Points out possible difficulties with the graphical representation of drape, based on the experience of the simple example.
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Aytac Arikoglu and Ibrahim Ozkol
To study the flow of a two‐dimensional, steady, incompressible and constant property fluid over a semi‐infinite flat plate represented by the well‐known Blasius equation.
Abstract
Purpose
To study the flow of a two‐dimensional, steady, incompressible and constant property fluid over a semi‐infinite flat plate represented by the well‐known Blasius equation.
Design/methodology/approach
Differential transform method was used to solve this equation. The solution is based on matching of the inner and outer solutions.
Findings
It is observed that the inner solution diverges at η≅5.5 and the outer solution developed for η≥5.5 should be matched with the inner one.
Originality/value
It is the first time, for the best of the authors' knowledge, the missing boundary condition f″(0), related to surface friction, is obtained in a closed form. Also, the matching technique implemented in this study can be used for this type of physical length sensitive problems.
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The purpose of this paper is to investigate the dynamic stability of liquid hydrogen turbopump rotor system in rocket engine under the effects of seal and internal rotor damping.
Abstract
Purpose
The purpose of this paper is to investigate the dynamic stability of liquid hydrogen turbopump rotor system in rocket engine under the effects of seal and internal rotor damping.
Design/methodology/approach
The dynamic modeling of a liquid hydrogen turbopump rotor system in rocket engine is presented in this paper with the aid of the finite element technique. The mathematical model takes into account the seal hydrodynamic forces described by Muszynska model and the internal rotor damping, viscous damping, and hysteretic damping. The shooting method and Floquet theory are employed to investigate the effects of seal and internal rotor damping on the nonlinear dynamic stability of two turbopump designs, the original and the modified design with a flexible bearing support.
Findings
The numerical results, which are in good agreement with test data, show that the destabilizing effect of internal rotor damping play a key role in the original design. In the modified design, the stability margins are enhanced and the vibration response levels are minimized. The onset speed of instability increases in original design and decreases in modified design as the effects of seal nonlinearities are considered. The predicted results indicate that the seals have a great destabilizing effect in the modified design and the turbine end bearing is the most dangerous hardware in both designs. The system stability analysis shows that the effect of seal length on the system stability is significant comparing with that of seal radius.
Practical implications
The results can be used in the design and operation of a liquid hydrogen turbopump rotor system to improve its stability performance and eliminate its subsynchronous problem.
Originality/value
Since seal and internal damping are two key destabilizing factors in liquid hydrogen turbopumps and the seal nonlinearities are inevitable, the use of nonlinear theory to study their effects on nonlinear stability and dynamic performance can lead to accurate prediction and explain the nature of the subsynchronous motion.
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Mohammad Ghalambaz, Natalia C. Roşca, Alin V. Roşca and Ioan Pop
This study aims to study the mixed convection flow and heat transfer of Al2O3-Cu/water hybrid nanofluid over a vertical plate. Governing equations for conservation of mass…
Abstract
Purpose
This study aims to study the mixed convection flow and heat transfer of Al2O3-Cu/water hybrid nanofluid over a vertical plate. Governing equations for conservation of mass, momentum and energy for the hybrid nanofluid over a vertical flat plate are introduced.
Design/methodology/approach
The similarity transformation approach is used to transform the set of partial differential equations into a set of non-dimensional ordinary differential equations. Finite-deference with collocation method is used to integrate the governing equations for the velocity and temperature profiles.
Findings
The results show that dual solutions exist for the case of opposing flow over the plate. Linear stability analysis was performed to identify a stable solution. The stability analysis shows that the lower branch of the solution is always unstable, while the upper branch of the solution is always stable. The results of boundary layer analysis are reported for the various volume fractions of composite nanoparticles and mixed convection parameter. The outcomes show that the composition of nanoparticles can notably influence the boundary layer flow and heat transfer profiles. It is also found that the trend of the variation of surface skin friction and heat transfer for each of the dual solution branches can be different. The critical values of the mixed convection parameter, λ, where the dual solution branches joint together, are also under the influence of the composition of hybrid nanoparticles. For instance, assuming a total volume fraction of 5 per cent for the mixture of Al2O3 and Cu nanoparticles, the critical value of mixing parameter of λ changes from −3.1940 to −3.2561 by changing the composition of nanofluids from Al2O3 (5 per cent) + Cu (0%) to Al2O3 (2.5%) + Cu (2.5 per cent).
Originality/value
The mixed convection stability analysis and heat transfer study of hybrid nanofluids for a stagnation-point boundary layer flow are addressed for the first time. The introduced hybrid nanofluid model and similarity solution are new and of interest in both mathematical and physical points of view.
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Golam Mortuja Sarkar, Suman Sarkar and Bikash Sahoo
This paper aims to theoretically and numerically investigate the steady two-dimensional (2D) Hiemenz flow with heat transfer of Reiner-Rivlin fluid over a linearly…
Abstract
Purpose
This paper aims to theoretically and numerically investigate the steady two-dimensional (2D) Hiemenz flow with heat transfer of Reiner-Rivlin fluid over a linearly stretching/shrinking sheet.
Design/methodology/approach
The Navier–Stokes equations are transformed into self-similar equations using appropriate similarity transformations and then solved numerically by using shooting technique. A simple but effective mathematical analysis has been used to prove the existence of a solution for stretching case (λ> 0). Moreover, an attempt has been laid to carry the asymptotic solution behavior for large stretching. The obtained asymptotic solutions are compared with direct numerical solutions, and the comparison is quite remarkable.
Findings
It is observed that the self-similar equations exhibit dual solutions within the range [λc, −1] of shrinking parameter λ, where λc is the turning point from where the dual solutions bifurcate. Unique solution is found for all stretching case (λ > 0). It is noticed that the effects of cross-viscous parameter L and shrinking parameter λ on velocity and thermal fields show opposite character in the dual solution branches. Thus, a linear temporal stability analysis is performed to determine the basic feasible solution. The stability analysis is based on the sign of the smallest eigenvalue, where positive or negative sign leading to a stable or unstable solution. The stability analysis reveals that the first solution is stable that describes the main flow. Increase in cross-viscous parameter L resulting in a significant increment in skin friction coefficient, local Nusselt number and dual solutions domain.
Originality/value
This work’s originality is to examine the combined effects of cross-viscous parameter and stretching/shrinking parameter on skin friction coefficient, local Nusselt number, velocity and temperature profiles of Hiemenz flow over a stretching/shrinking sheet. Although many studies on viscous fluid and nanofluid have been investigated in this field, there are still limited discoveries on non-Newtonian fluids. The obtained results can be used as a benchmark for future studies of higher-grade non-Newtonian flows with several physical aspects. All the generated results are claimed to be novel and have not been published elsewhere.
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