A Numerical Procedure for Coupled Integro-Differential Equations of Irreversible Quantum Dynamics

2003 ◽  
Author(s):  
K. Lendi
Keyword(s):  
Differential Equations ◽  
Quantum Dynamics ◽  
Numerical Procedure ◽  
Irreversible Quantum Dynamics

Influence of rotating magnetic field on Maxwell saturated ferrofluid flow over a heated stretching sheet with heat generation/absorption

2019 ◽  
Vol 20 (5) ◽  
pp. 502 ◽  
Author(s):  
Aaqib Majeed ◽  
Ahmed Zeeshan ◽  
Farzan Majeed Noori ◽  
Usman Masud
Keyword(s):  
Magnetic Field ◽  
Temperature Field ◽  
Velocity Field ◽  
Differential Equations ◽  
Heat Transport ◽  
Viscous Dissipation ◽  
Heat Generation ◽  
Fluid Motion ◽  
Numerical Procedure ◽  
The Impact

This article is focused on Maxwell ferromagnetic fluid and heat transport characteristics under the impact of magnetic field generated due to dipole field. The viscous dissipation and heat generation/absorption are also taken into account. Flow here is instigated by linearly stretchable surface, which is assumed to be permeable. Also description of magneto-thermo-mechanical (ferrohydrodynamic) interaction elaborates the fluid motion as compared to hydrodynamic case. Problem is modeled using continuity, momentum and heat transport equation. To implement the numerical procedure, firstly we transform the partial differential equations (PDEs) into ordinary differential equations (ODEs) by applying similarity approach, secondly resulting boundary value problem (BVP) is transformed into an initial value problem (IVP). Then resulting set of non-linear differentials equations is solved computationally with the aid of Runge–Kutta scheme with shooting algorithm using MATLAB. The flow situation is carried out by considering the influence of pertinent parameters namely ferro-hydrodynamic interaction parameter, Maxwell parameter, suction/injection and viscous dissipation on flow velocity field, temperature field, friction factor and heat transfer rate are deliberated via graphs. The present numerical values are associated with those available previously in the open literature for Newtonian fluid case (γ 1 = 0) to check the validity of the solution. It is inferred that interaction of magneto-thermo-mechanical is to slow down the fluid motion. We also witnessed that by considering the Maxwell and ferrohydrodynamic parameter there is decrement in velocity field whereas opposite behavior is noted for temperature field.

scholarly journals An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Süleyman Cengizci
Keyword(s):  
Differential Equations ◽  
Hybrid Method ◽  
Delay Differential Equations ◽  
Numerical Procedure ◽  
Taylor Series Expansion ◽  
Expansion Method ◽  
Singularly Perturbed ◽  
Convection Term ◽  
Linear Delay ◽  
Delay Differential

In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.

scholarly journals A New Numerical Procedure for Vibration Analysis of Beam under Impulse and Multiharmonics Piezoelectric Actuators

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Yassin Belkourchia ◽  
Lahcen Azrar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Methodological Approach ◽  
Numerical Procedure ◽  
Piezoelectric Actuators ◽  
Partial Differential ◽  
Simple Method ◽  
Regularization Procedure ◽  
Dirac Delta

The dynamic behavior of structures with piezoelectric patches is governed by partial differential equations with strong singularities. To directly deal with these equations, well adapted numerical procedures are required. In this work, the differential quadrature method (DQM) combined with a regularization procedure for space and implicit scheme for time discretization is used. The DQM is a simple method that can be implemented with few grid points and can give results with a good accuracy. However, the DQM presents some difficulties when applied to partial differential equations involving strong singularities. This is due to the fact that the subsidiaries of the singular functions cannot be straightforwardly discretized by the DQM. A methodological approach based on the regularization procedure is used here to overcome this difficulty and the derivatives of the Dirac-delta function are replaced by regularized smooth functions. Thanks to this regularization, the resulting differential equations can be directly discretized using the DQM. The efficiency and applicability of the proposed approach are demonstrated in the computation of the dynamic behavior of beams for various boundary conditions and excited by impulse and Multiharmonics piezoelectric actuators. The obtained numerical results are well compared to the developed analytical solution.

A Steepest-Ascent Method for Solving Optimum Programming Problems

1962 ◽  
Vol 29 (2) ◽  
pp. 247-257 ◽  
Author(s):  
A. E. Bryson ◽  
W. F. Denham
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Calculus Of Variations ◽  
Numerical Procedure ◽  
Nonlinear Ordinary Differential Equations ◽  
Point Boundary ◽  
Numerical Examples ◽  
Steepest Ascent ◽  
Maximum Range

A systematic and rapid steepest-ascent numerical procedure is described for solving two-point boundary-value problems in the calculus of variations for systems governed by a set of nonlinear ordinary differential equations. Numerical examples are presented for minimum time-to-climb and maximum altitude paths for a supersonic interceptor and maximum-range paths for an orbital glider.

NUMERICAL SOLUTIONS OF INTEGRO-DIFFERENTIAL EQUATIONS USING SOBOLEV GRADIENT METHODS

2012 ◽  
Vol 09 (04) ◽  
pp. 1250046 ◽  
Author(s):  
NAUMAN RAZA ◽  
SULTAN SIAL ◽  
JOHN W. NEUBERGER ◽  
MUHAMMAD OZAIR AHMAD
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Gradient Methods ◽  
Numerical Procedure ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Test Results ◽  
Volterra Type ◽  
Adomian Decomposition ◽  
Sobolev Gradient

A numerical procedure for solving a class of integro-differential equations of Volterra type using the Sobolev gradient method is presented. Results are compared with those from the variational iteration method (VIM) and Adomian decomposition method (ADM) (Batiha, B., Noorani, M. S. M. and Hashmi, I. [2008] "Numerical solutions of the nonlinear integro-differential equations," Int. J. Open Probl. Compt. Math.1, 34–42). The capabilities of our codes are briefly described and test results from some examples are presented.

Comparison of the Effects of Density and Sound Speed Variation in the Propagation of Planar Transient Waves

1992 ◽  
Vol 114 (3) ◽  
pp. 409-414
Author(s):  
J. H. Ginsberg
Keyword(s):  
Differential Equations ◽  
Material Properties ◽  
Sound Speed ◽  
Heterogeneous Media ◽  
Numerical Procedure ◽  
Mean Value ◽  
The Other ◽  
Transient Waves ◽  
One Dimensional ◽  
Spatial Grid

When expressed in the form of characteristic differential equations, the laws governing propagation of linear one-dimensional waves through heterogeneous media show that the only properties of significance are the sound speed c and the acoustic impedance ρc, either of which may vary spatially. The former occurs in the differential equations governing the (curved) characteristics, while the latter appears in the differential equations governing the evolution of particle velocity and stress along the characteristics. The present study employs an inherently stable finite difference representation of the characteristic equations, in which the spatial grid is obtained by evaluating the intersections in space-time of constant time lines with comparable increments of the characteristic variables. The numerical procedure is used to follow the propagation of a single-lobe sine pulse in cases where only ρ or c fluctuates spatially about a mean value while the other property is constant, and compares those results to the case were both material properties vary. Nonconstancy of c is shown to cause temporal shifts in waveforms, while spatial variation of ρc causes attenuation and distortion of the waveform.

Frictionless contact of a rigid stamp with a semi-plane in the presence of surface elasticity in the Steigmann–Ogden form

2017 ◽  
Vol 23 (8) ◽  
pp. 1140-1155 ◽  
Author(s):  
Anna Y Zemlyanova
Keyword(s):  
Differential Equations ◽  
Surface Elasticity ◽  
Numerical Procedure ◽  
Contact Problems ◽  
Mechanical Parameters ◽  
Mechanical Problem ◽  
Frictionless Contact ◽  
Rigid Stamp ◽  
The Fourier Transform ◽  
Curvature Dependence

In this paper, the surface elasticity in the form proposed by Steigmann and Ogden is applied to study a plane problem of frictionless contact of a rigid stamp with an elastic upper semi-plane. The results of this work generalize the results for contact problems with Gurtin–Murdoch elasticity by including additional dependency on the curvature of the surface. The mechanical problem is reduced to a system of singular integro-differential equations, which is further regularized using the Fourier transform. The size dependency of the solutions of the problem is highlighted. It is observed that the curvature dependence of the surface energy is increasingly important at small scales. The numerical procedure of the solution of the system of singular integro-differential equations is presented, and numerical results are obtained for different values of the mechanical parameters.

scholarly journals New Vertically Planed Pendulum Motion

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
A. I. Ismail
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
Numerical Procedure ◽  
Second Order ◽  
The Body ◽  
Physical Parameters ◽  
Pendulum Motion ◽  
Generalized Coordinates

This article is concerned about the planed rigid body pendulum motion suspended with a spring which is suspended to move on a vertical plane moving uniformly about a horizontal X-axis. This model depends on a system containing three generalized coordinates. The three nonlinear differential equations of motion of the second order are obtained to the elastic string length and the oscillation angles φ 1 and φ 2 which represent the freedom degrees for the pendulum motions. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity ω . The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the approximated fourth-order Runge–Kutta method through programming packages. These solutions are represented graphically to describe and discuss the behavior of the body at any instant for different values of the different physical parameters of the body. The obtained results have been discussed and compared with some previously published works. Some concluding remarks have been presented at the end of this work. The value of this study comes from its wide applications in both civil and military life. The main findings and objectives of the current study are obtaining periodic solutions for the problem and satisfying their accuracy and stabilities through the numerical procedure.

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