Search results for: Moment Equations

Authors: Ganesh D. Kale, Sheela N. Vadsola

Abstract:

Soil erosion is the most serious problem faced at global and local level. So planning of soil conservation measures has become prominent agenda in the view of water basin managers. To plan for the soil conservation measures, the information on soil erosion is essential. Universal Soil Loss Equation (USLE), Revised Universal Soil Loss Equation 1 (RUSLE1or RUSLE) and Modified Universal Soil Loss Equation (MUSLE), RUSLE 1.06, RUSLE1.06c, RUSLE2 are most widely used conventional erosion estimation methods. The essential drawbacks of USLE, RUSLE1 equations are that they are based on average annual values of its parameters and so their applicability to small temporal scale is questionable. Also these equations do not estimate runoff generated soil erosion. So applicability of these equations to estimate runoff generated soil erosion is questionable. Data used in formation of USLE, RUSLE1 equations was plot data so its applicability at greater spatial scale needs some scale correction factors to be induced. On the other hand MUSLE is unsuitable for predicting sediment yield of small and large events. Although the new revised forms of USLE like RUSLE 1.06, RUSLE1.06c and RUSLE2 were land use independent and they have almost cleared all the drawbacks in earlier versions like USLE and RUSLE1, they are based on the regional data of specific area and their applicability to other areas having different climate, soil, land use is questionable. These conventional equations are applicable for sheet and rill erosion and unable to predict gully erosion and spatial pattern of rills. So the research was focused on development of nonconventional (other than conventional) methods of soil erosion estimation. When these non-conventional methods are combined with GIS and RS, gives spatial distribution of soil erosion. In the present paper the review of literature on non- conventional methods of soil erosion estimation supported by GIS and RS is presented.

Keywords: Conventional methods, GIS, non-conventionalmethods, remote sensing, soil erosion modeling

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Authors: Shubha P Bhat, Krishnamurthy, T.C.Manjunath, C. Ardil

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In this paper, the effects of the restoring force device on the response of a space frame structure resting on sliding type of bearing with a restoring force device is studied. The NS component of the El - Centro earthquake and harmonic ground acceleration is considered for earthquake excitation. The structure is modeled by considering six-degrees of freedom (three translations and three rotations) at each node. The sliding support is modeled as a fictitious spring with two horizontal degrees of freedom. The response quantities considered for the study are the top floor acceleration, base shear, bending moment and base displacement. It is concluded from the study that the displacement of the structure reduces by the use of the restoring force device. Also, the peak values of acceleration, bending moment and base shear also decreases. The simulation results show the effectiveness of the developed and proposed method.

Keywords: DOF, Space structures, Acceleration, Excitation, Smart structure, Vibration, Isolation, Earthquakes.

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Authors: Rahul Bansal, Sudipta Majumdar

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This paper presents the closed form nonlinear expressions of bipolar junction transistor (BJT) differential amplifier (DA) using perturbation method. Circuit equations have been derived using Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL). The perturbation method has been applied to state variables for obtaining the linear and nonlinear terms. The implementation of the proposed method is simple. The closed form nonlinear expressions provide better insights of physical systems. The derived equations can be used for signal processing applications.

Keywords: Differential amplifier, perturbation method, Taylor series.

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Authors: Irina Eglite, Andrei A. Kolyshkin

Abstract:

Method of multiple scales is used in the paper in order to derive an amplitude evolution equation for the most unstable mode from two-dimensional shallow water equations under the rigid-lid assumption. It is assumed that shallow mixing layer is slightly curved in the longitudinal direction and contains small particles. Dynamic interaction between carrier fluid and particles is neglected. It is shown that the evolution equation is the complex Ginzburg-Landau equation. Explicit formulas for the computation of the coefficients of the equation are obtained.

Keywords: Shallow water equations, mixing layer, weakly nonlinear analysis, Ginzburg-Landau equation

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Authors: A. Pereira, S. Haffner, L. V. Gasperin

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This paper presents two simplified models to determine nodal voltages in power distribution networks. These models allow estimating the impact of the installation of reactive power compensations equipments like fixed or switched capacitor banks. The procedure used to develop the models is similar to the procedure used to develop linear power flow models of transmission lines, which have been widely used in optimization problems of operation planning and system expansion. The steady state non-linear load flow equations are approximated by linear equations relating the voltage amplitude and currents. The approximations of the linear equations are based on the high relationship between line resistance and line reactance (ratio R/X), which is valid for power distribution networks. The performance and accuracy of the models are evaluated through comparisons with the exact results obtained from the solution of the load flow using two test networks: a hypothetical network with 23 nodes and a real network with 217 nodes.

Keywords: Distribution network models, distribution systems, optimization, power system planning.

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Authors: A. Javed, K. Djidjeli, J. T. Xing

Abstract:

The concept of adaptive shape parameters (ASP) has been presented for solution of incompressible Navier Strokes equations using mesh-free local Radial Basis Functions (RBF). The aim is to avoid ill-conditioning of coefficient matrices of RBF weights and inaccuracies in RBF interpolation resulting from non-optimized shape of basis functions for the cases where data points (or nodes) are not distributed uniformly throughout the domain. Unlike conventional approaches which assume globally similar values of RBF shape parameters, the presented ASP technique suggests that shape parameter be calculated exclusively for each data point (or node) based on the distribution of data points within its own influence domain. This will ensure interpolation accuracy while still maintaining well conditioned system of equations for RBF weights. Performance and accuracy of ASP technique has been tested by evaluating derivatives and laplacian of a known function using RBF in Finite difference mode (RBFFD), with and without the use of adaptivity in shape parameters. Application of adaptive shape parameters (ASP) for solution of incompressible Navier Strokes equations has been presented by solving lid driven cavity flow problem on mesh-free domain using RBF-FD. The results have been compared for fixed and adaptive shape parameters. Improved accuracy has been achieved with the use of ASP in RBF-FD especially at regions where larger gradients of field variables exist.

Keywords: CFD, Meshless Particle Method, Radial Basis Functions, Shape Parameters

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Authors: M. Faraji, F. Berroug

Abstract:

A one-dimensional mathematical model was developed in order to analyze and optimize the latent heat storage wall. The governing equations for energy transport were developed by using the enthalpy method and discretized with volume control scheme. The resulting algebraic equations were next solved iteratively by using TDMA algorithm. A series of numerical investigations were conducted in order to examine the effects of the thickness of the PCM layer on the thermal behavior of the proposed heating system. Results are obtained for thermal gain and temperature fluctuation. The charging discharging process was also presented and analyzed.

Keywords: Phase change material, Building, Concrete, Latent heat, Thermal control.

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Authors: M. Hadi Mashinchi, M. Reza Mashinchi, Siti Mariyam H. J. Shamsuddin

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In this paper a genetic algorithms approach for solving the linear and quadratic fuzzy equations Ãx̃=B̃ and Ãx̃² + B̃x̃=C̃ , where Ã, B̃, C̃ and x̃ are fuzzy numbers is proposed by genetic algorithms. Our genetic based method initially starts with a set of random fuzzy solutions. Then in each generation of genetic algorithms, the solution candidates converge more to better fuzzy solution x̃_b . In this proposed method the final reached x̃_b is not only restricted to fuzzy triangular and it can be fuzzy number.

Keywords: Fuzzy coefficient, fuzzy equation, genetic algorithms.

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Authors: Hamdy M. Youssef, Eman A. Al-Lehaibi

Abstract:

Due to many applications and problems in the fields of plasma physics, geophysics, and other many topics, the interaction between the strain field and the magnetic field has to be considered. Adomian introduced the decomposition method for solving linear and nonlinear functional equations. This method leads to accurate, computable, approximately convergent solutions of linear and nonlinear partial and ordinary differential equations even the equations with variable coefficients. This paper is dealing with a mathematical model of generalized thermoelasticity of a half-space conducting medium. A magnetic field with constant intensity acts normal to the bounding plane has been assumed. Adomian’s decomposition method has been used to solve the model when the bounding plane is taken to be traction free and thermally loaded by harmonic heating. The numerical results for the temperature increment, the stress, the strain, the displacement, the induced magnetic, and the electric fields have been represented in figures. The magnetic field, the relaxation time, and the angular thermal load have significant effects on all the studied fields.

Keywords: Adomian’s Decomposition Method, magneto-thermoelasticity, finite conductivity, iteration method, thermal load.

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Authors: Dezhi Liu

Abstract:

In the paper, based on stochastic analysis theory and Lyapunov functional method, we discuss the mean square stability of impulsive stochastic delay differential equations with markovian switching and poisson jumps, and the sufficient conditions of mean square stability have been obtained. One example illustrates the main results. Furthermore, some well-known results are improved and generalized in the remarks.

Keywords: Impulsive, stochastic, delay, Markovian switching, Poisson jumps, mean square stability.

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Authors: Yiming Jin, Yuanhao Gao

Abstract:

In this paper, using the method of multiple scales, the second sub-harmonic resonance in vortex-induced vibrations (VIV) of a marine pipeline close to the seabed is investigated based on a developed wake oscillator model. The amplitude-frequency equations are also derived. It is found that the oscillation will increase all the time when both discriminants of the amplitude-frequency equations are positive while the oscillation will decay when the discriminants are negative.

Keywords: Vortex-induced vibrations, marine pipeline, seabed, sub-harmonic resonance.

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Authors: Behrooz Rezaie, Zahra Rahmani Cherati, Mohammad Reza Jahed Motlagh, Mohammad Farrokhi

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In this paper, a fuzzy controller is designed for stabilization of the Lorenz chaotic equations. A simple Mamdani inference method is used for this purpose. This method is very simple and applicable for complex chaotic systems and it can be implemented easily. The stability of close loop system is investigated by the Lyapunov stabilization criterion. A Lyapunov function is introduced and the global stability is proven. Finally, the effectiveness of this method is illustrated by simulation results and it is shown that the performance of the system is improved.

Keywords: Chaotic system, Fuzzy control, Lorenz equation.

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Authors: A.S.N.Murti, D.R.V.S.R.K. Sastry, P.K. Kameswaran, T. Poorna Kantha

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In this paper, the effects of radiation, chemical reaction and double dispersion on mixed convection heat and mass transfer along a semi vertical plate are considered. The plate is embedded in a Newtonian fluid saturated non - Darcy (Forchheimer flow model) porous medium. The Forchheimer extension and first order chemical reaction are considered in the flow equations. The governing sets of partial differential equations are nondimensionalized and reduced to a set of ordinary differential equations which are then solved numerically by Fourth order Runge– Kutta method. Numerical results for the detail of the velocity, temperature, and concentration profiles as well as heat transfer rates (Nusselt number) and mass transfer rates (Sherwood number) against various parameters are presented in graphs. The obtained results are checked against previously published work for special cases of the problem and are found to be in good agreement.

Keywords: Radiation, Chemical reaction, Double dispersion, Mixed convection, Heat and Mass transfer

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Authors: Mircea Pastrav, Horia Constantinescu

Abstract:

The construction industry is currently trying to develop sustainable reinforced concrete structures. In trying to aid in the effort, the research presented in this paper aims to prove the efficiency of modified special hybrid moment frames composed of discretely jointed precast and post-tensioned concrete members. This aim is due to the fact that current design standards do not cover the spatial design of moment frame structures assembled by post-tensioning with special hybrid joints. This lack of standardization is coupled with the fact that previous experimental programs, available in scientific literature, deal mainly with plane structures and offer little information regarding spatial behavior. A spatial model of a modified hybrid moment frame is experimentally analyzed. The experimental results of a natural scale model test of a corner column-beams sub-structure, cut from an actual multilevel building tested to seismic type loading are presented in order to highlight the behavior of this type of structure. The test is performed under alternative cycles of imposed lateral displacements, up to a storey drift ratio of 0.035. Seismic response of the spatial model is discussed considering the acceptance criteria for reinforced concrete frame structures designed based on experimental tests, as well as some of its major sustainability features. The results obtained show an overall excellent behavior of the system. The joint detailing allows for quick and cheap repairs after an accidental event and a self-balancing behavior of the system that ensures it can be used almost immediately after an accidental event it.

Keywords: Modified hybrid joint, seismic type loading response, self-balancing structure, acceptance criteria.

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Authors: N. Senu, I. A. Kasim, F. Ismail, N. Bachok

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In this paper zero-dissipative explicit Runge-Kutta method is derived for solving second-order ordinary differential equations with periodical solutions. The phase-lag and dissipation properties for Runge-Kutta (RK) method are also discussed. The new method has algebraic order three with dissipation of order infinity. The numerical results for the new method are compared with existing method when solving the second-order differential equations with periodic solutions using constant step size.

Keywords: Dissipation, Oscillatory solutions, Phase-lag, Runge- Kutta methods.

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Authors: Rekha P. Labade, Shankar B. Deosarkar, Narayan Pisharoty

Abstract:

In this article design and optimization of square printed monopole antenna for wireless application is proposed. Theory of characteristics mode (TCM) is used for analysis of current modes on the antenna. TCM analysis shows that beveled ground plane improves the impedance bandwidth. The antenna operates over the frequency range from 1.860 GHz to 5 GHz for a VSWR ≤ 2, covering the GSM (1900-1990MHz), IMT-2000(1920-2170MHz), Bluetooth (2.400-2484 MHz) and lower band of ultrawideband (UWB). Stable radiation pattern shows minimal pulse distortion. The radiation pattern is omni-directional along the H-plane and figure of eight along the E-plane. Size of proposed antenna is 39 mm x 29 mm x 1.6mm. Antenna is simulated using CAD FEKO suite (6.2) using method of moment. A prototype antenna is fabricated using FR4 dielectric substrate with a dielectric constant of 4.4 and loss tangent of 0.02 to validate the simulated and measured results of the proposed antenna. Measured results are in good agreement with simulated results.

Keywords: Destructive Ground Surface (DGS), Method of moment, Theory of characteristics mode, UWB, VSWR.

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Authors: Munawwar Mohabuth, Andrei Kotousov, Ching-Tai Ng

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On the basis of the theory of nonlinear elasticity, the effect of homogeneous stress on the propagation of Lamb waves in an initially isotropic hyperelastic plate is analysed. The equations governing the propagation of small amplitude waves in the prestressed plate are derived using the theory of small deformations superimposed on large deformations. By enforcing traction free boundary conditions at the upper and lower surfaces of the plate, acoustoelastic dispersion equations for Lamb wave propagation are obtained, which are solved numerically. Results are given for an aluminum plate subjected to a range of applied stresses.

Keywords: Acoustoelasticity, dispersion, finite deformation, lamb waves.

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Authors: Davod Khojasteh Salkuyeh, Sayyed Hasan Azizi

Abstract:

We study the semiconvergence of Gauss-Seidel iterative methods for the least squares solution of minimal norm of rank deficient linear systems of equations. Necessary and sufficient conditions for the semiconvergence of the Gauss-Seidel iterative method are given. We also show that if the linear system of equations is consistent, then the proposed methods with a zero vector as an initial guess converge in one iteration. Some numerical results are given to illustrate the theoretical results.

Keywords: rank deficient least squares problems, AOR iterativemethod, Gauss-Seidel iterative method, semiconvergence.

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Authors: J.S.C. Prentice

Abstract:

The RK1GL2X3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on the RK1GL2 method which, in turn, is a particular case of the general RKrGLm method. The RK1GL2X3 method is a fourth-order method, even though its underlying Runge-Kutta method RK1 is the first-order Euler method, and hence, RK1GL2X3 is considerably more efficient than RK1. This enhancement is achieved through an implementation involving triple-nested two-point Gauss- Legendre quadrature.

Keywords: RK1GL2X3, RK1GL2, RKrGLm, Runge-Kutta, Gauss-Legendre, initial value problem, local error, global error.

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Authors: Raphael de Oliveira Garcia, Samuel Rocha de Oliveira

Abstract:

We have developed a new computer program in Fortran 90, in order to obtain numerical solutions of a system of Relativistic Magnetohydrodynamics partial differential equations with predetermined gravitation (GRMHD), capable of simulating the formation of relativistic jets from the accretion disk of matter up to his ejection. Initially we carried out a study on numerical methods of unidimensional Finite Volume, namely Lax-Friedrichs, Lax-Wendroff, Nessyahu-Tadmor method and Godunov methods dependent on Riemann problems, applied to equations Euler in order to verify their main features and make comparisons among those methods. It was then implemented the method of Finite Volume Centered of Nessyahu-Tadmor, a numerical schemes that has a formulation free and without dimensional separation of Riemann problem solvers, even in two or more spatial dimensions, at this point, already applied in equations GRMHD. Finally, the Nessyahu-Tadmor method was possible to obtain stable numerical solutions - without spurious oscillations or excessive dissipation - from the magnetized accretion disk process in rotation with respect to a central black hole (BH) Schwarzschild and immersed in a magnetosphere, for the ejection of matter in the form of jet over a distance of fourteen times the radius of the BH, a record in terms of astrophysical simulation of this kind. Also in our simulations, we managed to get substructures jets. A great advantage obtained was that, with the our code, we got simulate GRMHD equations in a simple personal computer.

Keywords: Finite Volume Methods, Central Schemes, Fortran 90, Relativistic Astrophysics, Jet.

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Authors: V. Polenta, S. D. Garvey, D. Chronopoulos, A. C. Long, H. P. Morvan

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A parametric study on circular thin-walled pipes subjected to pure bending is performed. Both straight and curved pipes are considered. Ratio D/t, initial pipe curvature and internal pressure are the parameters varying in the analyses. The study is mainly FEA-based. It is found that negative curvatures (opposite to bending moment) considerably increase stiffness and buckling limit of the pipe when no internal pressure is acting and, similarly, positive curvatures decrease the stiffness and buckling limit. For internal pressurised pipes the effects of initial pipe curvature are less relevant. Results show that this phenomenon is in relationship with the cross-section deformation due to bending moment, which undergoes relevant ovalisation for no pressurised pipes and little ovalisation for pressurised pipes.

Keywords: Buckling, curved pipes, internal pressure, ovalisation, pure bending, thin-walled pipes.

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Authors: T. Watanabe

Abstract:

Combustion analysis of suspended sodium droplet is performed by solving numerically the Navier-Stokes equations and the energy conservation equations. The combustion model consists of the pre-ignition and post-ignition models. The reaction rate for the pre-ignition model is based on the chemical kinetics, while that for the post-ignition model is based on the mass transfer rate of oxygen. The calculated droplet temperature is shown to be in good agreement with the existing experimental data. The temperature field in and around the droplet is obtained as well as the droplet shape variation, and the present numerical model is confirmed to be effective for the combustion analysis.

Keywords: Combustion, analysis, sodium, droplet.

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Authors: M. Pasbani Khiavi, A. R. M. Gharabaghi, K. Abedi

Abstract:

The mechanical behavior of porous media is governed by the interaction between its solid skeleton and the fluid existing inside its pores. The interaction occurs through the interface of gains and fluid. The traditional analysis methods of porous media, based on the effective stress and Darcy's law, are unable to account for these interactions. For an accurate analysis, the porous media is represented in a fluid-filled porous solid on the basis of the Biot theory of wave propagation in poroelastic media. In Biot formulation, the equations of motion of the soil mixture are coupled with the global mass balance equations to describe the realistic behavior of porous media. Because of irregular geometry, the domain is generally treated as an assemblage of fmite elements. In this investigation, the numerical formulation for the field equations governing the dynamic response of fluid-saturated porous media is analyzed and employed for the study of transient wave motion. A finite element model is developed and implemented into a computer code called DYNAPM for dynamic analysis of porous media. The weighted residual method with 8-node elements is used for developing of a finite element model and the analysis is carried out in the time domain considering the dynamic excitation and gravity loading. Newmark time integration scheme is developed to solve the time-discretized equations which are an unconditionally stable implicit method Finally, some numerical examples are presented to show the accuracy and capability of developed model for a wide variety of behaviors of porous media.

Keywords: Dynamic analysis, Interaction, Porous media, time domain

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Authors: Abraham Varghese, Kannan Balakrishnan, Reji R. Varghese, Joseph S. Paul

Abstract:

Magnetic Resonance Imaging play a vital role in the decision-diagnosis process of brain MR images. For an accurate diagnosis of brain related problems, the experts mostly compares both T1 and T2 weighted images as the information presented in these two images are complementary. In this paper, rotational and translational invariant form of Local binary Pattern (LBP) with additional gray scale information is used to retrieve similar slices of T1 weighted images from T2 weighted images or vice versa. The incorporation of additional gray scale information on LBP can extract more local texture information. The accuracy of retrieval can be improved by extracting moment features of LBP and reweighting the features based on users feedback. Here retrieval is done in a single subject scenario where similar images of a particular subject at a particular level are retrieved, and multiple subjects scenario where relevant images at a particular level across the subjects are retrieved.

Keywords: Local Binary pattern (LBP), Modified Local Binary pattern (MOD-LBP), T1 and T2 weighted images, Moment features.

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Authors: Roslinda Nazar, Amin Noor, Khamisah Jafar, Ioan Pop

Abstract:

In this paper, the steady laminar three-dimensional boundary layer flow and heat transfer of a copper (Cu)-water nanofluid in the vicinity of a permeable shrinking flat surface in an otherwise quiescent fluid is studied. The nanofluid mathematical model in which the effect of the nanoparticle volume fraction is taken into account is considered. The governing nonlinear partial differential equations are transformed into a system of nonlinear ordinary differential equations using a similarity transformation which is then solved numerically using the function bvp4c from Matlab. Dual solutions (upper and lower branch solutions) are found for the similarity boundary layer equations for a certain range of the suction parameter. A stability analysis has been performed to show which branch solutions are stable and physically realizable. The numerical results for the skin friction coefficient and the local Nusselt number as well as the velocity and temperature profiles are obtained, presented and discussed in detail for a range of various governing parameters.

Keywords: Heat Transfer, Nanofluid, Shrinking Surface, Stability Analysis, Three-Dimensional Flow.

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Authors: Zarina Bibi Ibrahim, Mohamed Suleiman, Khairil Iskandar Othman

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In this paper, a direct method based on variable step size Block Backward Differentiation Formula which is referred as BBDF2 for solving second order Ordinary Differential Equations (ODEs) is developed. The advantages of the BBDF2 method over the corresponding sequential variable step variable order Backward Differentiation Formula (BDFVS) when used to solve the same problem as a first order system are pointed out. Numerical results are given to validate the method.

Keywords: Backward Differentiation Formula, block, secondorder.

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Authors: Dimitrios N. Gkritzapis, Elias E. Panagiotopoulos, Dionissios P. Margaris, Dimitrios G. Papanikas

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A full six degrees of freedom (6-DOF) flight dynamics model is proposed for the accurate prediction of short and long-range trajectories of high spin and fin-stabilized projectiles via atmospheric flight to final impact point. The projectiles is assumed to be both rigid (non-flexible), and rotationally symmetric about its spin axis launched at low and high pitch angles. The mathematical model is based on the full equations of motion set up in the no-roll body reference frame and is integrated numerically from given initial conditions at the firing site. The projectiles maneuvering motion depends on the most significant force and moment variations, in addition to wind and gravity. The computational flight analysis takes into consideration the Mach number and total angle of attack effects by means of the variable aerodynamic coefficients. For the purposes of the present work, linear interpolation has been applied from the tabulated database of McCoy-s book. The developed computational method gives satisfactory agreement with published data of verified experiments and computational codes on atmospheric projectile trajectory analysis for various initial firing flight conditions.

Keywords: Constant-Variable aerodynamic coefficients, low and high pitch angles, wind.

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Authors: M. Khoshab, M. J. Sedigh

Abstract:

Dynamic systems, which in mathematical point of view are those governed by differential equations, are much more difficult to study and to predict their behavior in comparison with static systems which are governed by algebraic equations. Economical systems such as market are among complicated dynamic systems. This paper tries to adopt a very simple mathematical model for market and to study effect of supply and demand function on behavior of the market while the supply side experiences a lag due to production restrictions.

Keywords: Dynamic System, Lag on Supply Demand, Market Stability, Supply Demand Model.

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Authors: Khosrow Maleknejad, Reza Mollapourasl, Parvin Torabi, Mahdiyeh Alizadeh,

Abstract:

Sinc-collocation scheme is one of the new techniques used in solving numerical problems involving integral equations. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. So, in this paper, some properties of the Sinc-collocation method required for our subsequent development are given and are utilized to reduce integral equation of the first kind to some algebraic equations. Then convergence with exponential rate is proved by a theorem to guarantee applicability of numerical technique. Finally, numerical examples are included to demonstrate the validity and applicability of the technique.

Keywords: Integral equation, Fredholm type, Collocation method, Sinc approximation.

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Authors: Ali Shojaei-Fard

Abstract:

In this paper we consider quantum motion integrals depended on the algebraic reconstruction of BPHZ method for perturbative renormalization in two different procedures. Then based on Bogoliubov character and Baker-Campbell-Hausdorff (BCH) formula, we show that how motion integral condition on components of Birkhoff factorization of a Feynman rules character on Connes- Kreimer Hopf algebra of rooted trees can determine a family of fixed point equations.

Keywords: Birkhoff Factorization, Connes-Kreimer Hopf Algebra of Rooted Trees, Integral Renormalization, Lax Pair Equation, Rota- Baxter Algebras.

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