Abstract: As we know, most differential equations concerning
physical phenomenon could not be solved by analytical method. Even if we use Series Method, some times we need an appropriate change of variable, and even when we can, their closed form solution may be
so complicated that using it to obtain an image or to examine the structure of the system is impossible. For example, if we consider Schrodinger equation, i.e.,
We come to a three-term recursion relations, which work with it takes, at least, a little bit time to get a series solution[6]. For this
reason we use a change of variable such as or when we consider the orbital angular momentum[1], it will be
necessary to solve. As we can observe, working with this equation is tedious. In this paper, after introducing Clenshaw method, which is a kind of Spectral method, we try to solve some of such equations.
Abstract: In the present analysis an unsteady laminar
forced convection water boundary layer flow is considered.
The fluid properties such as viscosity and Prandtl number are
taken as variables such that those are inversely proportional to
temperature. By using quasi-linearization technique the nonlinear
coupled partial differential equations are linearized and
the numerical solutions are obtained by using implicit finite
difference scheme with the appropriate selection of step sizes.
Non-similar solutions have been obtained from the starting
point of the stream-wise coordinate to the point where skin
friction value vanishes. The effect non-uniform mass transfer
along the surface of the cylinder through slot is studied on the
skin friction and heat transfer coefficients.
Abstract: In this paper, we establish existence and uniqueness of
solutions for a class of inverse problems of degenerate differential
equations. The main tool is the perturbation theory for linear operators.
Abstract: This paper examines the forced convection flow of
incompressible, electrically conducting viscous fluid past a sharp
wedge in the presence of heat generation or absorption with an
applied magnetic field. The system of partial differential equations
governing Falkner - Skan wedge flow and heat transfer is first
transformed into a system of ordinary differential equations using
similarity transformations which is later solved using an implicit
finite - difference scheme, along with quasilinearization technique.
Numerical computations are performed for air (Pr = 0.7) and
displayed graphically to illustrate the influence of pertinent physical
parameters on local skin friction and heat transfer coefficients and,
also on, velocity and temperature fields. It is observed that the
magnetic field increases both the coefficients of skin friction and heat
transfer. The effect of heat generation or absorption is found to be
very significant on heat transfer, but its effect on the skin friction is
negligible. Indeed, the occurrence of overshoot is noticed in the
temperature profiles during heat generation process, causing the
reversal in the direction of heat transfer.
Abstract: In this work, a radial basis function (RBF) neural network is developed for the identification of hyperbolic distributed parameter systems (DPSs). This empirical model is based only on process input-output data and used for the estimation of the controlled variables at specific locations, without the need of online solution of partial differential equations (PDEs). The nonlinear model that is obtained is suitably transformed to a nonlinear state space formulation that also takes into account the model mismatch. A stable robust control law is implemented for the attenuation of external disturbances. The proposed identification and control methodology is applied on a long duct, a common component of thermal systems, for a flow based control of temperature distribution. The closed loop performance is significantly improved in comparison to existing control methodologies.
Abstract: In this paper, a nonconforming mixed finite element method is studied for semilinear pseudo-hyperbolic partial integrodifferential equations. By use of the interpolation technique instead of the generalized elliptic projection, the optimal error estimates of the corresponding unknown function are given.
Abstract: 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.
Abstract: A dual-reciprocity boundary element method is presented
for the numerical solution of a class of axisymmetric elastodynamic
problems. The domain integrals that arise in the integrodifferential
formulation are converted to line integrals by using the
dual-reciprocity method together suitably constructed interpolating
functions. The second order time derivatives of the displacement
in the governing partial differential equations are suppressed by
using Laplace transformation. In the Laplace transform domain, the
problem under consideration is eventually reduced to solving a system
of linear algebraic equations. Once the linear algebraic equations are
solved, the displacement and stress fields in the physical domain can
be recovered by using a numerical technique for inverting Laplace
transforms.
Abstract: This paper presents a procedure of forming the
mathematical model of radial electric power systems for simulation
of both transient and steady-state conditions. The research idea has
been based on nodal voltages technique and on differentiation of
Kirchhoff's current law (KCL) applied to each non-reference node of
the radial system, the result of which the nodal voltages has been
calculated by solving a system of algebraic equations. Currents of the
electric power system components have been determined by solving
their respective differential equations. Transforming the three-phase
coordinate system into Cartesian coordinate system in the model
decreased the overall number of equations by one third. The use of
Cartesian coordinate system does not ignore the DC component
during transient conditions, but restricts the model's implementation
for symmetrical modes of operation only. An example of the input
data for a four-bus radial electric power system has been calculated.
Abstract: In this work, we obtain some analytic solutions for the (3+1)-dimensional breaking soliton after obtaining its Hirota-s bilinear form. Our calculations show that, three-wave method is very easy and straightforward to solve nonlinear partial differential equations.
Abstract: Avalanche release of snow has been modeled in the present studies. Snow is assumed to be represented by semi-solid and the governing equations have been studied from the concept of continuum approach. The dynamical equations have been solved for two different zones [starting zone and track zone] by using appropriate initial and boundary conditions. Effect of density (ρ), Eddy viscosity (η), Slope angle (θ), Slab depth (R) on the flow parameters have been observed in the present studies. Numerical methods have been employed for computing the non linear differential equations. One of the most interesting and fundamental innovation in the present studies is getting initial condition for the computation of velocity by numerical approach. This information of the velocity has obtained through the concept of fracture mechanics applicable to snow. The results on the flow parameters have found to be in qualitative agreement with the published results.
Abstract: In this paper, the criteria of Ψ-eventual stability have been established for generalized impulsive differential systems of multiple dependent variables. The sufficient conditions have been obtained using piecewise continuous Lyapunov function. An example is given to support our theoretical result.
Abstract: The Goursat partial differential equation arises in
linear and non linear partial differential equations with mixed
derivatives. This equation is a second order hyperbolic partial
differential equation which occurs in various fields of study such as
in engineering, physics, and applied mathematics. There are many
approaches that have been suggested to approximate the solution of
the Goursat partial differential equation. However, all of the
suggested methods traditionally focused on numerical differentiation
approaches including forward and central differences in deriving the
scheme. An innovation has been done in deriving the Goursat partial
differential equation scheme which involves numerical integration
techniques. In this paper we have developed a new scheme to solve
the Goursat partial differential equation based on the Adomian
decomposition (ADM) and associated with Boole-s integration rule to
approximate the integration terms. The new scheme can easily be
applied to many linear and non linear Goursat partial differential
equations and is capable to reduce the size of computational work.
The accuracy of the results reveals the advantage of this new scheme
over existing numerical method.
Abstract: This paper presents the vibrations suppression of a thermoelastic beam subject to sudden heat input by a distributed piezoelectric actuators. An optimization problem is formulated as the minimization of a quadratic functional in terms of displacement and velocity at a given time and with the least control effort. The solution method is based on a combination of modal expansion and variational approaches. The modal expansion approach is used to convert the optimal control of distributed parameter system into the optimal control of lumped parameter system. By utilizing the variational approach, an explicit optimal control law is derived and the determination of the corresponding displacement and velocity is reduced to solving a set of ordinary differential equations.
Abstract: Functional imaging procedures for the non-invasive assessment of tissue microcirculation are highly requested, but require a mathematical approach describing the trans- and intercapillary passage of tracer particles. Up to now, two theoretical, for the moment different concepts have been established for tracer kinetic modeling of contrast agent transport in tissues: pharmacokinetic compartment models, which are usually written as coupled differential equations, and the indicator dilution theory, which can be generalized in accordance with the theory of lineartime- invariant (LTI) systems by using a convolution approach. Based on mathematical considerations, it can be shown that also in the case of an open two-compartment model well-known from functional imaging, the concentration-time course in tissue is given by a convolution, which allows a separation of the arterial input function from a system function being the impulse response function, summarizing the available information on tissue microcirculation. Due to this reason, it is possible to integrate the open two-compartment model into the system-theoretic concept of indicator dilution theory (IDT) and thus results known from IDT remain valid for the compartment approach. According to the long number of applications of compartmental analysis, even for a more general context similar solutions of the so-called forward problem can already be found in the extensively available appropriate literature of the seventies and early eighties. Nevertheless, to this day, within the field of biomedical imaging – not from the mathematical point of view – there seems to be a trench between both approaches, which the author would like to get over by exemplary analysis of the well-known model.
Abstract: In this paper, the two-dimension differential transformation method (DTM) is employed to obtain the closed form solutions of the three famous coupled partial differential equation with physical interest namely, the coupled Korteweg-de Vries(KdV) equations, the coupled Burgers equations and coupled nonlinear Schrödinger equation. We begin by showing that how the differential transformation method applies to a linear and non-linear part of any PDEs and apply on these coupled PDEs to illustrate the sufficiency of the method for this kind of nonlinear differential equations. The results obtained are in good agreement with the exact solution. These results show that the technique introduced here is accurate and easy to apply.
Abstract: In this paper the Differential Quadrature Method (DQM) is employed to study the coupled lateral-torsional free vibration behavior of the laminated composite beams. In such structures due to the fiber orientations in various layers, the lateral displacement leads to a twisting moment. The coupling of lateral and torsional vibrations is modeled by the bending-twisting material coupling rigidity. In the present study, in addition to the material coupling, the effects of shear deformation and rotary inertia are taken into account in the definition of the potential and kinetic energies of the beam. The governing differential equations of motion which form a system of three coupled PDEs are solved numerically using DQ procedure under different boundary conditions consist of the combinations of simply, clamped, free and other end conditions. The resulting natural frequencies and mode shapes for cantilever beam are compared with similar results in the literature and good agreement is achieved.
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.
Abstract: In this paper, a self starting two step continuous block
hybrid formulae (CBHF) with four Off-step points is developed using
collocation and interpolation procedures. The CBHF is then used to
produce multiple numerical integrators which are of uniform order
and are assembled into a single block matrix equation. These
equations are simultaneously applied to provide the approximate
solution for the stiff ordinary differential equations. The order of
accuracy and stability of the block method is discussed and its
accuracy is established numerically.
Abstract: This paper considers the development of a two-point
predictor-corrector block method for solving delay differential
equations. The formulae are represented in divided difference form
and the algorithm is implemented in variable stepsize variable order
technique. The block method produces two new values at a single
integration step. Numerical results are compared with existing
methods and it is evident that the block method performs very well.
Stability regions of the block method are also investigated.