Second-order ordinary differential equations

The boundary conditions for the two simultaneous second-order ordinary differential equations, 9.2-28 and -29, may be chosen as ... 

We obtain a second-order ordinary differential equation ... 

Assertion 9. Let ansatz (53) reduce system (46) to a system of second-order ordinary differential equations. Then the reduced system is necessarily of the form... 

Systems (87) and (91) contain 12 nonlinear second-order ordinary differential equations with variable coefficients. That is why there is little hope for constructing their general solutions. Nevertheless, it is possible to obtain particular solutions of system (87), whose coefficients are given by formulas 2-4 from (91). 

Equation (140) is a second-order ordinary differential equation and standard methods [74] show that its solution must be of the form... 

This is a linear second-order ordinary differential equation that can be solved explicitly by using the characteristic equation... 

In Eq. (32), we have included the full spectrum of a second-order ordinary differential equation with negative bound state eigenvalues and the continuum being the positive real axis. The free-particle background is mbee = i /k. In this case, the full m-function becomes [in the equation below we have introduced a natural generalization of Js, the "jump" or imaginary part, see Eq. (36) for the general case]... 

This is a system of four coupled, second-order ordinary differential equations. The solution of each equation must satisfy two boundary conditions. The first is dictated by the symmetry of the crystallite (sphere) which requires the concentration gradients of all species to disappear at the center of the crystallite ... 

In equilibrium NVE Molecular Dynamics simulation new molecular positions are obtained solving by Newton s equation of motion numerically. To solve Equation Al we use Equation A3 and also specify the initial and boundary conditions of our d dimensional system. This results in a set of d x TV coupled second-order ordinary differential equations and a total of d x TV degrees of freedom. This set of equations are discretized and new positions and velocities for each atom is found numerically by integrating forward in time. Below we give the MD recipe ... 

This equation is a second-order ordinary differential equation. It is nonlinear when is other than zero- or first-order. 

The velocity distribution inside the EPR, z [0,h, can be obtained by means of a solution of the second-order ordinary differential equation (3.133). As the equation uses the quadratic law for the distributed drag force, it allows a numerical solution only. The following algorithm can be suggested ... 

Data can be generated and written out to a text file (i.e., a. txt file). For example, we can use Maple to solve the second order ordinary differential equation... 

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the steady state distribution of heat or concentration across the slab or the material in which the experiment is performed. This steady state process involves solving second order ordinary differential equations subject to boundary conditions at two ends. Whenever the problem requires the specification of boundary conditions at two points, it is often called a two point boundary value problem. Both linear and nonlinear boundary value problems will be discussed in this chapter. We will present analytical solutions for linear boundary value problems and numerical solutions for nonlinear boundary value problems. 

Pick s law of diffusion and Fourier s law of conduction are usually represented by second order ordinary differential equations (ODEs). In this chapter, we describe how one can obtain analytical solutions for linear boundary value problems using Maple and the matrix exponential. 

Steady state mass or heat transfer in solids and current distribution in electrochemical systems involve solving elliptic partial differential equations. The method of lines has not been used for elliptic partial differential equations to our knowledge. Schiesser and Silebi (1997)[1] added a time derivative to the steady state elliptic partial differential equation and applied finite differences in both x and y directions and then arrived at the steady state solution by waiting for the process to reach steady state. [2] When finite differences are applied only in the x direction, we arrive at a system of second order ordinary differential equations in y. Unfortunately, this is a coupled system of boundary value problems in y (boundary conditions defined at y = 0 and y = 1) and, hence, initial value problem solvers cannot be used to solve these boundary value problems directly. In this chapter, we introduce two methods to solve this system of boundary value problems. Both linear and nonlinear elliptic partial differential equations will be discussed in this chapter. We will present semianalytical solutions for linear elliptic partial differential equations and numerical solutions for nonlinear elliptic partial differential equations based on method of lines. 

Spatial coordinates and integrating numerically in time. In this chapter, we apply finite differences in one of the directions (x), convert the governing equation and boundary conditions in x to finite difference form. The resulting system of coupled nonlinear boundary values problems (second order ordinary differential equations in y) are then solved using Maple s dsolve numeric command for boundary value problems (see chapter 3.2.8). 

The Laplace transformation of Eq. (10) is a second-order ordinary differential equation in 2/. Using the transformed mobile phase boundary conditions, the mobile phase solution in the Laplace domain is obtained. 

Newton s second law relates the motion of a particle to the force acting on it. If we have N molecules, then Newton s second law yields 3N second-order, ordinary differential equations. For an isolated system of particles with position vectors r, and momenrnm p, the total energy of the system (potential energy plus kinetic energy) will be conserved. The total energy of the system will take the form ... 

A.3.C Second-Order Ordinary Differential Equation Methods of solving differential equations of the type... 

A first step to understand the nature of these fronts is to consider (4.16) on the infinite line and particularize it to the class of solutions (4.18). One obtains a second order ordinary differential equation for... 

Analysis of receptor/ligand interactions is made much more complicated by explicit consideration of these physical features. Without such considerations, we are often able to model receptor phenomena with first-order ordinary differential equations based on straightforward mass-action kinetic species balances. When diffusion and probabilistic effects are taken into account, the models can easily give rise to partial differential equations, second-order ordinary differential equations, extremely large sets of first-order ordinary differential equations, and/or probabilistic differential equations. 

The second order ordinary differential equation (5E.1) can be rewritten in form of a system of first order differential equations (Lohnstein 1906),... 

This results in a set of coupled hyperradial second-order ordinary differential equations (called coupled-channel equations) in the coefficients n"xlu(p p), which is cast as a second order differential equation in a matrix b7111 of the form... 

C. Zhu and H. Nakamura, Stokes constants for a certain class of second-order ordinary differential equations, J. Math Phys. 33 2697 (1992). 

Equation (6.53), whose second term on the left side is now constant, is a second-order ordinary differential equation (ODE). In order to solve this ODE for Q i), two initial conditions are necessary. They come in naturally, by realizing that at time t — 0, the size of crystals is L = Lq (seeds or stable nuclei size). The first initial condition is obtained from Eq. (6.52) as... 

---