Laplace transform technique equations

The Laplace transform technique also allows the reduction of the partial differential equation in two variables to one of a single variable In the present case. 

Solving equation (1-8) (using Laplace transform techniques) yields the time evolution of the current of a spherical electrode ... 

Pearson, E. M. Halicioglu, T. Tiller, W. A., Laplace-transform technique for deriving thermodynamic equations from the classical microcanonical ensemble, Phys. Rev. A 1988, 32, 3030-3039... 

The statement cA = c0/ (1 + K) in Eqs. (157a and b) above is tantamount to saying that cA + cB = Co, where c0 is the total concentration of both species of the dissolved solute. If the diffusivities SDA8 and DBs are assumed to be equal, then cB can be eliminated from Eqs. (155) and (156) and a fourth-order, linear partial-differential equation is obtained. The solution of this equation consistent with the conditions in Eq. (157) is obtainable by Laplace transform techniques (S9). Sherwood and Pigford discuss the results in terms of the behavior of the liquid-film mass transfer coefficient. 

Use of Computer Simulation to Solve Differential Equations Pertaining to Diffusion Problems. As shown earlier (Section 4.2.11), differential equations used in the solutions of Fick s second law can often be solved analytically by the use of Laplace transform techniques. However, there are some cases in which the equations can be solved more quickly by using an approximate technique known as the finite-difference method (Feldberg, 1968). 

Equation (122) together with the conditions (127)—(131) can be solved by the Laplace transform technique, details of which are given in Chap. 2, Sect. 3 and Appendix A. Collins and Kimball [4] showed that time-dependent recombination probability q(t r0, t0) was... 

This equation can be solved by Laplace transform techniques and Mt expressed as modified spherical Bessel functions [28]. However, because the boundary conditions on M are radically symmetric, only the / = 0 (i.e. S-wave) component is of interest. 

Equations (3-55) to (3-59) were rewritten in dimensionless form and solved analytically by the Laplace transformation technique. The final solution, the theoretical impulse response, can be obtained as follows 69... 

Fig. 2,17. Pressure and temperature dependence of reaction (24). The solid lines are a calculated fit using an inverse Laplace transform/Master Equation technique described in...

This IVP can be solved easily using the Laplace transform technique. The procedure presented above for the Laplace transformation technique can be used for solving this example. When solving a second order differential equation, the... 

Maple s dsolve command was used to solve linear ODEs in section 2.1.6. In our opinion, exponential matrix method is the best method to arrive at an elegant analytical solution. The Laplace transform technique illustrated in section 2.1.5 could be used for integro-differential equations. Maple s dsolve command has to be used if the exponential matrix method fails. 

The matrix exponential can also be obtained by using the Laplace transform technique. Taking the Laplace transform of the governing equations now written in lower case X (see Ogata page 725).[1]... 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a parabolic partial differential equation. For steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in semi-infinite domains using the Laplace transform technique, a similarity solution technique and Maple. In addition, we describe how numerical similarity solutions can be obtained for nonlinear partial differential equations in semi-infinite domains. 

Parabolic partial differential equations are solved using the Laplace transform technique in this section. Diffusion like partial differential equations are first order... 

In this chapter, analytical solutions were obtained for parabolic and elliptic partial differential equations in semi-infinite domains. In section 4.2, the given linear parabolic partial differential equations were converted to an ordinary differential equation boundary value problem in the Laplace domain. The dependent variable was then solved in the Laplace domain using Maple s dsolve command. The solution obtained in the Laplace domain was then converted to the time domain using Maple s inverse Laplace transform technique. Maple is not capable of inverting complicated functions. Two such examples were illustrated in section 4.3. As shown in section 4.3, even when Maple fails, one can arrive at the transient solution by simplifying the integrals using standard Laplace transform formulae. 

Both the Laplace transform and the similarity solution techniques are powerful techniques for partial differential equations in semi-infinite domains. The Laplace transform technique can be used for all linear partial differential equations with all possible boundary conditions. The similarity solution can be used only if the independent variables can be combined and if the boundary conditions in x and t can be converted to boundary conditions in the combined variable. In addition, unlike the Laplace transform technique, the similarity solution technique cannot handle partial differential equations in which the dependent variable appears explicitly. The Laplace transform cannot handle elliptic or nonlinear partial differential equations. The similarity solution can be used for elliptic and for a few nonlinear partial differential equations as shown in section 4.6. There are thirteen examples in this chapter. 

Laplace Transform Technique for Partial Differential Equations... 

Linear first order hyperbolic partial differential equations are solved using Laplace transform techniques in this section. Hyperbolic partial differential equations are first order in the time variable and first order in the spatial variable. The method involves applying Laplace transform in the time variable to convert the partial differential equation to an ordinary differential equation in the Laplace domain. This becomes an initial value problem (IVP) in the spatial direction with s, the Laplace variable, as a parameter. The boundary conditions in x are converted to the Laplace domain and the differential equation in the Laplace domain is solved by using the techniques illustrated in chapter 2.1 for solving linear initial value problems. Once an analytical solution is obtained in the Laplace domain, the solution is inverted to the time domain to obtain the final analytical solution (in time and spatial coordinates). This is best illustrated with the following example. 

---