Stochastic Finite Element Technique for Stochastic One-Dimension Time-Dependent Differential Equations with Random Coefficients

The stochastic finite element method (SFEM) is employed for solving stochastic onedimension time-dependent differential equations with random coefficients. SFEM is used to have a fixed form of linear algebraic equations for polynomial chaos coefficients of the solution process. Four fixed forms are obtained in the cases of stochastic heat equation with stochastic heat capacity or heat conductivity coefficients and stochastic wave equation with stochastic mass density or elastic modulus coefficients. The relation between the exact deterministic solution and the mean of solution process is numerically studied.

In this paper, SFEM is applied on stochastic heat and wave equations.The stochastic coefficients are decomposed by Karhunen-Loeve (K-L) expansion.The obtained set of ordinary differential equations is solved using the θ-dependent family.Then the solution process at every time step is projected on two-dimension first-order polynomial chaos.The mean of the solution process is obtained under different values of the point variance of stochastic coefficient.

The Karhunen-Loeve decomposition
The use of K-L expansion with orthogonal deterministic basis functions and uncorrelated random coefficients gained interest because of its biorthogonal property, that is, both the deterministic basis functions and the corresponding random coefficients are orthogonal.Let ω(x) denote the mean value of ω(x,θ) and C(x 1 ,x 2 ) its covariance function which is bounded and positive definite.It has spectral decomposition as [13] where λ n and f n (x) are the eigenvalues and the eigenvectors of the covariance kernel, respectively.They are the solutions of the homogeneous Fredholm integral equation of second kind given by Clearly, ω(x,φ) can be written as ω(x,φ) = ω(x) + α(x,φ), (2.3) where α(x,φ) is a process with zero mean and covariance function C(x 1 ,x 2 ).Finally, the K-L decomposition of the field α(x,φ) is given by where ζ n (φ) is a set of uncorrelated random variables.For example, consider a homogeneous Gaussian process with exponential covariance The eigenfunctions of this covariance kernel are where ω i is the solution of nonlinear equation Then the eigenvalues can be evaluated from the relation (2.8)

The polynomial chaos
The polynomial chaos is a particular basis of the random variables space based on Hermite polynomial of independent standard random variables ζ 1 , ζ 2 ,...,ζ ∞ .Classically, the one-dimension Hermite polynomial is defined by The multivariable Hermite polynomial can be defined as tensor product of Hermite polynomial.Consider the multi-index The multivariable Hermite polynomial associated with this sequence is Finally, any random variable k(φ) with finite variance can be expressed as [13] k where a i are deterministic constants and H i are enumeration of the H α .The expansion is convergent in the mean square sense.In the application of polynomial chaos, the dimension is selected according to the number of random variables in K-L expansion [14].

Stochastic heat equation
The unsteady stochastic heat equation for a spatially varying medium, in the absence of convection, is [15] c ∂U ∂t subjected to the following boundary and initial conditions The variational form of (4.1) over a typical element is where Let the approximation of solution over the element be given by where {ψ i } are linear interpolating functions.Substituting by (4.6) into (4.4) and by V = ψ j (x), we get (4.7) Let Assembling of the elements matrices, we obtain 4.1.Case 1. Stochastic heat capacity coefficient.Let the heat capacity coefficient be stochastic process; the two-dimension K-L expansion for that process is Equation (4.8) will be divided into the following parts: Hence, (4.11) will be in the form The previous equation is time-dependent system of ordinary differential equations which can be approximated to obtain a system of algebraic equations.Using the θ family of approximation which approximates a weighted average of time derivative of a dependent variable at two consecutive time steps [16], that is, where {•} n refers to the value of the enclosed vector quantity at time t = t n and Δt = t n+1 − t n is the time step.From (4.15), we can obtain a number of well-known difference schemes by choosing different values of θ like Projecting the solution at every time step on two-dimension first-order chaos polynomial, we get Substituting by (4.18) into (4.17) and making inner product with 6 Differential Equations and Nonlinear Mechanics Equation (4.19) is used to obtain polynomial chaos coefficients of the solution process over all nodes of the domain at every time step.To complete the applicability of this form, we project the initial condition (4.3) on polynomial chaos basis to get the coefficients {a i } 0 at t = 0. Hence [17], (4.20) The coefficients {a 1 } 0 and {a 2 } 0 are assumed to be zero because of deterministic initial condition.Therefore, (4.19) can be used to obtain the polynomial chaos coefficients of the solution {a i } 1 at time t = t 1 .Finally, the polynomial chaos coefficients at time t = t n+1 can be obtained in terms of the known coefficients at time t = t n .At this stage, the polynomial chaos coefficients of the solution are known for each node at every time step.Finally, the mean and variance of the solution process can be obtained from the relations (4.21)

Case 2. Stochastic heat conductivity coefficient. Let the heat conductivity coefficient be stochastic process with two-dimension K-L expansion in the form
Equation (4.9) will be divided into the following parts: Applying the time approximation (4.15) on (4.24), we obtain Substituting by (4.18) into (4.25),we get We can get the polynomial chaos coefficient of the solution process at every time step as in the previous case.

Stochastic wave equation
Consider the following stochastic wave equation that governs the axial motions in a rod with the mass density ρ, elastic modulus E, and unit cross-sectional area a = 1, namely [18], where c = ρa and A = Ea, subjected to boundary conditions and initial conditions (5.4) The variational form of (5.1) over a typical element is (5.9) By assembling the elements matrices, we get (5.10)

Case 1. Stochastic mass density.
Let the mass density coefficient be a stochastic process with two-dimension K-L expansion in the form (5.11) Equation (5.7) will be divided into the following parts: (5.12) Then (5.10) becomes (5.13) M. M. Saleh et al. 9 Applying Euler method for time approximation of the second derivative on (5.13) and rearranging the terms, we obtain (5.14) Projecting the solution on two-dimension first-order chaos polynomial as in (4.18) and making inner product with H i , we get (5.15) The coefficients of polynomial chaos {a i } 0 are defined by (4.20).Applying Euler method at t = 0 on initial velocity (5.4), we get then a 0 1 = Δt v x j + a 0 0 , j = 1,2,...,N + 1, (5.17) Therefore, (5.15) can be used to obtain the polynomial chaos coefficients of the solution at t = t 2 .Finally, the polynomial chaos coefficients of the solution at time t = t n+2 are obtained in terms of the known coefficients at times t = t n+1 and t = t n .

Case 2. Stochastic elastic modulus. Let the elastic modulus coefficient be a stochastic process with two-dimension K-L expansion in the form
(5.18) Equation (5.8) will be divided into the following parts: (5.19) Then (5.10) becomes (5.20) Applying Euler method and rearranging the terms, we get Projecting the solution on two-dimension first-order chaos polynomial and making inner product with H i , we get (5.22) We can get the chaos polynomial coefficients of the solution at every time step as in the previous case.

Numerical examples
In this section, we will apply the fixed forms on the following examples with studying the effect of stochastic parameters on the solution moments.The approach of the mean of stochastic solution to the exact deterministic one is studied numerically.
subjected to boundary and initial conditions Let the stochastic process be with mean one and exponential covariance in which λ i and f i are described by (2.6)-(2.8).Figures 6.1 From Figures 6.1 and 6.2, the type of stochastic coefficient affects only the standard deviation of the solution process.Figures 6.3 and 6.4 illustrate the effect of variation of σ 2 on the solution parameters.From Figures 6.3 and 6.4, the variation of σ 2 is more effective on standard deviation than the mean of the solution process.
The exact deterministic solution of this example at c = A = 1 is U = t exp(2x).( 6.3) Table 6.1 illustrates the approach of the mean of stochastic solution to this exact deterministic one.The mean of the stochastic solution approaches the exact deterministic solution as the point variance of stochastic coefficient decreases.Example 6.2.
subjected to boundary conditions and initial conditions

.6)
Let A be stochastic process with mean one and exponential covariance, in which λ i and f i are described by (2.6)-(2.8). Figure 6.5 shows the mean and standard deviation of the solution process at very small point variance σ 2 = .000001 of the stochastic process A. From Figure 6.6, the variation of σ 2 is more effective on standard deviation than the mean of the solution process.
The exact deterministic solution of this example at A = 1 is U = t Sin(2x).(6.7) Table 6.2 illustrates the approach of the mean of stochastic solution to this exact deterministic one.The mean of the stochastic solution approaches the exact deterministic solution as the point variance of stochastic coefficient decreases.

Conclusion
The stochastic finite element method based on K-L decomposition and projection of the solution on chaos polynomials is an effective and easy method for solving the stochastic one-dimension time-dependent partial differential equation.Two fixed forms are obtained for chaos polynomial coefficients of the solution in the case of stochastic heat equation with stochastic heat capacity (4.19) or stochastic heat conductivity (4.26) coefficients.Another two fixed forms are obtained for chaos polynomials coefficients of the solution in the case of stochastic wave equation with stochastic mass density (5.15) or stochastic elastic modulus (5.22) coefficients.The stochastic parameter σ 2 has a great effect on the standard deviation of the solution process but has a very small effect on the mean of solution process.The mean of the stochastic solution approaches the exact deterministic solution as the point variance of stochastic coefficient decreases.

Figure 6 . 3 .Figure 6 . 4 .
Figure 6.3.The effect of σ 2 on the solution moments at stochastic heat conductivity A. (a) Effect of σ 2 on the mean of solution at t = 1second.(b) Effect of σ 2 on SD of the solution at t = 1second.