On the solution of random linear difference equations with Laplace transform method

In this study, Laplace transformation, which is very important for solutions to initial value problems, is examined. To solve the initial value problem of a discrete‐time equation, Laplace implements the conversion method. Here, Laplace transformation is used to obtain an approach to the solutions of random difference equations formed by randomizing components of deterministic difference equations. For random behavior of linear difference equations under random effects, uniform, geometric, binomial, Poisson, and Bernouilli distributions are used, and approximate expected value, variance, standard deviation, and confidence interval of equations obtained by Laplace transformation are calculated. The results were obtained through the Maple package program.


| INTRODUCTION
Difference equations are one of the rich branches of mathematics that emerged with the discretization of differential equations and their numerical results [1][2][3].The transformation of the French mathematician Pierre Simon Laplace is a method used to study engineering and mathematical problems [4].Laplace transformation, one of the most powerful integral transformations, is considered a very good way to solve ordinary and partial differential equations [5].The role of Laplace transformation method in the solution of differential equations and Z-transformation method in the solution of difference equations is similar.In recent years, some mathematicians have implemented the Laplace transformation to solve different equations [6].In particular, many researchers such as Atici and Eloe [5], Holm [6,7], and Bohner and Guseinov [8] are involved in this transformation.The approximately expected values and variances are obtained by applying the Laplace conversion method to obtain an approach to the solution of the random linear difference equation by randomizing the difference equations with various probability distributions (Tables 1 and 2) [9][10][11][12][13][14].Some studies done in recent years in this are given in [15][16][17][18][19][20][21][22].Definition 1.Let n ℕ ¼ 0, 1, … f g be an independent variable, and then, the function with unknown x such that is called a difference Equation (1).
Definition 2. f , t > 0 get the single value function of the time variable and the s parameter.The Laplace transformation of f t ð Þ is defined as follows [23]: gis indicated by [4].
T A B L E 1 Laplace transform properties.

| DISCRETE TIME PROBABILITY DISTRIBUTIONS
In this section, definitions related to some probability concepts used are given [9,10,24].

| Discrete uniform distribution
Definition 3. Let k be a positive bit integer.A random variable X with probability function is called a discrete uniform random variable.
If there are only two results for an X random variable, X is called a Bernoulli random variable.Bernoulli variables are obtained with the probability mass function.
Let the total number of those who succeeded in n independent Bernoulli trials be the random variable X.For a single experiment, the probability of success is denoted by p, and the probability of failure is 1 À p ð Þ.The binomial random variable X has the following probability function Calculation of consecutive binomial probabilities is as follows: The number of experiments done to obtain the first desired result (success or unsuccessful) in a Bernoulli experiment repeated n times in succession is called a geometric random variable X.The distribution of this variable is called the geometric distribution and the probability function of the geometric random variable X, with probability of unsuccessfulness q ¼ 1 À p and probability of success p in a single experiment: The Taylor expansion of the function e y and the probability function gives e y ¼ P ∞ i¼0 y i i! :

| NUMERICAL RESULTS
Here are some numerical examples that are given for random linear difference equations randomized by various probability distributions.

| Example 1
Examine the solution behavior of the equation with the Laplace method, A, random variables with uniform distribution α ¼ 3, β ¼ 5, B, random variables with geometric distribution p ¼ 1  5 , q ¼ 4 5 .

| Solution
This equation is

If we consider this equation with the help of Laplace transformation f
and similarly obtain.Then, the equation and returns to the n variable, The solution is found.Higher order moments of random variables are needed to calculate the approximate expected value and variance.X random variable with uniform and geometric distribution Hence, the expected value and variance of the random variable X are To find the numerical characteristics of ( 5), we start with the expectation: The numerical characteristics of the approximate solutions of the random linear difference equation obtained from random Laplace transform were obtained by calculating as follows.
obtained and variance and the fact that the standard deviation equals to the square root of the variance, Similarly, the confidence intervals fort he expected values of the random variables could be obtained via these standard deviations as Confidence Interval equals to Þ .For K ¼ 3, this formula gives an confidence internal for the approximate expected value of the normally distributed random variable (Figure 1).

| Example 2
Examine the solution behavior of the equation with the Laplace transform method, including random variables p ¼ 2  3 and q ¼ 1 3 with binomial distribution A and B.
F I G U R E 1 Confidence interval and variance obtained from the Laplace transform of ( 5).[Colour figure can be viewed at wileyonlinelibrary.com] If we consider this equation with the help of Laplace transformation f 0 ¼ A,f 1 ¼ B with random initial conditions and let's get it using Equations ( 3) and (4) in Example 1 above.
Then, the equation g is divided into simple fractions and the result and returns to the n variable, The moment generating function of a binomial distributed random variable X $ Bin p,q ð Þ is given as Hence, the expected value and variance of the random variable X are To find the numerical characteristics of ( 6), we start with the expectation Approximate numerical characteristics of the random linear difference equation obtained from random Laplace transform are calculated as follows: Variance and the fact that the standard deviation equals to the square root of the variance, Similarly, it gives the confidence interval of the binomial distribution for K ¼ 3 is given in Figure 2 below.

| Example 3
x nþ Examine the solution behavior of the equation with random variables λ ¼ 3 with Poisson distribution A and B using the Laplace transform method.

| Solution
This equation is In this case, the equation becomes If we consider this equation with the help of Laplace transformation and similarly obtain.Then, the equation g is divided into simple fractions, and the result Confidence interval and variance obtained from the Laplace transform of ( 6).[Colour figure can be viewed at wileyonlinelibrary.com] and returns to the n variable, The moment generating function of a Poisson distributed random variable X $ Pois λ ð Þ is given as Hence, the expected value and variance of the random variable X are To find the numerical characteristics of (7), we start with the expectation Approximate numerical characteristics of the random linear difference equation obtained from random Laplace transform are calculated as follows: Variance and the fact that the standard deviation equals to the square root of the variance, Similarly, it gives the confidence interval of the Poisson distribution for K ¼ 3 is given in Figure 3 below.

| Example 4
Examine the solution behavior of the equation with the Laplace transform method, including random variables p ¼ 1  4 and q ¼ 3 4 with Bernouilli distribution A and B.

| Solution
This equation is If we consider this equation with the help of Laplace transformation f 0 ¼ A,f 1 ¼ B with random initial conditions and let's get it using Equations ( 3) and ( 4) in Example 1 above.
Then, the equation . Hence, and returns to the n variable, The moment generating function of a Bernouilli distributed random variable X $ B p,q ð Þ is given as Hence, the expected value and variance of the random variable X are To find the numerical characteristics of (8), we start with the expectation: Confidence interval and variance obtained from the Laplace transform of ( 8).[Colour figure can be viewed at wileyonlinelibrary . The numerical characteristics of the approximate solutions of the random linear difference equation obtained from random Laplace transform were obtained by calculating as follows.
obtained.Variance and the fact that the standard deviation equals to the square root of the variance, s Similarly, it gives the 99% confidence interval of the Bernouilli distribution is given in Figure 4 below.

| CONCLUSION
In this study, the most powerful Laplace transform method of transform is applied to solve random linear difference equations.Probability distributions are used for random behavior of difference equations.In the first example, the expected value, variance, standard deviation, and confidence interval were calculated by applying random parameters with uniform distribution and geometric distribution.In the second example, the expected value, variance, standard deviation, and confidence interval were calculated by converting the parameters into a random variable with a binomial distribution.In the third example, a random variable with a Poisson distribution was made, and the expected value, variance, standard deviation, and confidence interval were calculated.In the fourth example, the expected value, variance, standard deviation, and confidence interval were calculated by making it a random variable using the Bernouilli distribution.Thus, the Laplace transform method has been shown to be the appropriate method for the solutions of difference equations.

F G U R E 3
Confidence interval and variance obtained from the Laplace transform of(7).[Colour figure can be viewed at wileyonlinelibrary.com]