# Some Notes to Extend the Study on Random Non-Autonomous Second Order Linear Differential Equations Appearing in Mathematical Modeling

^{*}

## Abstract

**:**

**2018**, 392, 1–29, doi:10.1186/s13662-018-1848-8). We restate the main theorem therein that deals with the homogeneous case, so that the hypotheses are clearer and also easier to check in applications. Another novelty is that we tackle the non-homogeneous equation with a theorem of existence of mean square analytic solution and a numerical example. We also prove the uniqueness of mean square solution via a habitual Lipschitz condition that extends the classical Picard theorem to mean square calculus. In this manner, the study on general random non-autonomous second order linear differential equations with analytic data processes is completely resolved. Finally, we relate our exposition based on random power series with polynomial chaos expansions and the random differential transform method, the latter being a reformulation of our random Fröbenius method.

## 1. Introduction

## 2. Homogeneous Case

**Proposition**

**1**

**.**Let ${\left\{{Z}_{n}\right\}}_{n=1}^{\infty}$ and Z be second order random variables. If ${Z}_{n}$ converges to Z as $n\to \infty $ in ${\mathrm{L}}^{2}\left(\mathsf{\Omega}\right)$ (i.e., in the mean square sense), then the expectation and variance of Z can be approximated as follows:

**Proposition**

**2**

**.**Let $A\left(t\right)={\sum}_{n=0}^{\infty}{A}_{n}{(t-{t}_{0})}^{n}$ be a random power series in the ${\mathrm{L}}^{p}\left(\mathsf{\Omega}\right)$ setting ($p\ge 1$), for $t\in ({t}_{0}-r,{t}_{0}+r)$, $r>0$. Then, the random power series ${\sum}_{n=1}^{\infty}n{A}_{n}{(t-{t}_{0})}^{n-1}$ exists in ${\mathrm{L}}^{p}\left(\mathsf{\Omega}\right)$ for $t\in ({t}_{0}-r,{t}_{0}+r)$, and moreover, the ${\mathrm{L}}^{p}\left(\mathsf{\Omega}\right)$ derivative of $A\left(t\right)$ is equal to it: $\dot{A}\left(t\right)={\sum}_{n=1}^{\infty}n{A}_{n}{(t-{t}_{0})}^{n-1}$, for all $t\in ({t}_{0}-r,{t}_{0}+r)$.

**Proposition**

**3**

**.**Let $U={\sum}_{n=0}^{\infty}{U}_{n}$ and $V={\sum}_{n=0}^{\infty}{V}_{n}$ be two random series that converge in ${\mathrm{L}}^{2}\left(\mathsf{\Omega}\right)$. Suppose that one of the series converges absolutely, say ${\sum}_{n=0}^{\infty}{\parallel {V}_{n}\parallel}_{{\mathrm{L}}^{2}\left(\mathsf{\Omega}\right)}<\infty $. Then:

**Theorem**

**1**

**.**Let $A\left(t\right)={\sum}_{n=0}^{\infty}{A}_{n}{(t-{t}_{0})}^{n}$ and $B\left(t\right)={\sum}_{n=0}^{\infty}{B}_{n}{(t-{t}_{0})}^{n}$ be two random series in the ${\mathrm{L}}^{2}\left(\mathsf{\Omega}\right)$ setting, for $t\in ({t}_{0}-r,{t}_{0}+r)$, $r>0$ being finite and fixed. Assume that the initial conditions ${Y}_{0}$ and ${Y}_{1}$ belong to ${\mathrm{L}}^{2}\left(\mathsf{\Omega}\right)$. Suppose that there is a constant ${C}_{r}>0$, maybe dependent on r, such that $\parallel {A}_{n}{\parallel}_{{\mathrm{L}}^{\infty}\left(\mathsf{\Omega}\right)}\le {C}_{r}/{r}^{n}$ and $\parallel {B}_{n}{\parallel}_{{\mathrm{L}}^{\infty}\left(\mathsf{\Omega}\right)}\le {C}_{r}/{r}^{n}$, $n\ge 0$. Then, the stochastic process $X\left(t\right)={\sum}_{n=0}^{\infty}{X}_{n}{(t-{t}_{0})}^{n}$, $t\in ({t}_{0}-r,{t}_{0}+r)$, where:

**Theorem**

**2.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 3. Non-Homogeneous Case

**Theorem**

**3.**

**Proof.**

**Example**

**5.**

^{®}. Our code to build the partial sum ${X}^{N}\left(t\right)$ was the following one:

X[n_?NonPositive] := Y0;X[1] = Y1;X[n_] := 1/(n*(n - 1))*(-Sum[(m + 1)*A[n - 2 - m]*X[m + 1] +B[n - 2 - m]*X[m], {m, 0, n - 2}] + CC[n - 2]);seriesX[t_, t0_, N_] := X[0] + Sum[X[n]*(t - t0)^n, {n, 1, N}];

`N`, the functions $t\mapsto \mathbb{E}\left[{X}^{N}\left(t\right)\right]$ and $t\mapsto \mathbb{V}\left[{X}^{N}\left(t\right)\right]$ have been calculated with the built-in function

`Expectation`applied to

`seriesX[t, 0, N]`(with symbolic

`t`), by setting the desired probability distributions to

`A[n]`,

`B[n]`, and

`CC[n]`. In Table 1 and Table 2, we show $\mathbb{E}\left[{X}^{N}\left(t\right)\right]$ and $\mathbb{V}\left[{X}^{N}\left(t\right)\right]$ for $N=19$, $N=20$, and $0\le t\le 1.5$. Both orders of truncation produce similar results, which agrees with the theoretical convergence. Observe that, as we move away from the initial condition ${t}_{0}=0$, larger orders of truncation are needed. This indicates that the Fröbenius method might be computationally inviable for large t. The results have been compared with Monte Carlo simulation (with 100,000 and 200,000 realizations).

**Theorem**

**4.**

**Proof.**

## 4. Comparison with Other Methods

**Proposition**

**4**

**.**Let $F\left(t\right)$ and $G\left(t\right)$ be two second order stochastic processes, with mean square derivatives of k order ${F}^{\left(k\right)}\left(t\right)$ and ${G}^{\left(k\right)}\left(t\right)$. Then, the following results hold:

- (i)
- If $U\left(t\right)=F\left(t\right)\pm G\left(t\right)$, then $\widehat{U}\left(k\right)=\widehat{F}\left(k\right)\pm \widehat{G}\left(k\right)$.
- (ii)
- If $U\left(t\right)=\lambda F\left(t\right)$, where λ is a bounded random variable, then $\widehat{U}\left(k\right)=\lambda \widehat{F}\left(k\right)$.
- (iii)
- If $U\left(t\right)={G}^{\left(m\right)}\left(t\right)$, then $\widehat{U}\left(k\right)=(k+1)\cdots (k+m)\widehat{G}(k+m)$ (here, m is a nonnegative integer).
- (iv)
- If $U\left(t\right)=F\left(t\right)G\left(t\right)$, then $\widehat{U}\left(k\right)={\sum}_{n=0}^{k}\widehat{F}\left(n\right)\widehat{G}(k-n)$.

## 5. Summary, Conclusions, and Future Lines of Research

- Solve the open problem raised in this paper at the end of Section 2, concerning the necessity of the hypotheses of Theorem 2.
- Apply the technique of gPC expansions and stochastic Galerkin projections to general random second order linear differential equations.
- Extend Theorem 3 to higher order random linear differential equations. Probably, one would need to require all input stochastic processes to be random power series in an ${\mathrm{L}}^{\infty}$ sense, in analogy with the hypotheses of Theorem 3.
- Apply the random Fröbenius method to the random Riccati differential equation with the analytic input processes. In [35] (Section 3), the authors applied the random differential transform method (which is equivalent to a formal random Fröbenius method) to a particular case of the random Riccati differential equation with a random autonomous coefficient term. It would be interesting to apply the random Fröbenius method in the situation in which all input coefficients are analytic stochastic processes, by proving theoretical results and performing numerical experiments.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Goodwine, B. Engineering Differential Equations. Theory and Applications; Springer: New York, NY, USA, 2011. [Google Scholar]
- Schiesser, W.E. Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R; John Wiley & Sons Inc.: New York, NY, USA, 2014. [Google Scholar]
- Brown, C.M. Differential Equations: A Modeling Approach (Quantitative Applications in the Social Sciences); SAGE, Inc.: New York, NY, USA, 2007. [Google Scholar]
- Valée, O.; Soares, M. Airy Functions and Applications to Physics; Imperial College Press: London, UK, 2004. [Google Scholar]
- Fedoryuk, M.V. Encyclopedia of Mathematics; Springer Science+Business Media B.V./Kluwer Academic Publisher: New York, NY, USA, 2001. [Google Scholar]
- Spain, B.; Smith, M.G. Functions of Mathematical Physics; Van Nostrand Reinhold Company: London, UK, 1970. [Google Scholar]
- Johnson, C.S.; Pedersen, L.G. Problems and Solutions in Quantum Chemistry and Physics; Dover Publ.: New York, NY, USA, 1986. [Google Scholar]
- Cortés, J.-C.; Jódar, L.; Camacho, J.; Villafuerte, L. Random Airy type differential equations: Mean square exact and numerical solutions. Comput. Math. Appl.
**2010**, 60, 1237–1244. [Google Scholar] [CrossRef] - Calbo, G.; Cortés, J.-C.; Jódar, L. Random Hermite differential equations: Mean square power series solutions and statistical properties. Appl. Math. Comput.
**2011**, 218, 3654–3666. [Google Scholar] [CrossRef][Green Version] - Calbo, G.; Cortés, J.-C.; Jódar, L.; Villafuerte, L. Solving the random Legendre differential equation: Mean square power series solution and its statistical functions. Comput. Math. Appl.
**2011**, 61, 2782–2792. [Google Scholar] [CrossRef] - Calatayud, J.; Cortés, J.-C.; Jornet, M. Improving the approximation of the first and second order statistics of the response process to the random Legendre differential equation. arXiv, 2018; arXiv:1807.03141. [Google Scholar]
- Cortés, J.-C.; Jódar, L.; Company, R.; Villafuerte, L. Laguerre random polynomials: Definition, differential and statistical properties. Util. Math.
**2015**, 98, 283–295. [Google Scholar] - Cortés, J.-C.; Jódar, L.; Villafuerte, L. Mean square solution of Bessel differential equation with uncertainties. J. Comput. Appl. Math.
**2017**, 309, 383–395. [Google Scholar] [CrossRef] - Golmankhaneh, A.A.; Porghoveh, N.A.; Baleanu, D. Mean square solutions of second-order random differential equations by using homotopy analysis method. Rom. Rep. Phys.
**2013**, 65, 350–362. [Google Scholar] - Khudair, A.R.; Ameen, A.A.; Khalaf, S.L. Mean square solutions of second-order random differential equations by using variational iteration method. Appl. Math. Sci.
**2011**, 5, 2505–2519. [Google Scholar] - Khudair, A.R.; Ameen, A.A.; Khalaf, S.L. Mean square solutions of second-order random differential equations by using Adomian decomposition method. Appl. Math. Sci.
**2011**, 5, 2521–2535. [Google Scholar] - Calatayud, J.; Cortés, J.-C.; Jornet, M.; Villafuerte, L. Random non-autonomous second order linear differential equations: Mean square analytic solutions and their statistical properties. Adv. Differ. Equ.
**2018**, 2018, 392. [Google Scholar] [CrossRef] - Kadry, S. On the generalization of probabilistic transformation method. Appl. Math. Comput.
**2007**, 190, 1284–1289. [Google Scholar] [CrossRef] - Kloeden, P.E.; Platen, E. Numerical Solution of Stochastic Differential Equations; Springer: New York, NY, USA, 1992. [Google Scholar]
- Díaz-Infante, S.; Jerez, S. The linear Steklov method for SDEs with non-globally Lipschitz coefficients: Strong convergence and simulation. J. Comp. Appl. Math.
**2017**, 309, 408–423. [Google Scholar] [CrossRef] - Soong, T.T. Random Differential Equations in Science and Engineering; Academic Press: New York, NY, USA, 1973. [Google Scholar]
- Smith, R.C. Uncertainty Quantification. Theory, Implementation, and Applications; SIAM in the Computational Science and Engineering Series; CS12; SIAM: New York, NY, USA, 2014. [Google Scholar]
- Strand, J.L. Random ordinary differential equations. J. Differ. Equ.
**1970**, 7, 538–553. [Google Scholar] [CrossRef] - Henderson, D.; Plaschko, P. Stochastic Differential Equations in Science And Engineering; World Scientific Pub Co. Inc.: Singapore, 2006. [Google Scholar]
- Xiu, D. Numerical Methods for Stochastic Computations. A Spectral Method Approach; Princeton University Press: Princeton, NJ, USA, 2010. [Google Scholar]
- Xiu, D.; Karniadakis, G.E. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput.
**2002**, 24, 619–644. [Google Scholar] [CrossRef] - Chen-Charpentier, B.-M.; Cortés, J.-C.; Licea, J.-A.; Romero, J.-V.; Roselló, M.-D.; Santonja, F.-J.; Villanueva, R.-J. Constructing adaptive generalized polynomial chaos method to measure the uncertainty in continuous models: A computational approach. Math. Comput. Simul.
**2015**, 109, 113–129. [Google Scholar] [CrossRef][Green Version] - Cortés, J.-C.; Romero, J.-V.; Roselló, M.-D.; Villanueva, R.-J. Improving adaptive generalized polynomial chaos method to solve nonlinear random differential equations by the random variable transformation technique. Commun. Nonlinear Sci. Numer. Simul.
**2017**, 50, 1–15. [Google Scholar] [CrossRef] - Cortés, J.-C.; Romero, J.-V.; Roselló, M.-D.; Santonja, F.-J.; Villanueva, R.-J. Solving continuous models with dependent uncertainty: A computational approach. Abstr. Appl. Anal.
**2013**, 2013, 1–10. [Google Scholar] [CrossRef] - Calatayud, J.; Cortés, J.-C.; Jornet, M.; Villanueva, R.-J. Computational uncertainty quantification for random time-discrete epidemiological models using adaptive gPC. Math. Meth. Appl. Sci.
**2018**, 1–10. [Google Scholar] [CrossRef] - Chen-Charpentier, B.-M.; Cortés, J.-C.; Romero, J.-V.; Roselló, M.-D. Some recommendations for applying gPC (generalized polynomial chaos) to modeling: An analysis through the Airy random differential equation. Appl. Math. Comput.
**2013**, 219, 4208–4218. [Google Scholar] [CrossRef][Green Version] - Chen-Charpentier, B.-M.; Cortés, J.-C.; Romero, J.-V.; Roselló, M.-D. Do the generalized polynomial chaos and Fröbenius methods retain the statistical moments of random differential equations? Appl. Math. Lett.
**2013**, 26, 553–558. [Google Scholar] [CrossRef] - Chen-Charpentier, B.-M.; Stanescu, D. Epidemic models with random coefficients. Math. Comput. Mod.
**2010**, 52, 1004–1010. [Google Scholar] [CrossRef] - Calatayud, J.; Cortés, J.-C.; Jornet, M. On the convergence of adaptive gPC for non-linear random difference equations: Theoretical analysis and some practical recommendations. J. Nonlinear Sci. Appl.
**2018**, 11, 1077–1084. [Google Scholar] [CrossRef] - Villafuerte, L.; Chen-Charpentier, B.-M. A random differential transform method: Theory and applications. Appl. Math. Lett.
**2012**, 25, 1490–1494. [Google Scholar] [CrossRef]

**Table 1.**Approximation of $\mathbb{E}\left[X\right(t\left)\right]$ with $N=19$, $N=20$, and Monte Carlo simulations. Example 5.

t | $\mathbb{E}\left[{\mathit{X}}^{19}\left(\mathit{t}\right)\right]$ | $\mathbb{E}\left[{\mathit{X}}^{20}\left(\mathit{t}\right)\right]$ | MC 100,000 | MC 200,000 |
---|---|---|---|---|

$0.00$ | 1 | 1 | $0.995893$ | $1.00266$ |

$0.25$ | $1.14231$ | $1.14231$ | $1.13899$ | $1.14544$ |

$0.50$ | $1.28890$ | $1.28890$ | $1.28672$ | $1.29236$ |

$0.75$ | $1.49183$ | $1.49183$ | $1.49130$ | $1.49547$ |

$1.00$ | $1.85892$ | $1.85892$ | $1.86087$ | $1.86246$ |

$1.25$ | $2.62573$ | $2.62574$ | $2.63173$ | $2.62863$ |

$1.50$ | $4.34772$ | $4.34784$ | $4.36111$ | $4.34892$ |

**Table 2.**Approximation of $\mathbb{V}\left[X\right(t\left)\right]$ with $N=19$, $N=20$, and Monte Carlo simulations. Example 5.

t | $\mathbb{V}\left[{\mathit{X}}^{19}\left(\mathit{t}\right)\right]$ | $\mathbb{V}\left[{\mathit{X}}^{20}\left(\mathit{t}\right)\right]$ | MC 100,000 | MC 200,000 |
---|---|---|---|---|

$0.00$ | $0.5$ | $0.5$ | $0.493124$ | $0.504501$ |

$0.25$ | $0.520298$ | $0.520298$ | $0.514702$ | $0.524803$ |

$0.50$ | $0.597008$ | $0.597008$ | $0.593603$ | $0.601376$ |

$0.75$ | $0.790556$ | $0.790556$ | $0.790161$ | $0.794549$ |

$1.00$ | $1.27425$ | $1.27425$ | $1.27702$ | $1.27759$ |

$1.25$ | $2.60694$ | $2.60694$ | $2.60987$ | $2.60982$ |

$1.50$ | $6.94095$ | $6.94100$ | $6.92663$ | $6.94787$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Calatayud Gregori, J.; Cortés López, J.C.; Jornet Sanz, M. Some Notes to Extend the Study on Random Non-Autonomous Second Order Linear Differential Equations Appearing in Mathematical Modeling. *Math. Comput. Appl.* **2018**, *23*, 76.
https://doi.org/10.3390/mca23040076

**AMA Style**

Calatayud Gregori J, Cortés López JC, Jornet Sanz M. Some Notes to Extend the Study on Random Non-Autonomous Second Order Linear Differential Equations Appearing in Mathematical Modeling. *Mathematical and Computational Applications*. 2018; 23(4):76.
https://doi.org/10.3390/mca23040076

**Chicago/Turabian Style**

Calatayud Gregori, Julia, Juan Carlos Cortés López, and Marc Jornet Sanz. 2018. "Some Notes to Extend the Study on Random Non-Autonomous Second Order Linear Differential Equations Appearing in Mathematical Modeling" *Mathematical and Computational Applications* 23, no. 4: 76.
https://doi.org/10.3390/mca23040076