December  2020, 7(2): 183-208. doi: 10.3934/jcd.2020008

Solving the inverse problem for an ordinary differential equation using conjugation

1. 

Departamento de Ciência da Computação, Universidade Federal do Rio de Janeiro, Caixa Postal 68.530, CEP 21941-590, Rio de Janeiro, RJ, Brazil

2. 

Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, CEP 36036-900, MG, Brazil

3. 

Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago de Chile, Chile

4. 

Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, CEP 22460-320, Rio de Janeiro, RJ, Brazil

* Corresponding author: Daniel G. Alfaro Vigo

Received  October 2019 Published  July 2020

Fund Project: The second author's work was partially supported by IMPA/CAPES. The third author was partially supported by FAPEMIG under Grant APQ 01377/15 and CNPq under grant 303245/2019-0. The fourth author was partially supported by DICYT grant 041933GM from VRIDEI-USACH

We consider the following inverse problem for an ordinary differential equation (ODE): given a set of data points $ P = \{(t_i,x_i),\; i = 1,\dots,N\} $, find an ODE $ x^\prime(t) = v (x) $ that admits a solution $ x(t) $ such that $ x_i \approx x(t_i) $ as closely as possible. The key to the proposed method is to find approximations of the recursive or discrete propagation function $ D(x) $ from the given data set. Afterwards, we determine the field $ v(x) $, using the conjugate map defined by Schröder's equation and the solution of a related Julia's equation. Moreover, our approach also works for the inverse problems where one has to determine an ODE from multiple sets of data points.

We also study existence, uniqueness, stability and other properties of the recovered field $ v(x) $. Finally, we present several numerical methods for the approximation of the field $ v(x) $ and provide some illustrative examples of the application of these methods.

Citation: Daniel G. Alfaro Vigo, Amaury C. Álvarez, Grigori Chapiro, Galina C. García, Carlos G. Moreira. Solving the inverse problem for an ordinary differential equation using conjugation. Journal of Computational Dynamics, 2020, 7 (2) : 183-208. doi: 10.3934/jcd.2020008
References:
[1]

A. C. AlvarezP. G. BedrikovetskyG. HimeA. O. MarchesinD. Marchesin and J. R. Rodrigues, A fast inverse solver for the filtration function for flow of water with particles in porous media, Inverse Problems, 22 (2006), 69-88.  doi: 10.1088/0266-5611/22/1/005.  Google Scholar

[2]

A. C. Alvarez, G. Hime, J. D. Silva and D. Marchesin, Analytic regularization of an inverse filtration problem in porous media, Inverse Problems, 29 (2013), 025006, 20 pp. doi: 10.1088/0266-5611/29/2/025006.  Google Scholar

[3]

V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, vol. 250, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1037-5.  Google Scholar

[4]

P. Bedrikovetsky, D. Marchesin, G. Hime, A. Alvarez, A. O. Marchesin, A. G. Siqueira, A. L. S. Souza, F. S. Shecaira and J. R. Rodrigues, Porous Media Deposition Damage from Injection of Water with Particles, in ECMOR Ⅷ-8th European Conference on the Mathematics of Oil Recovery, 2002. doi: 10.3997/2214-4609.201405931.  Google Scholar

[5]

C. F. Borges, A full-Newton approach to separable nonlinear least squares problems and its application to discrete least squares rational approximation, Electron. Trans. Numer. Anal., 35 (2009), 57-68.   Google Scholar

[6]

S. L. BruntonJ. L. Proctor and J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci. USA, 113 (2016), 3932-3937.  doi: 10.1073/pnas.1517384113.  Google Scholar

[7]

R. Burckel, A history of complex dynamics from Schroeder to Fatou and Julia (Daniel S. Alexander), SIAM Review, 36 (1994), 663-664.   Google Scholar

[8]

T. Curtright, X. Jin and C. Zachos, Approximate solutions of functional equations, J. Phys. A, 44 (2011), 405205, 12 pp. doi: 10.1088/1751-8113/44/40/405205.  Google Scholar

[9]

T. Curtright and C. Zachos, Evolution profiles and functional equations, J. Phys. A, 42 (2009), 485208, 16 pp. doi: 10.1088/1751-8113/42/48/485208.  Google Scholar

[10]

T. L. Curtright and C. K. Zachos, Chaotic maps, hamiltonian flows and holographic methods, J. Phys. A, 43 (2010), 445101, 15 pp. doi: 10.1088/1751-8113/43/44/445101.  Google Scholar

[11]

T. L. Curtright and C. K. Zachos, Renormalization group functional equations, Physical Review D, 83 (2011), 065019. doi: 10.1103/PhysRevD.83.065019.  Google Scholar

[12]

M. L. Heard, A change of variables for functional differential equations, J. Differential Equations, 18 (1975), 1-10.  doi: 10.1016/0022-0396(75)90076-5.  Google Scholar

[13]

B. Hofmann, A. Leitão and J. P. Zubelli, New Trends in Parameter Identification for Mathematical Models, Springer, 2018. doi: 10.1007/978-3-319-70824-9.  Google Scholar

[14]

J. B. KellerI. Kay and J. Shmoys, Determination of the potential from scattering data, Phys. Rev., 102 (1956), 557-559.  doi: 10.1103/PhysRev.102.557.  Google Scholar

[15]

M. Kuczma, Functional Equations in a Single Variable, Monografie Matematyczne, Tom 46 Państwowe Wydawnictwo Naukowe, Warsaw, 1968.  Google Scholar

[16] M. KuczmaB. Choczewski and R. Ger, Iterative Functional Equations, vol. 32, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9781139086639.  Google Scholar
[17]

H. Kunze and S. Vasiliadis, Using the collage method to solve ODEs inverse problems with multiple data sets, Nonlinear Anal., 71 (2009), e1298–e1306. doi: 10.1016/j.na.2009.01.167.  Google Scholar

[18]

H. E. Kunze and E. R. Vrscay, Solving inverse problems for ordinary differential equations using the Picard contraction mapping, Inverse Problems, 15 (1999), 745-770.  doi: 10.1088/0266-5611/15/3/308.  Google Scholar

[19]

H. KunzeD. La Torre and E. R. Vrscay, Solving inverse problems for DEs using the collage theorem and entropy maximization, Appl. Math. Lett., 25 (2012), 2306-2311.  doi: 10.1016/j.aml.2012.06.021.  Google Scholar

[20]

F. LuD. Xu and G. Wen, Estimation of initial conditions and parameters of a chaotic evolution process from a short time series, Chaos: An Interdisciplinary Journal of Nonlinear Science, 14 (2004), 1050-1055.  doi: 10.1063/1.1811548.  Google Scholar

[21]

W. H. Miller, WKB solution of inversion problems for potential scattering, J. Chem. Phys., 51 (1969), 3631-3638.  doi: 10.1063/1.1672572.  Google Scholar

[22]

T. G. Müller and J. Timmer, Parameter identification techniques for partial differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 2052-2060.  doi: 10.1142/S0218127404010424.  Google Scholar

[23]

T. G. Müller and J. Timmer, Fitting parameters in partial differential equations from partially observed noisy data, Phys. D, 171 (2002), 1-7.  doi: 10.1016/S0167-2789(02)00546-8.  Google Scholar

[24]

Y. Nakatsukasa, O. Sète and L. N. Trefethen, The AAA algorithm for rational approximation, SIAM J. Sci. Comput., 40 (2018), A1494–A1522. doi: 10.1137/16M1106122.  Google Scholar

[25]

E. B. Nelson, Nonlinear Regression Methods for Estimation, Technical report, Air Force Inst. of Tech. Wright-Patterson, 2005. Google Scholar

[26]

M. Pachter and O. R. Reynolds, Identification of a discrete-time dynamical system, IEEE Transactions on Aerospace and Electronic Systems, 36 (2000), 212-225.  doi: 10.1109/7.826323.  Google Scholar

[27]

M. Peifer and J. Timmer, Parameter estimation in ordinary differential equations for biochemical processes using the method of multiple shooting, IET Systems Biology, 1 (2007), 78-88.  doi: 10.1049/iet-syb:20060067.  Google Scholar

[28] S. S. Roy, Dynamic System Identification Using Adaptive Algorithm, Scholars Press, 2017.   Google Scholar
[29]

C. G. Small, Functional Equations and How to Solve Them, Springer, 2007. doi: 10.1007/978-0-387-48901-8.  Google Scholar

[30]

W.-B. Zhang, Discrete Dynamical Systems, Bifurcations and Chaos in Economics, Elsevier, 2006. Google Scholar

show all references

References:
[1]

A. C. AlvarezP. G. BedrikovetskyG. HimeA. O. MarchesinD. Marchesin and J. R. Rodrigues, A fast inverse solver for the filtration function for flow of water with particles in porous media, Inverse Problems, 22 (2006), 69-88.  doi: 10.1088/0266-5611/22/1/005.  Google Scholar

[2]

A. C. Alvarez, G. Hime, J. D. Silva and D. Marchesin, Analytic regularization of an inverse filtration problem in porous media, Inverse Problems, 29 (2013), 025006, 20 pp. doi: 10.1088/0266-5611/29/2/025006.  Google Scholar

[3]

V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, vol. 250, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1037-5.  Google Scholar

[4]

P. Bedrikovetsky, D. Marchesin, G. Hime, A. Alvarez, A. O. Marchesin, A. G. Siqueira, A. L. S. Souza, F. S. Shecaira and J. R. Rodrigues, Porous Media Deposition Damage from Injection of Water with Particles, in ECMOR Ⅷ-8th European Conference on the Mathematics of Oil Recovery, 2002. doi: 10.3997/2214-4609.201405931.  Google Scholar

[5]

C. F. Borges, A full-Newton approach to separable nonlinear least squares problems and its application to discrete least squares rational approximation, Electron. Trans. Numer. Anal., 35 (2009), 57-68.   Google Scholar

[6]

S. L. BruntonJ. L. Proctor and J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci. USA, 113 (2016), 3932-3937.  doi: 10.1073/pnas.1517384113.  Google Scholar

[7]

R. Burckel, A history of complex dynamics from Schroeder to Fatou and Julia (Daniel S. Alexander), SIAM Review, 36 (1994), 663-664.   Google Scholar

[8]

T. Curtright, X. Jin and C. Zachos, Approximate solutions of functional equations, J. Phys. A, 44 (2011), 405205, 12 pp. doi: 10.1088/1751-8113/44/40/405205.  Google Scholar

[9]

T. Curtright and C. Zachos, Evolution profiles and functional equations, J. Phys. A, 42 (2009), 485208, 16 pp. doi: 10.1088/1751-8113/42/48/485208.  Google Scholar

[10]

T. L. Curtright and C. K. Zachos, Chaotic maps, hamiltonian flows and holographic methods, J. Phys. A, 43 (2010), 445101, 15 pp. doi: 10.1088/1751-8113/43/44/445101.  Google Scholar

[11]

T. L. Curtright and C. K. Zachos, Renormalization group functional equations, Physical Review D, 83 (2011), 065019. doi: 10.1103/PhysRevD.83.065019.  Google Scholar

[12]

M. L. Heard, A change of variables for functional differential equations, J. Differential Equations, 18 (1975), 1-10.  doi: 10.1016/0022-0396(75)90076-5.  Google Scholar

[13]

B. Hofmann, A. Leitão and J. P. Zubelli, New Trends in Parameter Identification for Mathematical Models, Springer, 2018. doi: 10.1007/978-3-319-70824-9.  Google Scholar

[14]

J. B. KellerI. Kay and J. Shmoys, Determination of the potential from scattering data, Phys. Rev., 102 (1956), 557-559.  doi: 10.1103/PhysRev.102.557.  Google Scholar

[15]

M. Kuczma, Functional Equations in a Single Variable, Monografie Matematyczne, Tom 46 Państwowe Wydawnictwo Naukowe, Warsaw, 1968.  Google Scholar

[16] M. KuczmaB. Choczewski and R. Ger, Iterative Functional Equations, vol. 32, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9781139086639.  Google Scholar
[17]

H. Kunze and S. Vasiliadis, Using the collage method to solve ODEs inverse problems with multiple data sets, Nonlinear Anal., 71 (2009), e1298–e1306. doi: 10.1016/j.na.2009.01.167.  Google Scholar

[18]

H. E. Kunze and E. R. Vrscay, Solving inverse problems for ordinary differential equations using the Picard contraction mapping, Inverse Problems, 15 (1999), 745-770.  doi: 10.1088/0266-5611/15/3/308.  Google Scholar

[19]

H. KunzeD. La Torre and E. R. Vrscay, Solving inverse problems for DEs using the collage theorem and entropy maximization, Appl. Math. Lett., 25 (2012), 2306-2311.  doi: 10.1016/j.aml.2012.06.021.  Google Scholar

[20]

F. LuD. Xu and G. Wen, Estimation of initial conditions and parameters of a chaotic evolution process from a short time series, Chaos: An Interdisciplinary Journal of Nonlinear Science, 14 (2004), 1050-1055.  doi: 10.1063/1.1811548.  Google Scholar

[21]

W. H. Miller, WKB solution of inversion problems for potential scattering, J. Chem. Phys., 51 (1969), 3631-3638.  doi: 10.1063/1.1672572.  Google Scholar

[22]

T. G. Müller and J. Timmer, Parameter identification techniques for partial differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 2052-2060.  doi: 10.1142/S0218127404010424.  Google Scholar

[23]

T. G. Müller and J. Timmer, Fitting parameters in partial differential equations from partially observed noisy data, Phys. D, 171 (2002), 1-7.  doi: 10.1016/S0167-2789(02)00546-8.  Google Scholar

[24]

Y. Nakatsukasa, O. Sète and L. N. Trefethen, The AAA algorithm for rational approximation, SIAM J. Sci. Comput., 40 (2018), A1494–A1522. doi: 10.1137/16M1106122.  Google Scholar

[25]

E. B. Nelson, Nonlinear Regression Methods for Estimation, Technical report, Air Force Inst. of Tech. Wright-Patterson, 2005. Google Scholar

[26]

M. Pachter and O. R. Reynolds, Identification of a discrete-time dynamical system, IEEE Transactions on Aerospace and Electronic Systems, 36 (2000), 212-225.  doi: 10.1109/7.826323.  Google Scholar

[27]

M. Peifer and J. Timmer, Parameter estimation in ordinary differential equations for biochemical processes using the method of multiple shooting, IET Systems Biology, 1 (2007), 78-88.  doi: 10.1049/iet-syb:20060067.  Google Scholar

[28] S. S. Roy, Dynamic System Identification Using Adaptive Algorithm, Scholars Press, 2017.   Google Scholar
[29]

C. G. Small, Functional Equations and How to Solve Them, Springer, 2007. doi: 10.1007/978-0-387-48901-8.  Google Scholar

[30]

W.-B. Zhang, Discrete Dynamical Systems, Bifurcations and Chaos in Economics, Elsevier, 2006. Google Scholar

Figure 1.  Iteration function $ D $ (solid line), points of the data set with a noise level $ \sigma = 0.5 $ (blue points) and the recovered iteration function (red circle) as discussed in Example 1 (Subsection 6.1). Notice that the exact and recovered iteration functions are basically indistinguishable
Figure 2.  The exact field $ v(x) $ (solid blue line) and the recovered field corresponding to Example 1 (Subsection 6.1), for different values of $ \sigma $. For $ \sigma = 0.1 $ (dashed blue line), $ \sigma = 0.5 $ (dotted black), $ \sigma = 0.9 $ (dash-dot green line), $ \sigma = 1.5 $ (dashed red blue) and $ \sigma = 2.9 $ (dotted magenta line). Notice that for $ \sigma<1 $ exact and recovered fields are indistinguishable
Figure 3.  Multiple sets of data points (upper plot) and the corresponding synthetic data (lower plot) used in Example 2 (Subsection 6.2). The set 1 (blue points) is used in both cases (a) and (b), whereas the other sets (red circles) are only used in case (b)
Figure 4.  Exact and approximate iteration function and its derivative (upper plot) and the associated approximation errors (lower plot) in the interval $ (0,1) $, corresponding to case (a) of Example 2 (Subsection 6.2). Notice that the exact functions and their approximations are indistinguishable
Figure 5.  Exact and approximated field $ v(x) $ (upper plot) and approximation error (lower plot) in the interval $ (0,1) $ corresponding to case (a) of Example 2 (Subsection 6.2). Observe that the exact field and its approximation are indistinguishable
Figure 6.  Exact and approximated iteration function and its derivative (upper plot) and the associated approximation errors (lower plot) corresponding to case (b) of Example 2 (Subsection 6.2). Notice that the exact functions and their approximations are almost indistinguishable
Figure 7.  Exact and approximated field $ v(x) $ (upper plot) and approximation error (lower plot) in the interval $ (-2.04,2.04) $ corresponding to case (b) of Example 2 (Subsection 6.2). Notice that the exact field and its approximation are indistinguishable
Figure 8.  Exact and approximated iteration function and its derivative (upper plot), and approximation errors for the iteration function and its derivative (lower plot); corresponding to Example 3 (Subsection 6.3)
Figure 9.  Exact and approximated field (upper plot) and the approximation error (lower plot) corresponding to Example 3 (Subsection 6.3). Notice that the difference between the field and its approximation is only noticeable around $ x = 1.2 $
Algorithm 1: Implementation of formula (35)
Require: $ x_0 $, $ \epsilon $, functions $ D $ and $ D^{\prime} $
Ensure: $ g(x_0) = q_n $
1: $ x_n=x_0 $, error=1, lim=1, $ q_n=1 $
2: while error $ > \epsilon $ do
3:    last=lim
4:    $ q_n=q_n D(x_n)/(x_n D^{\prime}(x_n)) $
5:    $ x_n=D(x_n) $
6:    lim=$ q_n $
7:    error=$ | $lim-last$ | $/$ | $last$ | $
8: end while
9: $ q_n = x_o q_n $
10: return $ q_n $
Algorithm 1: Implementation of formula (35)
Require: $ x_0 $, $ \epsilon $, functions $ D $ and $ D^{\prime} $
Ensure: $ g(x_0) = q_n $
1: $ x_n=x_0 $, error=1, lim=1, $ q_n=1 $
2: while error $ > \epsilon $ do
3:    last=lim
4:    $ q_n=q_n D(x_n)/(x_n D^{\prime}(x_n)) $
5:    $ x_n=D(x_n) $
6:    lim=$ q_n $
7:    error=$ | $lim-last$ | $/$ | $last$ | $
8: end while
9: $ q_n = x_o q_n $
10: return $ q_n $
Algorithm 2: Implementation of formula (37)
Require: $ x_0,\dots,x_m $, $ \epsilon $, functions $ D $ and $ D^{\prime} $
Ensure: $ g(x_0) = g_0, \dots, g(x_m) = g_m $
1: $ g_0 = \cdots = g_m = 1 $, error=1, lim=1
2: $ q_0 = D(x_0)/(x_0 D^{\prime}(x_0)), \dots, q_m = D(x_m)/(x_m D^{\prime}(x_m)) $
3: while error $ > \epsilon $ do
4:    $ gl_0 = g_0, \dots, gl_m = g_m $
5:    Compute function $ g(x) $ interpolating data : $ (x_0,g_0), \dots, (x_m,g_m) $
6:    $ g_0= q_0 g(D(x_0)), \dots, g_m= q_m g(D(x_m)) $
7:    error=max($ |gl_j-g_j| $)/max($ |gl_j| $)
8: end while
9: $ g_0 = x_o g_0, \dots, g_m = x_m g_m $
10: return $ g_0, \dots, g_m $
Algorithm 2: Implementation of formula (37)
Require: $ x_0,\dots,x_m $, $ \epsilon $, functions $ D $ and $ D^{\prime} $
Ensure: $ g(x_0) = g_0, \dots, g(x_m) = g_m $
1: $ g_0 = \cdots = g_m = 1 $, error=1, lim=1
2: $ q_0 = D(x_0)/(x_0 D^{\prime}(x_0)), \dots, q_m = D(x_m)/(x_m D^{\prime}(x_m)) $
3: while error $ > \epsilon $ do
4:    $ gl_0 = g_0, \dots, gl_m = g_m $
5:    Compute function $ g(x) $ interpolating data : $ (x_0,g_0), \dots, (x_m,g_m) $
6:    $ g_0= q_0 g(D(x_0)), \dots, g_m= q_m g(D(x_m)) $
7:    error=max($ |gl_j-g_j| $)/max($ |gl_j| $)
8: end while
9: $ g_0 = x_o g_0, \dots, g_m = x_m g_m $
10: return $ g_0, \dots, g_m $
Table 1.  Values of the relative errors for the iteration function $ D $, its derivative $ D' $ and the field $ v $, and the stability constant $ C_v $ corresponding to example 6.1
$ \sigma $ $ \epsilon_{D} $ $ \epsilon_{D^{'}} $ $ \epsilon_{v} $ $ C_v $
[0.5ex] 0.1 2.51 0.026 0.019 0.0075
0.5 2.46 0.0883 0.06 0.026
0.9 2.65 0.45 0.25 0.080
1.9 2.3 0.67 0.49 0.1661
2.9 2.77 0.822 0.38 0.107
3.9 2.38 0.3137 0.27 0.10
4.5 2.59 0.263 0.1614 0.05
$ \sigma $ $ \epsilon_{D} $ $ \epsilon_{D^{'}} $ $ \epsilon_{v} $ $ C_v $
[0.5ex] 0.1 2.51 0.026 0.019 0.0075
0.5 2.46 0.0883 0.06 0.026
0.9 2.65 0.45 0.25 0.080
1.9 2.3 0.67 0.49 0.1661
2.9 2.77 0.822 0.38 0.107
3.9 2.38 0.3137 0.27 0.10
4.5 2.59 0.263 0.1614 0.05
[1]

Gianni Gilioli, Sara Pasquali, Fabrizio Ruggeri. Nonlinear functional response parameter estimation in a stochastic predator-prey model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 75-96. doi: 10.3934/mbe.2012.9.75

[2]

Tarik Mohammed Touaoula. Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models). Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4391-4419. doi: 10.3934/dcds.2018191

[3]

David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319

[4]

Xiao Tang, Yingying Zeng, Weinian Zhang. Interval homeomorphic solutions of a functional equation of nonautonomous iterations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020214

[5]

Chun-Gil Park. Stability of a linear functional equation in Banach modules. Conference Publications, 2003, 2003 (Special) : 694-700. doi: 10.3934/proc.2003.2003.694

[6]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159

[7]

Marcus A. Khuri. On the local solvability of Darboux's equation. Conference Publications, 2009, 2009 (Special) : 451-456. doi: 10.3934/proc.2009.2009.451

[8]

Tzung-shin Yeh. S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-Ⅲ functional response. Communications on Pure & Applied Analysis, 2017, 16 (2) : 645-670. doi: 10.3934/cpaa.2017032

[9]

Lei Qiao. Matsaev's type theorems for solutions of the stationary Schrödinger equation and its applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5709-5720. doi: 10.3934/dcds.2016050

[10]

Leonid V. Bogachev, Gregory Derfel, Stanislav A. Molchanov. Analysis of the archetypal functional equation in the non-critical case. Conference Publications, 2015, 2015 (special) : 132-141. doi: 10.3934/proc.2015.0132

[11]

Chunhong Li, Jiaowan Luo. Stochastic invariance for neutral functional differential equation with non-lipschitz coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3299-3318. doi: 10.3934/dcdsb.2018321

[12]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281

[13]

Tomás Caraballo, María J. Garrido–Atienza, Björn Schmalfuss, José Valero. Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 439-455. doi: 10.3934/dcdsb.2010.14.439

[14]

Wafa Hamrouni, Ali Abdennadher. Random walk's models for fractional diffusion equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2509-2530. doi: 10.3934/dcdsb.2016058

[15]

Kevin Zumbrun. L resolvent bounds for steady Boltzmann's Equation. Kinetic & Related Models, 2017, 10 (4) : 1255-1257. doi: 10.3934/krm.2017048

[16]

Darya V. Verveyko, Andrey Yu. Verisokin. Application of He's method to the modified Rayleigh equation. Conference Publications, 2011, 2011 (Special) : 1423-1431. doi: 10.3934/proc.2011.2011.1423

[17]

Szandra Beretka, Gabriella Vas. Stable periodic solutions for Nazarenko's equation. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3257-3281. doi: 10.3934/cpaa.2020144

[18]

J. Leonel Rocha, Sandra M. Aleixo. Dynamical analysis in growth models: Blumberg's equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 783-795. doi: 10.3934/dcdsb.2013.18.783

[19]

Laurent Desvillettes, Clément Mouhot, Cédric Villani. Celebrating Cercignani's conjecture for the Boltzmann equation. Kinetic & Related Models, 2011, 4 (1) : 277-294. doi: 10.3934/krm.2011.4.277

[20]

S. Jiménez, Pedro J. Zufiria. Characterizing chaos in a type of fractional Duffing's equation. Conference Publications, 2015, 2015 (special) : 660-669. doi: 10.3934/proc.2015.0660

 Impact Factor: 

Article outline

Figures and Tables

[Back to Top]