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Infinite product expansion of the Fokker-Planck equation with steady-state solution.

Martin RJ, Craster RV, Kearney MJ - Proc. Math. Phys. Eng. Sci. (2015)

Bottom Line: We present an analytical technique for solving Fokker-Planck equations that have a steady-state solution by representing the solution as an infinite product rather than, as usual, an infinite sum.This method has many advantages: automatically ensuring positivity of the resulting approximation, and by design exactly matching both the short- and long-term behaviour.The efficacy of the technique is demonstrated via comparisons with computations of typical examples.

View Article: PubMed Central - PubMed

Affiliation: Apollo Global Management International LLP , 25 St George Street, London W1S 1FS, UK ; Department of Mathematics , Imperial College London , South Kensington, London SW7 2AZ, UK.

ABSTRACT

We present an analytical technique for solving Fokker-Planck equations that have a steady-state solution by representing the solution as an infinite product rather than, as usual, an infinite sum. This method has many advantages: automatically ensuring positivity of the resulting approximation, and by design exactly matching both the short- and long-term behaviour. The efficacy of the technique is demonstrated via comparisons with computations of typical examples.

No MeSH data available.


Numerics and leading-order expansion compared for the model (1.7). Panels (a,b) are for Y0=0 and (c,d) are for Y0=−2. (a,c) The numerical solution for fY(τ,y) and (b,d) the numerical solution (solid) and the leading-order (N=0) expansion (dots) for τ=0.1,1,5.
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RSPA20150084F8: Numerics and leading-order expansion compared for the model (1.7). Panels (a,b) are for Y0=0 and (c,d) are for Y0=−2. (a,c) The numerical solution for fY(τ,y) and (b,d) the numerical solution (solid) and the leading-order (N=0) expansion (dots) for τ=0.1,1,5.

Mentions: We are now in a position to explore the efficacy of the expansion scheme versus full numerical solutions and we proceed to do so in figures 6–8. Each shows a numerical simulation of the full PDE, using the same parameters as in §3c, i.e. , for Y0=0,−2; the shift in the source position illustrates that, in each case, the solution drifts back to the origin as it simultaneously diffuses outward. In each case, we illustrate for f, on log axes to accentuate any error, the numerical solution of the PDE for τ=0.1,1,5 versus the N=0 (just the leading-order) solution (3.23). The solutions are virtually indistinguishable. As a more stringent, and demanding, test upon the methodology the double-well example (1.9) is also evaluated both numerically and via the expansion; the results shown in figure 9 are again remarkably accurate, particularly considering that it is just the N=0 approximation that is shown.Figure 6.


Infinite product expansion of the Fokker-Planck equation with steady-state solution.

Martin RJ, Craster RV, Kearney MJ - Proc. Math. Phys. Eng. Sci. (2015)

Numerics and leading-order expansion compared for the model (1.7). Panels (a,b) are for Y0=0 and (c,d) are for Y0=−2. (a,c) The numerical solution for fY(τ,y) and (b,d) the numerical solution (solid) and the leading-order (N=0) expansion (dots) for τ=0.1,1,5.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4528656&req=5

RSPA20150084F8: Numerics and leading-order expansion compared for the model (1.7). Panels (a,b) are for Y0=0 and (c,d) are for Y0=−2. (a,c) The numerical solution for fY(τ,y) and (b,d) the numerical solution (solid) and the leading-order (N=0) expansion (dots) for τ=0.1,1,5.
Mentions: We are now in a position to explore the efficacy of the expansion scheme versus full numerical solutions and we proceed to do so in figures 6–8. Each shows a numerical simulation of the full PDE, using the same parameters as in §3c, i.e. , for Y0=0,−2; the shift in the source position illustrates that, in each case, the solution drifts back to the origin as it simultaneously diffuses outward. In each case, we illustrate for f, on log axes to accentuate any error, the numerical solution of the PDE for τ=0.1,1,5 versus the N=0 (just the leading-order) solution (3.23). The solutions are virtually indistinguishable. As a more stringent, and demanding, test upon the methodology the double-well example (1.9) is also evaluated both numerically and via the expansion; the results shown in figure 9 are again remarkably accurate, particularly considering that it is just the N=0 approximation that is shown.Figure 6.

Bottom Line: We present an analytical technique for solving Fokker-Planck equations that have a steady-state solution by representing the solution as an infinite product rather than, as usual, an infinite sum.This method has many advantages: automatically ensuring positivity of the resulting approximation, and by design exactly matching both the short- and long-term behaviour.The efficacy of the technique is demonstrated via comparisons with computations of typical examples.

View Article: PubMed Central - PubMed

Affiliation: Apollo Global Management International LLP , 25 St George Street, London W1S 1FS, UK ; Department of Mathematics , Imperial College London , South Kensington, London SW7 2AZ, UK.

ABSTRACT

We present an analytical technique for solving Fokker-Planck equations that have a steady-state solution by representing the solution as an infinite product rather than, as usual, an infinite sum. This method has many advantages: automatically ensuring positivity of the resulting approximation, and by design exactly matching both the short- and long-term behaviour. The efficacy of the technique is demonstrated via comparisons with computations of typical examples.

No MeSH data available.