Beyond Linear Use of Equation Superposition (BLUES)

Collaborators: Joseph Indekeu (KU Leuven, Belgium)

The BLUES (Beyond Linear Use of Equation Superposition) function method is a powerful approach for analytically solving (nonlinear) differential equations. It employs an iterative procedure to find approximations to the solution and is effective for nonlinear (fractional) ordinary [1, 2] and partial differential equations [3, 4]. The analytical approximations obtained usually converge exponentially to the exact solutions. Even the zeroth-order approximation provides useful results, while the first-order approximation remains accurate and relatively easy to compute. The method also offers flexibility in handling different types of nonlinearities.

My PhD research on the BLUES method involved extending the formalism beyond ordinary differential equations to tackle coupled systems and partial differential equations. One particular example is in epidemiology; the method accurately approximates the time evolution of population fractions in a SIRS model with vaccination, outperforming alternative methods in terms of rapid and global convergence. It accurately predicts the infection peak time and generates analytic expressions closely matching numerical solutions [5].

References:

[1] Jonas Berx and Joseph O. Indekeu, "Analytic iteration procedure for solitons and traveling wavefronts with sources", J. Phys. A: Math. Theor. 52, 38LT01 (2019)

[2] Jonas Berx and Joseph O. Indekeu, "BLUES iteration applied to nonlinear ordinary differential equations for wave propagation and heat transfer", J. Phys. A: Math. Theor. 54, 025702 (2020)

[3] Jonas Berx and Joseph O. Indekeu, "BLUES function method applied to partial differential equations and analytic approximants for interface growth under shear", Phys. Rev. Research 3, 033113 (2021)

[4] Jonas Berx and Joseph O. Indekeu, "The BLUES function method for second-order partial differential equations: Application to a nonlinear telegrapher equation", PADIFF 5, 100392 (2022)

[5] Jonas Berx and Joseph O. Indekeu, "Epidemic processes with vaccination and immunity loss studied with the BLUES function method", Physica A 590, 126724 (2022)