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Collocation method for a functional equation arising in behavioral sciences

Josefa Caballero, Hanna Okrasińska-Płociniczak, Łukasz Płociniczak, Kishin Sadarangani

Abstract

We consider a nonlocal functional equation that is a generalization of the mathematical model used in behavioral sciences. The equation is built upon an operator that introduces a convex combination and a nonlinear mixing of the function arguments. We show that, provided some growth conditions of the coefficients, there exists a unique solution in the natural Lipschitz space. Furthermore, we prove that the regularity of the solution is inherited from the smoothness properties of the coefficients. As a natural numerical method to solve the general case, we consider the collocation scheme of piecewise linear functions. We prove that the method converges with the error bounded by the error of projecting the Lipschitz function onto the piecewise linear polynomial space. Moreover, provided sufficient regularity of the coefficients, the scheme is of the second order measured in the supremum norm. A series of numerical experiments verify the proved claims and show that the implementation is computationally cheap and exceeds the frequently used Picard iteration by orders of magnitude in the calculation time.

Collocation method for a functional equation arising in behavioral sciences

Abstract

We consider a nonlocal functional equation that is a generalization of the mathematical model used in behavioral sciences. The equation is built upon an operator that introduces a convex combination and a nonlinear mixing of the function arguments. We show that, provided some growth conditions of the coefficients, there exists a unique solution in the natural Lipschitz space. Furthermore, we prove that the regularity of the solution is inherited from the smoothness properties of the coefficients. As a natural numerical method to solve the general case, we consider the collocation scheme of piecewise linear functions. We prove that the method converges with the error bounded by the error of projecting the Lipschitz function onto the piecewise linear polynomial space. Moreover, provided sufficient regularity of the coefficients, the scheme is of the second order measured in the supremum norm. A series of numerical experiments verify the proved claims and show that the implementation is computationally cheap and exceeds the frequently used Picard iteration by orders of magnitude in the calculation time.

Paper Structure

This paper contains 8 sections, 4 theorems, 62 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Existence, uniqueness, and regularity
  3. Collocation method
  4. Numerical examples
  5. Paradise fish equation
  6. Exact solution (smooth)
  7. Exact solution (nonsmooth)
  8. Conclusion and future work

Key Result

Theorem 1

Assume eqn:Assumptions and suppose that $(1+\|\varphi\|)(\|\varphi_1\|+\|\varphi_2\|)<1$. Then, there exists an exactly one solution of eqn:MainEqGeneral in $H_0^1$ satisfying

Figures (6)

  • Figure 1: An exemplary of of the solution to \ref{['eqn:ParadiseFish']} obtained collocation method with the step $h=10^{-2}$.
  • Figure 2: Error (solid line) of the collocation method applied to the problem \ref{['eqn:ExactProblem']} with respect to the number of subdivisions of the interval $n$. The reference line $n^{-2}$ is depicted with a dashed line and $\alpha = 0.3$.
  • Figure 3: Computation time as a function of the $L^2$ (RMS) error on the log-log scale. Lines represent the fitted trends.
  • Figure 4: Computation time as a function of the number of degrees of freedom $n$. The solid line represents the fitted trend $\propto n^{2.07}$.
  • Figure 5: Exact solution to \ref{['eqn:ExactProblemNonsmooth']} (solid line) obtained collocation method with the step $h=2^{-8}$ (dashed line) and $\alpha = 0.45$.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Theorem 1: Existence and uniqueness
  • proof
  • Proposition 1: Regularity
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Theorem 2: Convergence
  • proof
  • Remark 2