Linear Neutral Delay Differential Equations solver

This function solves the first-order, homogeneous, linear Neutral Delay Differential Equation (NDDE) with constant coefficients and delay

About the function

sol=nddesolver(dydt,delay,preshape,interval,N,s) integrates a linear, homogeneous, delay differential equations of neutral type with constant coefficients and constant delay given by

where

• t is the independent variable representing time,
• is the solution with delay h, and
• is the derivative of the solution with delay h.
• is the history/preshape function

This function can be solved explicitly using a generalized Lambert W function called the Lambert W function [1]. This solution is given by

,

where

• λ(i) is a complex value dependent on the value of the Lambert W function, and
• is a real valued coefficient of dependent on the preshape/history function.

Examples

Included in the files are two examples comparing the solutions of a first-order linear, homogeneous NDDE obtained using nddesolver, with its analytical solution.

Input Arguments

• dydt is the vector containing the coefficients of (1). (dydt = [])
• delay is the constant delay h in (1). The value of h must be positive.
• preshape is the history function $\phi(t)$ on
• interval is the interval of integration . The first element, , is the initial value of t. The second element, , is the final value of t. The value of must be less than
• N is the number of complex roots of the Lambert W function with positive imaginary part. (See the function rlambertw below for .)
• s is the step size of the values of t

Output Arguments

sol, returned as a structure containing the following fields:

• sol.x is the mesh on
• sol.y is the approximation to at the mesh points
• sol.c is a vector containing the coefficients from (2)
• sol.l is a vector containing all computed using the function lambda
• sol.histx is the mesh on
• sol.histy is the preshape function evaluated at sol.histx
• sol.psi is a matrix with elements
• sol.cond is the condition number of sol.psi

Other functions

Also included in the files are the following functions:

• RA_solver finds the values of r and a for the Lambert W function
• lambda computes a vector of $\lambda(i)s$, where each satisfies the equation

(See [1] for derivation of the formula.)

• rlambertw computes a vector of the values of the Lambert W function for some r and a(denoted) where each satisfies

.

(See [2] for reference.)

Reference:

[1] Jamilla, C., Mendoza, R., & Mező, I. (2020). Solutions of neutral delay differential equations using a generalized Lambert W function. Applied Mathematics and Computation, 382, 125334.https://doi.org/10.1016/j.amc.2020.125334

[2] Mező, I., & Baricz, Á. (2017). On the generalization of the Lambert 𝑊 function. Transactions of the American Mathematical Society, 369(11), 7917-7934. https://doi.org/10.1090/tran/6911