# Simultaneous Numerical Determination of Two Time-dependent Coefficients in Second Order Parabolic Equation With Nonlocal Initial and Boundary Conditions

## TL;DR

This paper presents a method to accurately determine two time-dependent coefficients in a mathematical model using a stable numerical approach and regularization.

## Contribution

A novel framework for simultaneously identifying two time-dependent coefficients in a parabolic equation with nonlocal conditions using regularization and optimization.

## Key findings

- The Crank-Nicolson FDM ensures stability and accuracy in solving the direct problem.
- Tikhonov regularization effectively reduces errors and stabilizes the inverse problem solution.
- Numerical experiments confirm the method's reliability under exact and noisy data.

## Abstract

This study establishes a mathematically consistent and computational framework for the simultaneous identification of two time-dependent coefficients in a one-dimensional second-order parabolic partial differential equation. The considered problem is governed by nonlocal initial, boundary, and integral overdetermination conditions.

The direct problem is solved using the Crank-Nicolson finite difference method (FDM), which ensures unconditional stability and second-order accuracy in both spatial and temporal discretizations. The corresponding inverse problem is reformulated as a nonlinear regularized least-squares optimization problem and efficiently solved used the MATLAB subroutine
lsqnonlin from the optimization Toolbox. Due to the intrinsic, ill-posedness of the inverse formulation, small input data errors lead to big output errors. Then, Tikhonov regularization, is employed to enhance numerical stability and robustness.

Extensive numerical experiments are carried out under exact and noisy data to evaluate the numerical accuracy and convergence behavior of the method. The results confirm that the regularization technique effectively damps numerical oscillations, minimizes reconstruction error, and ensures reliable recovery of the unknown coefficients. Sensitivity analysis further reveals the essential role of the regularization parameter in controlling the trade-off between stability and accuracy.

The proposed approach provides an accurate and computationally efficient tool for IP in heat transfer, diffusion processes, and related applied sciences.