## News Update on Heat Equation : April 21

**[1] Motion planning for the heat equation**

The paper gives an explicit open‐loop control, able to approximately steer the one‐dimensional heat equation with control on the boundary from any state to any other state. The control is obtained thanks to a parametrization of the solutions of the heat equation by a series involving infinitely many derivatives of the system ‘flat output’. Copyright © 2000 John Wiley & Sons, Ltd.

**[2] Existence and non-existence of global solutions for a semilinear heat equation**

The existence and non-existence of global solutions and theL p blow-up of non-global solutions to the initial value problemu′(t)=Δu(t)+u(t)γ onR n are studied. We consider onlyγ>1. In the casen(γ − 1)/2=1, we present a simple proof that there are no non-trivial global non-negative solutions. Ifn(γ−1)/2≦1, we show under mild technical restrictions that non-negativeL p solutions always blow-up inL p norm in finite time. In the casen(γ−1)/2>1, we give new sufficient conditions on the initial data which guarantee the existence of global solutions.

**[3] On the support of solutions to the heat equation with noise**

Let be 2-parameter white noise. Let satisfy and suppose that is bounded, nonnegative, with compact support and not identically 0. We show that with probability for all . This complements results of Iscoe and Shiga, who show that for y has compact support.

**[4] Solving Fuzzy Heat Equation by Using Numerical Methods**

This research proposes an explicit method to solve fuzzy heat equation with integral boundary conditions. The necessary materials and preliminaries are expressed, and a finite difference scheme for one dimensional heat equation is considered. Here, boundary conditions include integral equations which are approximated by the composite trapezoid rule. Finally, an example in order to illustrate the numerical results is given. In this example, the Hausdor distance between exact solution and approximate solution is obtained.

**[5] Modelling, Simulation and Visualization of Heat Equation Dynamics**

Aims/ Objectives: To show modelling, simulation and visualization of the dynamics of heat equation in a rod. Other PDE models and Numerical approaches are also discussed.

Study Design: This paper is simply about how to use MATLAB software to solve PDE model.

Methodology: A Class of PDE was uniquely modelled, simulated, and visualized using certain algorithm and MATLAB routine for Elliptic/Hyperbolic PDE. Different diagrams showing the nature of the solution are discussed.

Results: The nature of the PDE governing heat dynamic in a rod are shown in the figures. Other PDE models are also presented.

Conclusion: A simple demostration of numerical simulation of PDE governing the popular heat equation is presented. Three other PDE models are considered. The results herein can be adapted and applied to other more complex PDE models.

**Reference**

**[1] **Laroche, B., Martin, P. and Rouchon, P., 2000. Motion planning for the heat equation. *International Journal of Robust and Nonlinear Control: IFAC**‐**Affiliated Journal*, *10*(8), pp.629-643.

**[2] **Weissler, F.B., 1981. Existence and non-existence of global solutions for a semilinear heat equation. *Israel Journal of Mathematics*, *38*(1-2), pp.29-40.

**[3] **Mueller, C., 1991. On the support of solutions to the heat equation with noise. *Stochastics: An International Journal of Probability and Stochastic Processes*, *37*(4), pp.225-245.

**[4] **Hosseinpour, A., 2018. Solving Fuzzy Heat Equation by Using Numerical Methods. *Asian Research Journal of Mathematics*, pp.1-7.

**[5] **Oluleye, H.B., 2014. Modelling, Simulation and Visualization of Heat Equation Dynamics. *Journal of Advances in Mathematics and Computer Science*, pp.2155-2169.