Research Article  Open Access
P. G. Dlamini, M. Khumalo, "On the Computation of Blowup Solutions for Semilinear ODEs and Parabolic PDEs", Mathematical Problems in Engineering, vol. 2012, Article ID 162034, 15 pages, 2012. https://doi.org/10.1155/2012/162034
On the Computation of Blowup Solutions for Semilinear ODEs and Parabolic PDEs
Abstract
We introduce an adaptive numerical method for computing blowup solutions for ODEs and wellknown reactiondiffusion equations. The method is based on the implicit midpoint method and the implicit Euler method. We demonstrate that the method produces superior results to the adaptive PECEimplicit method and the MATLAB solver of comparable order.
1. Introduction
Reactiondiffusion equations model a wide range of problems in physics, biology, and chemistry. They explain how the concentration of one or more substances distributed in space changes under the influence of two processes: chemical reactions and diffusion. These substances can be basic particles in physics, bacteria, molecules, cells, and so forth. The substances reside in a region .
The reactiondiffusion equation is a semilinear parabolic partial differential equation of the form
Equation (1.1) can be viewed as a heat conduction problem, where is the temperature of a substance in a bounded domain and represents a heat source. is referred to as the diffusion term and as the reaction term. In this case, convection does not take place, so does not depend on .
For sufficiently large initial function the solution of (1.1) will blowup in finite time. Blowup occurs when the solution of the partial differential equation ceases to exist in finite time; that is, there is (blowup time) so that
Bebernes and Eberly [1] state that a necessary condition for blowup in finite time is if Kaplan [2] showed that for convex source terms satisfying (1.3) diffusion cannot prevent blowup if the initial state is large enough. In most papers, blowup properties are discussed in the case where the nonlinear term in (1.1) is of the form where is bounded and , where is finite. In most of the work the case where has been studied. However, more recently, Brunner et al. [3] studied the numerical solution of blowup problems within the context of unbounded domains.
Stuart and Floater [4] showed that fixed step methods, both explicit and implicit, fail to reproduce blowup time for a scalar ODE. They also examined variable step methods. They used time stepping strategies which are based on a rescaling of the time variable in the underlying differential equation. They also apply these ideas to a PDE. Bandle and Brunner [5] present a survey of the theory and the numerical analysis of blowup solutions for quasilinear reactiondiffusion equations. Budd et al. [6] proposed the use of moving mesh partial differential equation (MMPDE) methods for solving (1.1). In this method the function is discretized to give the solution values defined on a moving mesh . A more general study of the MMPDE is presented in [7] and the references therein. More recently Ma et al. [8, 9] have used the moving mesh methods to numerically study blowup in nonlocal reaction diffusion equations and partial integrodifferential equations in general.
In this paper we will use the method of lines (MOLs) to solve (1.1). In this method the PDE is discretized in space, which leads to a system of ODEs with initial conditions. The numerical solution can then be obtained by solving the ODE initial value problems (see [10]). We introduce an adaptive method based on the implicit midpoint method and the implicit Euler method to solve the resulting system of ODEs. More work has been done on PsDEs with autonomous nonlinear reaction term. In this work we also give numerical results for PDEs with nonautonomous nonlinear term.
2. Description of Methods Used
In this work we use variable step methods to compute the blowup time. We use onepoint collocation methods and compare the results with MATLAB solvers ode45 and ode15s. In our procedures we specify the acceptable error per step, and if it is not met, the procedure adjusts the step size so that each step introduces an error that is not more than the acceptable error. At each step we solve the problem using two different algorithms, giving two different solutions, say and . We adjust the stepsize in accordance with .
2.1. OnePoint Collocation Methods
Onepoint collocation methods are a family of methods of the form where . The specified cases are (i) corresponds to explicit Euler method; (ii) corresponds to implicit midpoint method; (iii) corresponds to implicit Euler method.
Note that all the onepoint collocation methods are of order 1; however, the implicit midpoint method () achieves order 2 local superconvergence (see [11]).
2.2. Adaptive PECEImplicit Euler Method
This method is based on the implicit Euler method. We compute using a predictorcorrector method in which is obtained using explicit Euler's method so that we have
To get , we use Newton's method as the solver to deal with the implicit nature of the implicit Euler method.
2.3. Adaptive Implicit MidpointImplicit Euler Method
We use the implicit Euler method with Newton's method as the solver to get . To get we use midpoint Euler method given by
First we get using the implicit Euler method and substitute in (2.3) to get , that is, then,
2.4. MATLAB Solvers (ode45 and ode15s)
ode45 is a onestep Matlab solver that is based on an explicit RungeKutta (4,5) scheme. It varies the size of the step of the independent variable in order to meet the accuracy specified. On the other hand, ode15s is a multistep Matlab variable order solver based on implicit methods. We use these solvers to compare with the adaptive implicit midpointimplicit Euler method we are introducing.
3. Blowup for ODEs
We will first consider this simple case with and .
3.1. Analytic Solution
Theorem 3.1. Given the system (3.1), its solution will blowup in finite time for and at (see Brunner [11]).
Proof. Note that (3.1) is a Bernoulli equation, whose solution is Then, when Thus β is finite if Thus as required.
We compute the blowup time for , and . The results are shown in Table 1, and a graph showing the analytic solution of (3.1) is shown in Figure 1.

3.2. Numerical Computation
We solve (3.1) with , , , and . Tables 2 and 3 show the blowup times and the errors of each method, respectively, and Figure 2 shows the graphs of the solution of (3.1). The tolerance used for the computations is .


(a) By PECEimplicit Euler
(b) By midpointimplicit Euler
(c) By ode45
(d) By ode15s
3.3. Discussion
The blowup results for the different methods are very close to the analytic value as shown in Table 2. From Table 3, we see that ode45, which is of higher order than the other three methods, gives the best results than the other three methods. The adaptive implicit midpointimplicit Euler gives a better result than the other two methods of comparable order, that is, adaptive PECEimplicit Euler method and ode15s. The adaptive PECEimplicit Euler method gives a quite large error and requires a very small tolerance to get a result which is close to the exact value. From the results, we observe that as the value of in (3.1) is increased the blowup time occurs earlier and is much later for values of much closer to 1.
We seek to determine whether the performance of the methods is the same in the case where we have a reactiondiffusion equation.
4. ReactionDiffusion Equation: One Space Dimension
In this section we compute the blowup time for a one space dimension reactiondiffusion equation with an autonomous reaction term and a nonautonomous reaction term.
4.1. Autonomous Reaction Term
We solve the system (1.1) with the autonomous reaction term , where and the domain is just the real line, that is, and . The system becomes
We use the method of lines (MOLs) to discretize (4.1) in space. For the spatial discretization, we choose a uniform mesh (with ) on and replace by the standard central difference approximation. We use the function as the initial function with different values of . We get the following system of ODEs for :
We solve the system (4.2) with and . Tables 4, 5, and 6 show the blowup results obtained with , and 200, respectively, and with different values of in the initial function . Figures 3, 4, 5, and 6 show the graphs of the solution of (4.1) for and .



(a) with
(b) with
(a) With
(b) With
(a) With
(b) With
(a) With
(b) With
4.2. Nonautonomous Reaction Term
We now solve the system (1.1) with the nonautonomous reaction term , where and the domain is just the real line, that is, and . The system becomes
As in (4.1) we use the method of lines to discretize (4.3) to obtain the following system:
We solve (4.4) for and , Tables 7 and 8 show the blowup times obtained from the different values of and .


4.3. Discussion
We observe that as we increase the amplitude of the initial function, , the blowup time tends to occur earlier. We also observe that for smaller values of , there is no blowup.
Considering the reactiondiffusion equation, and comparing the autonomous against the nonautonomous reaction term cases, we observe that the introduction of the nonautonomous term ensures that a much larger amplitude, , in the initial function, is required for blowup to occur. We also note that increasing for fixed or increasing for fixed increases the minimum amplitude for blowup to occur.
On the performance of the methods, we note a similar trend to what we observed in the ODE case. The adaptive implicit midpointimplicit Euler method gives results that are significantly superior to the adaptive PECEimplicit Euler method and ode15s. In fact, its performance is comparable to ode45.
5. ReactionDiffusion Equation: Two Space Dimensions
We solve the system (1.1) with the reaction term , where with . The system becomes
We use the method of lines (MOLs) to discretize (5.1) in space. For the spatial discretization, we choose uniform meshes for and , and , respectively, (with and ) on . We replace and () by the standard central difference approximation. We use the function as the initial function with different values of . We get the following system of ODEs for :
We solve the system (5.2) with and . Figure 7 shows the solution of (5.2), and Table 9 shows the blowup times obtained using the four methods.

(a) By PECEimplicit Euler
(b) By midpointimplicit Euler
(c) By ode45
(d) By ode15s
5.1. Discussion
We observe similar results for the two space dimensions reactiondiffusion equation to the one space dimension case; that is, the adaptive implicit midpointimplicit Euler method gives significantly better results than the adaptive PECEimplicit Euler method and ode15s. Its results are comparable with the higher order, computationally costly ode45.
6. Future Work
We would like to extend the blowup computations to the Volterra integrodifferential equations, that is, cases where the reaction term is nonlocal.
Given the relative simplicity of this new method, and cheaper computational expense, we conclude that it is much better than the higherorder RKbased solver, for implementation on problems of the nature studied in this paper. We would further like to test the new method on other problems in engineering and applied science, with the objective of proposing it for wider implementation.
Acknowledgment
The authors gratefully acknowledge the valuable discussions with Professor Hermann Brunner on finitetime blowup problems.
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Copyright
Copyright © 2012 P. G. Dlamini and M. Khumalo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.