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In this paper, the authors show that the general linear second order ordinary Differential Equation can be formulated as an optimization problem and that evolutionary algorithms for solving optimization problems can also be adapted for solving the formulated problem. The authors propose a polynomial based scheme for achieving the above objectives. The coefficients of the proposed scheme are approximated by an evolutionary algorithm known as Differential Evolution (DE). Numerical examples with good results show the accuracy of the proposed method compared with some existing methods.

For centuries, Differential Equations (DEs) have been an important concept in many branches of science. They arise spontaneously in physics, engineering, chemistry, biology, economics and a lot of fields in between. Many Ordinary Differential Equations (ODEs) have been solved analytically to obtain solutions in a closed form. However, the range of Differential Equations that can be solved by straightforward analytical methods is relatively restricted. In many cases, where a Differential Equation and known boundary conditions are given, an approximate solution is often obtainable by the application of numerical methods.

Several numerical methods (see [

Since many evolutionary optimization techniques are methods that optimizing a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality (see [

Nikos [

In this paper we show that the Differential Evolution (DE) algorithm can also be used to find very accurate approximate solutions of second order Initial Value Problems (IVPs) of the form

Formally, let

Let

• Initialize all agents

• Until a termination criterion is met (e.g. number of iterations performed, or adequate fitness reached), repeat the following.

• For each agent

* Pick three agents

* Pick a random index

* Compute the agent’s potentially new position

• For each i, pick a uniformly distributed number

• If

• (In essence, the new position is outcome of binary crossover of agent

* If

• Pick the agent from the population that has the highest fitness or lowest cost and return it as the best found candidate solution.

Note that

In this section, we show the steps involved in formulating the general linear second order initial value problem (1) as an optimization problem and then use the Differential Evolution algorithm to obtain approximate solution of the ODE.

Consider the second order initial value problem (1), in this work we assume a polynomial solution of the form

where

Using the initial conditions we have the constraint that

Using (3), at each node point

To solve the above problem, we need to find the set of coefficients

where

following way:

Equations (8) and (9) together is the formulated optimization problem of the IVP (1). The next objective of this work is to solve Equations (8) and (9) using the Differential Evolution algorithm.

Using the Differential Evolution algorithm we are able to obtain the set

ODEs (DEODEs)”.

We now perform some numerical experiments confirming the theoretical expectations regarding the method we have proposed. The propose scheme is compared with the Runge-Kutta scheme for solving (1).

The table of “CPU-time” and the maximum error of all computations are also given.

The following parameters are used for all computations.

Differential Evolution:

Cross Probability = 0.5;

Initial Points = Automatic;

Penalty Function = Automatic;

Post Process = Automatic;

Random Seed = 0;

Scaling Factor = 0.6;

Search Points = Automatic;

Tolerance = 0.001.

All computations were carried out on a “Core i3 Intel” processor machine.

We examine the following linear equation

with the exact solution

Implementing the proposed scheme with

Consider the equation

with the exact solution

Implementing the proposed scheme with

From the results obtained in

Maximum Absolute Error | CPU-Time (Seconds) | |||
---|---|---|---|---|

i | Runge-Kutta Method | DEODEs | Runge-Kutta Method | DEODEs |

3 | 4.984042E−6 | 5.573320E−14 | 5.210430E−3 | 4.056000E−4 |

4 | 3.281185E−7 | 6.594725E−14 | 1.014006E−2 | 6.864000E−4 |

5 | 2.104785E−8 | 7.016610E−14 | 2.009293E−2 | 1.248010E−3 |

6 | 1.332722E−9 | 7.105427E−14 | 3.996746E−2 | 2.464820E−3 |

7 | 8.383871E−11 | 7.149836E−14 | 8.018451E−2 | 4.836030E−3 |

8 | 5.258460E−12 | 7.149836E−14 | 1.608682E−1 | 1.023367E−2 |

9 | 3.286260E−13 | 7.149836E−14 | 3.238269E−1 | 2.162174E−2 |

Maximum Absolute Error | CPU-Time (Seconds) | |||
---|---|---|---|---|

i | Runge-Kutta Method | DEODEs | Runge-Kutta Method | DEODEs |

3 | 0 | 0 | 3.182420E−3 | 2.184000E−4 |

4 | 0 | 0 | 6.146440E−3 | 4.056000E−4 |

5 | 0 | 0 | 1.207448E−2 | 7.488000E−4 |

6 | 0 | 0 | 2.421136E−2 | 1.435210E−3 |

7 | 0 | 0 | 4.842271E−2 | 2.870420E−3 |

8 | 0 | 0 | 9.640862E−2 | 6.115240E−3 |

9 | 0 | 0 | 1.964989E−1 | 1.332249E−2 |

We see that the Differential Evolution algorithm for solving ODEs gave better approximate results for different steplengths (h) compared with the Runge-Kutta Nystrom method. The proposed solution process also gave better CPU-Time for both problems solved.

In this paper, we have been able to formulate the general linear second order ODE as an optimization problem, and we have also been able to solve the formulated optimization problem using the Differential Evolution algorithm. Numerical examples also show that the method gives better approximate solutions. Other evolutionary techniques can be exploited as well.

Bakre OmolaraFatimah,Wusu AshiriboSenapon,Akanbi MosesAdebowale, (2015) Solving Ordinary Differential Equations with Evolutionary Algorithms. Open Journal of Optimization,04,69-73. doi: 10.4236/ojop.2015.43009