# Improving Particle Swarm Optimization Analysis Using Differential Models
## 4 Differential Evolution PSO
### Differential Evolutionary PSO Based on Different Numerical Computation Methods  
The standard PSO algorithm (Equation (26)) can be viewed as a system of first-order differential equations containing two differential equations. By applying different numerical calculation methods, different difference forms of PSO can be obtained. Since these PSO algorithms of different difference forms are all differened using the differential model (Equation (26)). Therefore, they are called differential evolutionary PSO (differential evolutionary PSO, DPSO). According to the difference in the order of the local truncation error, the following three typical numerical calculation methods are selected as representatives to analyze the corresponding differential evolution PSO algorithm.  
The differential model (Equation (26)) can be simply viewed as:  
\[\left\{\begin{array}{l}\frac{dv_{k}(t)}{dt}=f_{1}\Big{(}t,v_{jk},x_{jk}\Big{)} \\ \frac{dx_{jk}(t)}{dt}=f_{2}\Big{(}t,v_{jk},x_{jk}\Big{)}\end{array}\right. \tag{38}\]  
1. Euler Method It is one of the most basic differential numerical calculation methods, and the order of the local truncation error is 1. Applying it to the differential equation system (Equation (38)), the corresponding PSO evolution formula (Equation (27)) can be obtained.
2. Improved Euler Method It is also a second-order Runge-Kutta method with a local truncation error of order 2. Applying it to the system of differential equations (Equation (38)), the corresponding evolution formula can be obtained as: \[\left\{\begin{array}{l}v_{jk}^{p}(t+1)=[1+h(\omega-1)]v_{jk}(t)-h\varphi x_{ jk}(t)+h\Big{(}\varphi_{1}p_{jk}+\varphi_{2}p_{gk}\Big{)}\\ x_{jk}^{p}(t+1)=h\omega v_{jk}(t)+(1-h\varphi)x_{jk}(t)+h\Big{(}\varphi_{1}p_{ jk}+\varphi_{2}p_{gk}\Big{)}\\ v_{jk}^{c}(t+1)=[1+h(\omega-1)]v_{jk}^{p}(t+1)-h\varphi x_{jk}^{p}(t+1)+h\Big{(} \varphi_{1}p_{jk}+\varphi_{2}p_{gk}\Big{)}\\ v_{jk}^{c}(t+1)=h\omega v_{jk}^{p}(t+1)+(1-h\varphi)x_{jk}^{p}(t+1)+h\Big{(} \varphi_{1}p_{jk}+\varphi_{2}p_{gk}\Big{)}\\ v_{jk}(t+1)=\frac{1}{2}\Big{(}v_{jk}^{p}(t+1)+v_{jk}^{c}(t+1)\Big{)}\\ x_{jk}(t+1)=\frac{1}{2}\Big{(}x_{jk}^{p}(t+1)+x_{jk}^{c}(t+1)\Big{)}\end{array}\right.\] (39)
3. Fourth-Order Runge-Kutta Method Its local truncation error is of order 4. Applying it to the system of differential equations (Equation (38)), the following evolution formula (Equation (40)) can be obtained: \[\left\{\begin{array}{l}v_{jk}(t+1)=v_{jk}(t)+\frac{h}{6}(K_{1}+2K_{2}+2K_{3} +K_{4})\\ x_{jk}(t+1)=x_{jk}(t)+\frac{h}{6}(L_{1}+2L_{2}+2L_{3}+KL_{4})\end{array}\right.\] (40)  
It also contains Equations (41)-(48):  
\[K_{1}=(\omega-1)v_{jk}(t)-\varphi x_{jk}(t)+\Big{(}\varphi_{1}p_{jk}+\varphi_ {2}p_{gk}\Big{)} \tag{41}\]  
\[L_{1}=\omega v_{jk}(t)-\varphi x_{jk}(t)+\Big{(}\varphi_{1}p_{jk}+\varphi_{2} p_{gk}\Big{)} \tag{42}\]  
\[K_{2}=(\omega-1)\bigg{(}v_{jk}(t)+\frac{h}{2}K_{1}\bigg{)}-\varphi\bigg{(}x_{ jk}(t)+\frac{h}{2}L_{1}\bigg{)}+\Big{(}\varphi_{1}p_{jk}+\varphi_{2}p_{gk}\Big{)} \tag{43}\]  
\[L_{2}=\omega\bigg{(}v_{jk}(t)+\frac{h}{2}K_{1}\bigg{)}-\varphi\bigg{(}x_{jk}(t) +\frac{h}{2}L_{1}\bigg{)}+\Big{(}\varphi_{1}p_{jk}+\varphi_{2}p_{gk}\Big{)} \tag{44}\]  
\[K_{3}=(\omega-1)\bigg{(}v_{jk}(t)+\frac{h}{2}K_{2}\bigg{)}-\varphi\bigg{(}x_{ jk}(t)+\frac{h}{2}L_{2}\bigg{)}+\Big{(}\varphi_{1}p_{jk}+\varphi_{2}p_{gk}\Big{)} \tag{45}\]\[L_{3}=\omega\bigg{(}v_{jk}(t)+\frac{h}{2}K_{2}\bigg{)}-\varphi\bigg{(}x_{jk}(t)+ \frac{h}{2}L_{2}\bigg{)}+\Big{(}\varphi_{1}p_{jk}+\varphi_{2}p_{gk}\Big{)} \tag{46}\]  
\[K_{4}=(\omega-1)\Big{(}v_{jk}(t)+hK_{3}\Big{)}-\varphi\Big{(}x_{jk}(t)+hL_{3} \Big{)}+\Big{(}\varphi_{1}p_{jk}+\varphi_{2}p_{gk}\Big{)} \tag{47}\]  
\[L_{4}=\omega\Big{(}v_{jk}(t)+hK_{3}\Big{)}-\varphi\Big{(}x_{jk}(t)+hL_{3} \Big{)}+\Big{(}\varphi_{1}p_{jk}+\varphi_{2}p_{gk}\Big{)} \tag{48}\]