first order forward Euler methodstep sizeThe leapfrog method is popular because of its good stability when solving partial differential equations with oscillatory solutions. "It has the disadvantage that the solution at odd time steps tends to drift farther and farther from the solution for even ...
Clearly, the stages of the method (9) are simply linear combinations of forward Euler steps with a modified time step Δtβi,kαi,k. Consistency requires that ∑k=0i−1αi,k=1, so that as long as all the coefficients αi, k and βi, k are nonnegative, we have a convex comb...
Furthermore, we give the necessary condition for it to be observed as a stable state, which in some instances takes the form of a Master Stability Function (MSF), a method initially developed in ref. 57 for pairwise coupled systems, and later extended in many ways to complex networks58 ...
The NLESO in Equation (17) can be discretized using the Euler's forward method and expressed as follows: ê(k) = x̂1(k) − y(k), ( ) () x̂1(k + 1) = x̂1(k) + Ts x̂2 + b0u(k) − Mc1 ê(k) , () x̂2(k + 1) = x̂2(k) − Mc2(...
The range of contraction rates in Fig. 3a is obtained by varying s between 2 and 100. The noise standard deviation (σ) is set to 0.2. Numerical solution of Eq. (7) is obtained using the Euler-Maruyama method with a time step of Δt=5×10−3. The stochastic Lorenz attractor is ...
dimensions, nonlinear stability results until very recently either required a positive cosmological constant or a restriction to the massless case. a recent series of works then established the stability of minkowski space for the einstein–vlasov system by a vector-field-method approach [ fjs15 , ...
In order to apply Theorem 1.2 for the study of exponential stability of a dynamic equation (1.1), one can replace condition (1.18) by the necessary and sufficient condition of exponential stability of an exponential function on a time scale given in [12]. Example 1.9. Consider the Euler ...
Then we can determine under what conditions |g|⩽1 to find a range of stability. In the example above where A=(0−110), the eigenvalues are ±i. Using the growth rate for forward Euler, we get gFE=1±iΔt, which implies |gFE|=1+Δt2. This result means that for any Δt, ...
[15] introduced concepts on FFC of variation and proposed necessary conditions to get the fractional Euler-Lagrange equation for fractional variational problems in the fuzzy sense. Hoa [17] studied the problems of the existence of solutions to FFDEs with delay, and the numerical method to solve ...
This paper deals with thenumerical stabilityof implicit Euler method for nonlinear pantograph equation in which constant stepsize and variable stepsize are applied. 讨论非线性比例延迟微分方程隐式Euler法的数值稳定性,其中步长采用定步长和变步长两种方式。