Study on interval Volterra integral equations via parametric approach of intervals

We consider a SEIR model for the spread (transmission) of an infectious disease. The model has played an important role due to world pandemic disease spread cases. Our contributions in this paper are three folds. Our first contribution is to provide successive approximation and variational iteration methods to obtain analytical approximate solutions to the SEIR model. Our second contribution is to prove that for solving the SEIR model, the variational iteration and successive approximation methods are identical when we have some particular values of Lagrange multipliers in the variational iteration formulation. Third, we propose a new multistage-analytical method for solving the SEIR model. Computational experiments show that the successive approximation and variational iteration methods are accurate for small size of time domain. In contrast, our proposed multistage-analytical method is successful to solve the SEIR model very accurately for large size of time domain. Furthermore, the order of accuracy of the multistage-analytical method can be made higher simply by taking more number of successive iterations in the multistage evolution.

Based on the method of successive Krylov-Chernousko approximations, a procedure for the synthesis of trajectory control of observations for a multi-position measuring system with a variable topology is obtained. Particular search procedures are applied sequentially to the trajectories of the corresponding mobile reception points. The method of successive approximations is implemented both with a partial update of the trajectory control law for observations of each of the IFR, and with a partial update of the topology during the transition from one mobile receiving point to another.

Applications of the methods of averaging and successive approximations for studying nonlinear oscillations

The conventional method of joint rate and power control (JRPC) relies on high signal-to-interference ratio (SIR) assumption which achieves only suboptimal results. By using a novel successive convex approximations method, we can attain the global optimal source rates and link powers in a distributed fashion exploiting message passing. Through simulations, our method converges faster than the previous work based on logarithm successive convex approximations.

Power flow analysis using successive approximation and adomian decomposition methods with a new power flow formulation

This paper addresses the Optimal Power Flow (OPF) problem in Direct Current (DC) networks by considering the integration of Distributed Generators (DGs). In order to model said problem, this study employs a mathematical formulation that has, as the objective function, the reduction in power losses associated with energy transport and that considers the set of constraints that compose DC networks in an environment of distributed generation. To solve this mathematical formulation, a master–slave methodology that combines the Salp Swarm Algorithm (SSA) and the Successive Approximations (SA) method was used here. The effectiveness, repeatability, and robustness of the proposed solution methodology was validated using two test systems (the 21- and 69-node systems), five other optimization methods reported in the specialized literature, and three different penetration levels of distributed generation: 20%, 40%, and 60% of the power provided by the slack node in the test systems in an environment with no DGs (base case). All simulations were executed 100 times for each solution methodology in the different test scenarios. The purpose of this was to evaluate the repeatability of the solutions provided by each technique by analyzing their minimum and average power losses and required processing times. The results show that the proposed solution methodology achieved the best trade-off between (minimum and average) power loss reduction and processing time for networks of any size.

Modified successive approximation methods for the nonlinear eigenvalue problems

An algorithm for the method of successive approximations in optimal control problems

The classical Method of Successive Approximations (MSA) is an iterative method for solving stochastic control problems and is derived from Pontryagin’s optimality principle. It is known that the MSA may fail to converge. Using careful estimates for the backward stochastic differential equation (BSDE) this paper suggests a modification to the MSA algorithm. This modified MSA is shown to converge for general stochastic control problems with control in both the drift and diffusion coefficients. Under some additional assumptions the rate of convergence is shown. The results are valid without restrictions on the time horizon of the control problem, in contrast to iterative methods based on the theory of forward-backward stochastic differential equations.

The Method of Successive Approximation in Descriptive Ecology

The paper contains a survey of results devoted to one of the numerical methods of optimal control—the method of successive approximations. This method is based on Pontryagin's maximum principle and is known in the English literature as the min‐H method. Various modifications of the method and some theoretical results on its convergence are presented. Examples of applications of the method for the calculation of optimal trajectories are given. The method of small parameters which is close to the method of successive approximations, is also described.

In this article we present the existence, uniqueness and stability of mild solutions for impulsive stochastic functional integro differential equations with non-Lipschitz condition. The mild solution is obtained by using a resolvent operator in a different sense and the results are proved by using the method of successive approximation and Bihari’s inequality.

A modified Pagano method for the 3D dynamic responses of functionally graded magneto-electro-elastic plates

Purpose. New algorithms were consider for functional equations solving using the Feigenbaum equation as an example. This equation is of great interest in the theory of deterministic chaos and is a good illustrative example in the class of functional equations with superposition. Methods. The article proposes three new effective methods for solving functional equations — the method of successive approximations, the method of successive approximations using the fast Fourier transform and the numerical-analytical method using a small parameter. Results. Three new methods for solving functional equations were presented, considered on the example of the Feigenbaum equation. For each of them, the features of their application were investigated, as well as the complexity of the resulting algorithms was estimated. The methods previously used by researchers to solve functional equations are compared with those described in this article. In the description of the latter, the numerical-analytical method, several coefficients of expansions of the universal Feigenbaum constants were written out. Conclusion. The obtained algorithms, based on simple iteration methods, allow solving functional equations with superposition without the need to reverse the Jacobi matrix. This feature greatly simplifies the use of computer memory and gives a gain in the operating time of the algorithms in question, compared with previously used ones. Also, the latter, numerically-analytical method made it possible to obtain sequentially the coefficients of expansions of the universal Feigenbaum constants, which in fact can be an analytical representation of these constants.

Medical image registration is widely used in the clinical field. A simple and effective registration method for medical images is proposed, which is not only used images’ hierarchical information, but also the successive approximate method is used. It mainly includes two steps, which are the coarse registration and the fine registration. In the coarse registration procedure, the couple images’ edge features are extracted. Further more, the contour line features are extracted based on the edge features. The contour lines are selected as registration objects to accomplish the coarse registration based on the Principal Axis method. In the second registration procedure, i.e. the fine registration, the hierarchical information of the images are obtained by the convolution operator and pooling method firstly. Then these information are matched from the last scale to the first scale to obtain the last registration result. The proposed method’s validity is proved by different registration experiments, and the proposed method can also be applied to incomplete images’ registration. The error of registration is less than one pixel unit in the different registration experiments of the proposed method.