During the last decade, a big progress has been achieved in the analysis and numerical treatment of Initial Value Problems (IVPs) in Differential Algebraic Equations (DAEs) and Ordinary Differential Equations (ODEs). In spite of the rich variety of results available in the literature, there are still many specific problems that require special attention. Two of such, which are considered in this work, are the optimization of order of accuracy and reduction of cost of functional evaluations of Explicit Runge - Kutta (ERK) methods.

Traditionally, the maximum attainable order p of an s-stage ERK method for advancing the solution of an IVP is such that
p(s)
 4
 
In 1999, Goeken presented an s-stage ERK Method of order p(s)=s +1,s>2.
However, this work focuses on the design of a new approximation algorithm that reduces the cost of functional evaluations and yet increases the attainable order higher 
U n and Jonhson [94]; and the classical ERK methods. The order p of 
the new scheme called Multiderivative Explicit Runge-Kutta (MERK) Methods is such that p(s) 2. The stability, convergence and implementation for the optimization of IVPs in DAEs and ODEs systems are also considered.

Numerical solution of initial value problems in differential - algebraic equations

Differential-algebraic equations (DAEs) are important tools to model complex problems in various application fields easily. Those DAEs with an index-3, even the linear ones, are known to cause problems when solving them numerically. The present article proposes a new algorithm together with its multistage form to efficiently solve a class of nonlinear implicit Hessenberg index-3 DAEs. This algorithm is based on the idea of applying the differential transform method (DTM) directly to the DAE without applying the traditional index reduction methods, which can be complex and often result in violations of the DAE constraints. Also, to deal with the nonlinear terms in the DAE, we approximate them using the Adomian polynomials. This new idea has given us a simple and efficient algorithm, which involves the solution of linear algebraic systems except for the initial recursion terms. This algorithm is easy to implement in Maple or Mathematica. Furthermore, to enlarge the interval of convergence of the power series solution obtained from the DTM, an algorithm for the multistage DTM is also given. Both algorithms are applied to solve two examples of highly nonlinear implicit index-3 Hessenberg DAEs. Numerical results show that the DTM can determine the exact solution in convergent power series, while the multistage DTM can compute accurate numerical solutions over large intervals. 

/pdf/the-differential-transform-method-as-an-effective-tool-to-1xtoa9a8.pdf

The Differential Transform Method as an Effective Tool to Solve Implicit Hessenberg Index-3 Differential-Algebraic Equations

This research investigates differential-algebraic equations with higher index (index four). Specifically, a functional analytic approach is proposed to find the solution of (index four) Hessenberg differential-algebraic equations (DAEs). The approximate solution of the proposed functional approach for a suitable separable Hilbert space is obtained with the help of the direct optimisation technique and Ritz basis functions method. Illustrations have been ranked from an 8 × 8 test system of index-four Hessenberg linear DAEs to a 4 × 4 DAE of rotating masses as well as an 8 × 8 differential-algebraic generator model, where it reformulated index-four linear Hessenberg DAEs. Their approximate solutions were obtained using the present approach with comparisons. The numerical results demonstrate the simplicity of the proposed approach and express suitable accuracy and efficiency.

Functional Approach for Solving Reduced Order of Index-Four Hessenberg Differential-Algebraic Control System

Abstract 
    In this paper, the solutions to class of robust non-linear semi-explicit descriptor control systems with matching condition via optimal control strategy are obtained. The optimal control strategy  has been introduced and  developed in the sense that, the optimal control  solution is robust solution to the given non-linear uncertain semi-explicit descriptor control system. The necessary mathematical proofs and remarks as well as  discussions are also proposed. The present approach is step-by-step illustrated by application example to show its effectiveness a and efficiency to compensate  the structure uncertainty in the given semi-explicit (descriptor) control system

Optimal Control Problem To Robust Nonlinear Descriptor control Systems with Matching Condition

Index-3 differential-algebraic equations (DAEs) are mathematical models for the dynamics of constrained multibody mechanical systems that arise in many applications. These DAEs are known to pose a challenge to numerical methods. The purpose of this paper is to propose a new approach to efficiently solve these equations. This approach relies on an effective combination of the power series method (PSM) with the Adomian polynomials. Here, the PSM is directly applied to these DAEs without using the usual index reduction techniques, which are costly and often lead to nonphysical solutions. We expand the nonlinear terms in a series form using the Adomian polynomials to overcome the limitation of the PSM in collecting the coefficients of the power series solution. This technique has led to a simple and efficient algorithm. The domain of convergence of the power series solution is expanded by developing a multistage PSM (MSPSM). To demonstrate the efficiency of the MSPSM and show its applicability, an index-3 nonlinear DAE problem describing a two-link planar robot arm is solved. The numerical results show that the MSPSM is a powerful tool for solving the index-3 DAEs arising from constrained multibody mechanical systems.

A New Numerical Technique for Index-3 DAEs Arising from Constrained Multibody Mechanical Systems

In this paper, a modified numerical algorithm for solving the fractional diffusion equation is proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev polynomials in space are utilized respectively. The problem is reduced to the solution of a system of linear algebraic equations. From the computational point of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.

/pdf/a-tau-approach-for-solving-fractional-diffusion-equations-568kr01eme.pdf

A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Chebyshev Polynomial Method

In this paper, we provide sufficient conditions for the perturbed Sobolev-type system’s controllability in real Hilbert space. The results are obtained using the Banach fixed-point theorem, perturbation theory, and a strongly continuous semigroup. 

Controllability of perturbed Sobolev-type of control systems using semigroup theory

Sequential unconstrained approach for solving index one constrained differential-algebraic optimal control problems

Input-Output Stabilization Problem For Some Nonlinear Dynamical Systems With Bifurcation

Monte Carlo-Bernstein polynomials simulation method for solving fractional Volterra integral equations

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