We extend the operational matrices technique to design a spectral solution of nonlinear fractional differential equations (FDEs). The derivative is considered in the Caputo sense. The coupled system of two FDEs is considered, subjected to more generalized integral type conditions. The basis of our approach is the most simple orthogonal polynomials. Several new matrices are derived that have strong applications in the development of computational scheme. The scheme presented in this article is able to convert nonlinear coupled system of FDEs to an equivalent S-lvester type algebraic equation. The solution of the algebraic structure is constructed by converting the system into a complex Schur form. After conversion, the solution of the resultant triangular system is obtained and transformed back to construct the solution of algebraic structure. The solution of the matrix equation is used to construct the solution of the related nonlinear system of FDEs. The convergence of the proposed method is investigated analytically and verified experimentally through a wide variety of test problems.
The analysis ofarchitectural computer constructions is made mainly by technical characteristics of time and hardware complexities. The usage of SH-model for computer devices allows an extreme enlargement of their characteristics' list. Significantly new informational and quantitative characteristics are added to this list. Those are structure and program complexities. They are used in evaluating the amount of information in the software-hardware of computer devices. Those two characteristics add a physical sense to the conception of "artificial intelligence ". Using the SH-model, the theories of algorithms complexity became a good tool for analysis and optimization ofcomputer devices. The central part of the report is the usage of that theory in construction of operational devices. Three schemes of the devices which realize the same calculations are considered. The multiplication is taken as an example. Technical solutions of multi-digit numbers' multiplication devices are well known. The purpose of returning to this topic is to demonstrate the possibilities of the complexity theory for non-abstract algorithm. Let us point at some information about SH-model complexity characteristics.During synthesis, the analysis and optimization of SHmodels it is advised to use four characteristics of complexity (hardware, program, structural and capacity).Hardware complexity is the quantity (amount) of elementary converters and temporary memory cells of some hierarchical level of SH-model's hardware:of hardware complexity (now it reaches 100 billion of transistors) gives us great opportunities for expansion the functional characteristics, for improving practically all processor's usage characteristics including the calculations' precision and productivity growth (without enlarging the physical size of the computer, expansion of functions).This hardware complexity definition doesn't contradict "amount of equipment" conception which is used in computing methods.Time complexity. In the metric theory, time complexity is known to be the number of basic operations such as the Turing machine's steps. In software/hardware theory time complexity is identified in a little bit different way.SH-model's time complexity is evaluated by the number of elements of the scheme located along the maximal critical way of distribution of the signal: L =ImaxXil (2) where max Xi is SH-model's series of elements which belong to the most critical way of signal distribution, including repeated passages of elements to a cycle.The unit of time complexity is the elementary converter of some hierarchical level of the scheme. The switch from time complexity to define the time of the scheme's operation is resulted in the following formula: T= ZT(e) Tei cmaxl ei (3) where X is the set of all elements in the circuit of hardware. This definition verifies the hierarchal structure of computer devices. If we consider SH-model to be an operational device than elementary converters would be one-digit cells or gates, at the level of register transmissions elements are ...
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