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Homotopy Algorithms for Engineering Analysis
Published in Hojjat Adeli, Supercomputing in Engineering Analysis, 2020
Layne T. Watson, Manohar P. Kamat
Continuation is a well known and established procedure in numerical analysis. The idea is to continuously deform a simple (easy) problem into the given (hard) problem, while solving the family of deformed problems. The solutions to the deformed problems are related, and can be tracked as the deformation proceeds. The function describing the deformation is called a homotopy map. Homotopies are a traditional part of topology, and have found significant application in nonlinear functional analysis and differential geometry. Similar ideas, such as incremental loading, are also widely used in engineering.
Systems with nonlinear control
Published in Simon Serovajsky, Optimization and Differentiation, 2017
We shall use an important result of the nonlinear functional analysis, the Implicit function theorem, for proving the differentiability of the map y[·]:V→Y $ y[\cdot ]:V\rightarrow Y $ , i.e., implicit operator. Consider an operator A that is determined on a neighborhood O of the point (v0,y0) $ (v_0,y_0) $ with the domain V×Y $ V\times Y $ and the codomain Z. Suppose the equality A(v0,y0)=0 $ A(v_0,y_0)=0 $ and the existence of the partial derivatives of the operator A on the set O that are continuous at the point (v0,y0) $ (v_0,y_0) $ .
setting and preliminary applications
Published in Mircea Sofonea, Stanisław Migórski, Variational-Hemivariational Inequalities with Applications, 2017
Mircea Sofonea, Stanisław Migórski
In this section we present preliminary material from functional analysis which will be used in subsequent chapters. We assume that the reader is familiar with the theory of linear normed spaces and, therefore, we restrict ourselves to recall the main results on nonlinear functional analysis we need. Most of the results are stated without proofs, since they are standard and can be found in many books and surveys, including [24,69,91,162].
On new updated concept for delay differential equations with piecewise Caputo fractional-order derivative
Published in Waves in Random and Complex Media, 2023
Khursheed Jamal Ansari, Fatima Ilyas, Kamal Shah, Aziz Khan, Thabet Abdeljawad
Short memory fractional-order differential equations play significant roles in the description of many real-world problems. Therefore, in this work, we have considered a class of delay fractional differential equations under the concept of piecewise derivative in the Caputo sense. We have expanded some results about the uniqueness, existence, and stability analysis for our considered problem. The results were obtained by utilizing the fixed point concept and nonlinear functional analysis tools. Sufficient circumstances have been set up to ensure the existence of at least one solution and its uniqueness to the suggested problem. Nonlinear tools of analysis were also used to determine its stability. We can observe that these derivatives can better describe an abrupt behavioral change in the dynamics of various processes of reals problems. As a result, we concluded that this sort of calculus has now attracted researchers more. More research into how to deal with piecewise fractional-order differential boundary value problems will be conducted in the future.
Qualitative analysis and numerical simulation of fractal-fractional COVID-19 epidemic model with real data from Pakistan
Published in Waves in Random and Complex Media, 2022
Rahat Zarin, Amir Khan, Ramashis Banerjee
The fractional-order system has been established with dimensional consistency by taking the corresponding arbitrary-order parameter on all dimension quantities given in the proposed problem. The theory of Fixed Point used in the analyzed manuscript supports the conditions about the non-integer-order system for FFP sense to have existence and unique solution. So, the key outcomes of the current research work pointed out the infectious system chosen into consideration in the format of arbitrary-order ordinary differential equations in the sense of FFP. Incorporating the fractal-fractional approach, we analyzed the COVID-19 model. It has been observed that the fractal-fractional derivative is the best tool to predict the global dynamics of the epidemic model. The existence, as well as uniqueness of the solution of the model, are carried out by using the Banach and Leray-Schauder theorems. U-H stability using nonlinear functional analysis has been used for the proposed mathematical model of COVID-19. It is observed from the sensitivity analysis that the μ and Λ have a high impact on basic reproduction number . Also, the fractional and fractal parameters have a significant impact on the dynamics of the epidemic model. In the future, one can extend our work by using harmonic mean and convex-type incidence rates. To find the optimal solution to our model one can also use optimal control theory.
Tykhonov well-posedness of a mixed variational problem
Published in Optimization, 2022
Dong-ling Cai, Mircea Sofonea, Yi-bin Xiao
The rest of this paper is structured as follows. In Section 2 we list the assumptions on the data and present our main results in the study of Problems and , Theorems 2.2, 2.4, 2.6, 2.8. Besides their novelty and their mathematical interest, these results are important since they provide tools useful in the variational and numerical analysis of various contact problems with unilateral constraints. To provide an example, in Section 3 we present two mathematical models which describe the contact between an elastic body and a rigid foundation covered by a layer of soft material. We list the assumptions on the data and derive their variational formulation which is in the form of Problems and , respectively. Finally, in Section 4 we apply our abstract results in the study of these problems and provide the corresponding mechanical interpretations. In this way we illustrate the cross fertilization between models and applications, on one hand, and the nonlinear functional analysis, on the other hand.