Positive and Negative Solutions of a Class of Fractional Schrödinger Equation
Issue:
Volume 7, Issue 6, December 2021
Pages:
85-91
Received:
22 November 2021
Accepted:
16 December 2021
Published:
29 December 2021
Abstract: In this paper, we study the existence of positive and negative solutions for a class of fractional Schrodinger equations. Firstly, we give the definition of fractional Laplace operator and the conditions satisfied by nonlinear terms. This paper introduces the previous progress in this field, and gives the definitions of space and energy functional and the positive and negative parts of function. Then we introduce the main results of this paper. Next, we give the embedding relationship between workspace and Lp space and give the definition of inner product and norm of space. In order to obtain the existence of positive and negative solutions of the equation, we give the definitions of functions u+, u− and functional weak solutions. This paper mainly uses mountain pass lemma to prove. Firstly, according to the embedding relationship of workspace and the condition of nonlinear term f, it is proved that functional I satisfies mountain road structure. Secondly, we need to prove that functional I satisfies the (Cc) condition, we first prove that the sequence un is bounded, then prove that UN has convergent subsequence by the definition of inner product and holder inequality. Therefore, we prove that functional I satisfies the (Cc) condition. Then, we define functional I± and its inner product form to verify that functional I± also has mountain path structure and satisfies (Cc) condition. Finally, taking u+ and u− as experimental functions respectively, it is verified that they are the solutions of functional I. It is obtained that both u+ and u− are the solutions of functional I. Therefore, we get the conclusion.
Abstract: In this paper, we study the existence of positive and negative solutions for a class of fractional Schrodinger equations. Firstly, we give the definition of fractional Laplace operator and the conditions satisfied by nonlinear terms. This paper introduces the previous progress in this field, and gives the definitions of space and energy functional ...
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Stochastic Modelling of Claim Frequency in the Ethiopian Motor Insurance Corporation: A Case Study of Hawassa District in Ethiopia
Mikiyas Gebresamuel Gebru
Issue:
Volume 7, Issue 6, December 2021
Pages:
92-103
Received:
18 February 2021
Accepted:
11 November 2021
Published:
5 February 2022
Abstract: The objective of this study is to model seasonal variations in claim intensities and to evaluate the dependency of covariates on claim rates. The data for this study are obtained from claimants registered during September 2009 to August 2011, both inclusive at the Ethiopian Insurance Corporation in Hawassa. We present a procedure for consistent estimation of the claim frequency for motor vehicles in the Ethiopian Insurance Corporation, Hawassa District. The seasonal variation is modeled with a non-homogeneous Poisson process with a time varying intensity function. Covariates of the policy holders, like gender and age, is corrected for in the average claim rate by Poisson regression in a GLM setting. An approximate maximum likelihood criterion is used when estimating the model parameters. Numerical simulations are carried out applying the numerical software Matlab. These simulations show the reliability of the estimator. The seasonal parameters are found to be ±25% and to be statistically significant. February has highest while August has lowest claim rate. Only age group 36 - 45 has significantly lower claim rate than age group 20 - 25. The rate is about one third. Lastly female is not found to have significantly lower claim rates than males, however, there are indications that might be slightly so.
Abstract: The objective of this study is to model seasonal variations in claim intensities and to evaluate the dependency of covariates on claim rates. The data for this study are obtained from claimants registered during September 2009 to August 2011, both inclusive at the Ethiopian Insurance Corporation in Hawassa. We present a procedure for consistent est...
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