In this work, some class of the fractional differential equations under fractional boundary conditions with the Katugampola derivative is considered. By proving the Lyapunov-type inequality, there are deduced the conditions for existence, and non-existence of the solutions to the considered boundary problem. Moreover, we present some examples to demonstrate the effectiveness and applications of the new results.
Citation: Barbara Łupińska, Ewa Schmeidel. Analysis of some Katugampola fractional differential equations with fractional boundary conditions[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 7269-7279. doi: 10.3934/mbe.2021359
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In this work, some class of the fractional differential equations under fractional boundary conditions with the Katugampola derivative is considered. By proving the Lyapunov-type inequality, there are deduced the conditions for existence, and non-existence of the solutions to the considered boundary problem. Moreover, we present some examples to demonstrate the effectiveness and applications of the new results.
Landslides are a common hazard on Earth [1]. In 2003 alone, the total reported deaths by landslides were about 18,200 throughout the world [2]. Landslides may affect small areas but often lead to severe financial losses and human casualties [3]. Landslides have significantly affected people's lives, livestock, infrastructures, life-lines, housing and agricultural lands throughout the world [4]. Like other parts of the world, Pakistan is also faced with landslides hazards, especially in the mountainous parts of the country (Table 1). One of such landslide prone region of the country is the Pakistani Himalayan region [5,6]. This region includes one of the worst landslide affected areas of Pakistan, the Murree hills. Murree is positioned on a lateral spur of the sub-Himalayan Mountains [7]. Murree hills receive highest average annual rainfall (1640 mm) during monsoon season i.e., from July to August [8]. Therefore, landslides mostly occur during this season. Murree hills are composed of rocks of fragile nature, inter-bedded with soft calcareous shale [9] and are therefore, prone to landslides [10,11]. Besides fragility of the natural settings, human mismanagements including irrational land use and deforestation have also paved the way for landslides in the area. Analysis of risk perception of disasters and natural hazards is imperative for devising policies for preparedness and mitigation actions [12].
| Year | No. of lives lost | Location |
| November 3, 2017 | 08 | Bajaur Agency (FATA), KP |
| January 4, 2010 | 20 | Attabad, Gilgit Baltistan |
| March, 2007 | 80 | District Dir, KP |
| January, 2007 | 20 | District Kotli (Kashmir) |
| September, 2006 | 04 | Murree Hills, KP |
| July, 2006 | 29 | Ghaeel Village (Kalam area), KP |
| May, 2003 | 12 | Ronala Village (Kohistan), KP |
| July, 2001 | 16 | Karachi, Hyderabad, Sukkur, Sindh |
| July, 2001 | 15 | Chitta Katha, Kaghan Valley, KP |
DownLoad: CSVPrevious studies conducted about landslides in Pakistan were focused on landslide hazards and policy response in Murree [13], impacts of landslides on housing and socio-economic characteristics in Murree [7], landslides triggered by the 8 October 2005 Kashmir earthquake [14], evolution of earthquake-triggered landslides in the Kashmir [6], causes and extent of landslides in Murree hills [15] and landslide causes and damages in Kashmir area [16]. To our knowledge, there has not been a single attempt made in Pakistan to empirically assess the landslide risk perceptions of the households living in the landslide prone regions of Pakistan. Based on the above, we made an attempt to determine the risk perceptions of the households in the landslide prone Murree hill station of Pakistan.
Source: [17]
Disasters are considered as sociological processes by researchers [18]. Social conditions of the people can affect the vulnerability factors and may thus turn a hazard in to a disaster [19]. Social vulnerability or socio-economic vulnerability is defined as the inability of the people, organizations and societies to combat the impacts of natural disasters [18,19,20]. Social vulnerability to natural disasters is usually determined based on individual characteristics of the people including socio-economic status, gender, race, age, employment, occupation, residential property, infrastructure and lifelines, education and family structure [20,21,22]. Earlier, qualitative assessments were preferred for measuring social vulnerability to natural disasters [21,23]. Social vulnerability attracted less attention by researchers due to difficulties faced in its quantification [23]. Researchers then tried to quantify social vulnerability to natural disasters using proxy indices and indicators [19,20,21]. Since then social vulnerability indices have been used by researchers to find vulnerability to natural hazards [23,24,25,26], coastal hazards [27], earthquake losses [28], landslides [19] and floods [29].
Risk perception studies originated with the studies of Gilbert White about adaptations to floods in United States during 1940s. Two decades later, risk perceptions were used to understand the views of public about nuclear technologies. Analysis of risk relies on measurement of probabilities and consequence [30,31]. Therefore, risk can be quantified through objective measurement of probabilities and consequences [32]. To be responsive to a natural hazard, an individual's perception of its risk is very important [33]. An individual's perception of risk is affected by his knowledge, personnel beliefs, experiences of previous events as well as environmental circumstances [34]. The perceived risk tells us about an individual's attitude, cognition and vulnerability [35,36]. Psychometric and cultural theories are the only two approaches being followed by the researchers for studying risk. As the names suggest, psychometric approach is related to studies in the field of psychology and cultural theory is related to the fields of anthropology and sociology. Questionnaires and factor-analytic techniques are employed to study risk communication, gender, race and demographics in psychometric approach [36]. Supporters of the cultural theory on the other hand believe that risk perceptions and risk acceptance have their origin in the social and cultural values in the society. Both these approaches faced severe criticism in the research community due to qualitative nature and operationalization in their measurements. Despite severe criticism these approaches have been widely used for studies of risk perception worldwide.
Murree is located in the sub-Himalayan Mountains with geographical position extending from 33°52' to 33°59' North and 73°24' to 73°31' East (Figure 1). It is one of the severe slide-affected area of Pakistan. Mountains of Murree reach an altitude of about 5000 to 7500 feet above sea level [7]. Murree, being located on fragile rocks of the Himalayas, faces frequent landslides. During monsoon season, the landslides become more frequent due to high precipitation received over Murree Mountains. Geologists are of the view that Murree is built on a lateral spur of the Himalayas. The study area has temperate climate. Summers are cool and winters are cold. June is the hottest month (with mean maximum and mean minimum temperatures of 80°F and 56°F, respectively). January and February are the coldest months (with mean maximum and mean minimum temperatures of 43.4°F and 31.1°F, respectively) [7]. This area receives highest average annual rainfall (66 inches) and more number of rainy days (85 per year) than any other area in Pakistan. Snowfall begins at the end of December and lasts up to the end of February and covers the ground with about 5 to 6 feet of snow. As soon as March begins, the snow starts to melt but still remains in freezing condition at some peaks on northern aspects.
Lives and properties are often at risk due to fragile nature of the rocks and high rainfall. Besides these natural conditions, deforestation, accessibility, growing population and unplanned urban development have paved the way for frequent landslides in the area. People living in Murree are therefore frequently faced with disrupted roads, broken communication lines, damaged houses and damages to scarce agricultural lands. Electricity and water supply are also affected by recurrent landslides in the area. Livestock and humans are also lost due to severe damages caused by landslides in the study area.
Murree has a total population of about 21,371 [37]. We determined the sample households with formula for sampling devised by the analysis introduced by Yamane T [38]. The formula suggested a sample of 200 households with 93% confidence level. To equally represent socio-economic and cultural differences among the respondents and their perceptions about landslide risk, the study area was divided in to (ⅰ) Inner city (ⅱ) Urban fringe and (ⅲ) rural fringe. The inner city includes Kashmir point and Station area. The people here are mostly engaged in retail activities. The Urban fringe consists of Kashmiri Mohalla, Chitta More, Abbasi Mohalla, Dhobi Ghat and Dhok Jabar Topa. The people belonging to this site are engaged in agricultural and retail activities. The rural fringe looks like a semi-urbanized area and presents the rural settings. It includes the areas of Dhok Jabar, Bari Nakkar, Hill Dholu, Dhak, Bangan, Chawana, Choora, Mohra, Maula Dohongi, Batnara, Ihata Noor Khan and Murree Brewery. In the rural fringe, the main occupation of the people is agricultural activities.
The data were collected from May to August 2017. The respondents were selected for survey through simple random sampling from the three sites. Primary data were collected from the respondents through field survey. Questionnaire was used for primary data collection and contained information regarding the socio-economic and demographic characteristics of the respondents and their perceptions about the indicators of risk perception. Household heads were interviewed during survey. In absence of household head, the elder family member was interviewed.
Both natural environment and socio-economic conditions of the people are responsible for causing landslides and influence perception of risk from landslide [19]. Therefore, literature review was also performed to find out the variables affecting landslide risk perceptions. These include age, income, level of education, past experience and location (Table 2). The dependent variable used was dichotomous variable i.e., "perception of landslide risk (PRCPLSRSK)". The PRCPLSRSK was given a value of 1 if the household perceived that landslide would occur within the next coming five years, otherwise 0.
| S.No. | Indicator with direction of influence | Unit for measurement | Justification | Source |
| 1 | Age (+) | Years (Continuous) | The more the age of the respondents, the more their perception of landslides based on experiences | [22,39,40] |
| 2 | Income (+) | Rupees/year (Continuous) | The more the income of households, the more they may be concerned with their safety against landslides | [3,22,39,41,42] |
| 3 | Education level (+) | People with or above higher secondary school education (Continuous) | The more the educated members in the family, the more they have knowledge about landslides | [22,39,40,41,43] |
| 4 | Location (+) | Yes/No (Dummy, 1 if house is located on or near slope, otherwise 0) | The more the people make houses on or near slopes, the more they perceive the risk | [31,40] |
| 5 | Past experience (+) | Yes/No (Dummy, 1 if past experience, 0 otherwise) | The more the past experiences with landslides, the more they are concerned about occurrences of landslides | [3,39,40] |
DownLoad: CSVThe primary data collected in the field survey were entered in to SPSS version 16 for analysis. Pearson correlation was first performed to check correlation among independent variables. Logistic regression analysis was performed using the enter method.
About 81 percent perceived landslide risk in the next five years. Average size of family was almost 10 in the area. Average age of the household heads was 46. About 41% of the surveyed population were literate. Average income per year per family was about 66,000 Pakistani rupees. Almost 32% households had built their houses at risky locations. Nearly 55% of the surveyed household reported to have experienced landslides in the past.
As discussed earlier, socio-economic characteristics of the households greatly affects their perceptions about landslides. Through literature review, we identified five variables that affect landslide risk perceptions including age, income, educational level, location and past experiences (Table 2). The nature of the dependent variable in this study was binary as explained above in the methodology section. Other models of regression like probit and discriminant analysis could be used, but logit model is easy to use. Therefore, we used logit regression model for this study. Pearson correlation was performed on the independent variables before application of regression model, to check if correlation exists among the independent variables. As no correlation was seen among independent variables, we applied logistic regression model using the enter method.
Of the five variables, three variables including, past experience, location and educational level were found to have significant positive effect on the dependent variable of PRCPLSRSK (Table 3). The household size, respondent's age and income per year showed positive association with landslide risk perception but were not significant. [44] in their study have also shown that household size has no significant relationship with perception of natural hazards. The variable age of the household head was also found to have no significant impact on risk perceptions about natural hazards [45,46]. Income was also found non-significant by [47] for risk perception of natural hazards.
| Independent variable | B | Wald | Significance | df | Odds ratio |
| Age | 0.002 | 0.013 | 0.908 | 1 | 1.002 |
| Income | 0.000 | 2.504 | 0.114 | 1 | 1.000 |
| Education level | 0.176 | 4.848 | 0.028* | 1 | 1.192 |
| Location | 0.862 | 3.660 | 0.056* | 1 | 2.368 |
| Past experience | 1.021 | 7.111 | 0.008** | 1 | 2.775 |
| Constant | −0.510 | 0.644 | 0.422 | 1 | 0.600 |
| Note: *, ** represent significance at 95% and 99% confidence level, respectively. −2log likelihood = 182.71, Cox & Snell R2 = 0.096, Nagelkerke R2 = 0.0151. | |||||
DownLoad: CSVThe variable past experience had positive and significant association with PRCPLSRSK. It was found that a unit increase in the past experience increases the perception of landslide risk by a factor of 2.78. This may be because the study area is one of the worst slide-affected areas of the Punjab province of Pakistan and the people of this area have experienced recurrent landslides due to fragile nature of the rocks and high rainfall in the area as mentioned earlier in this study. The findings of this study are in line with the studies of [3] and [48], who found that past experience is a good predictor for risk perceptions of natural hazards. Variable location was found to have significant positive correlation with the landslide risk perception of the households. This may be because the people who have built houses on or near slopes might have experienced more damages to their houses than the people who lived far away from such places.
Results showed that a unit increase in the location variable increases the odds of PRCPLSRSK by a factor of 2.37. Studies of [46] and [47] were also of the view that location near a risky place has significant impact on household's perception of natural hazards. The third significant variable of this study was educational level of the respondents. This may be because education plays a key role in making the people aware about their environmental settings and natural hazards. Results showed that a unit increase in education variable increases odds of landslides risk perception by 1.19 factor. Studies of [47] and [3] also showed that education of the households had the significance impact on their risk perception.
The perceptions of the local people are rarely incorporated in the preparedness and control programme for landslide risk management. This study analyzed the landslide risk perceptions of the households in Murree area of Punjab province of Pakistan. Disaster risk reduction of landslides depends upon perceptions of the local people of the affected area and is closely associated with their economic and social conditions. Results of the study showed that risk perceptions of the people in the area about landslides were affected by educational level, past experience and location. Education plays a key role in enhancing risk perception of the people about landslides, therefore, the government organizations and non-governmental organizations should try to provide training and awareness programmers to safeguard communities in case of landslide hazards. The people should be made aware of the need to make their dwellings at safe locations. This study therefore has policy implications for reducing landslide risk in the area.
The authors are thankful to the editors of the journal and two anonymous reviewers for constructive comments on the manuscript.
All authors declare no conflicts of interest in this paper.
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Figures(1)
Barbara Łupińska, Ewa Schmeidel. Analysis of some Katugampola fractional differential equations with fractional boundary conditions[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 7269-7279. doi: 10.3934/mbe.2021359
| Year | No. of lives lost | Location |
| November 3, 2017 | 08 | Bajaur Agency (FATA), KP |
| January 4, 2010 | 20 | Attabad, Gilgit Baltistan |
| March, 2007 | 80 | District Dir, KP |
| January, 2007 | 20 | District Kotli (Kashmir) |
| September, 2006 | 04 | Murree Hills, KP |
| July, 2006 | 29 | Ghaeel Village (Kalam area), KP |
| May, 2003 | 12 | Ronala Village (Kohistan), KP |
| July, 2001 | 16 | Karachi, Hyderabad, Sukkur, Sindh |
| July, 2001 | 15 | Chitta Katha, Kaghan Valley, KP |
DownLoad: CSV| S.No. | Indicator with direction of influence | Unit for measurement | Justification | Source |
| 1 | Age (+) | Years (Continuous) | The more the age of the respondents, the more their perception of landslides based on experiences | [22,39,40] |
| 2 | Income (+) | Rupees/year (Continuous) | The more the income of households, the more they may be concerned with their safety against landslides | [3,22,39,41,42] |
| 3 | Education level (+) | People with or above higher secondary school education (Continuous) | The more the educated members in the family, the more they have knowledge about landslides | [22,39,40,41,43] |
| 4 | Location (+) | Yes/No (Dummy, 1 if house is located on or near slope, otherwise 0) | The more the people make houses on or near slopes, the more they perceive the risk | [31,40] |
| 5 | Past experience (+) | Yes/No (Dummy, 1 if past experience, 0 otherwise) | The more the past experiences with landslides, the more they are concerned about occurrences of landslides | [3,39,40] |
DownLoad: CSV| Independent variable | B | Wald | Significance | df | Odds ratio |
| Age | 0.002 | 0.013 | 0.908 | 1 | 1.002 |
| Income | 0.000 | 2.504 | 0.114 | 1 | 1.000 |
| Education level | 0.176 | 4.848 | 0.028* | 1 | 1.192 |
| Location | 0.862 | 3.660 | 0.056* | 1 | 2.368 |
| Past experience | 1.021 | 7.111 | 0.008** | 1 | 2.775 |
| Constant | −0.510 | 0.644 | 0.422 | 1 | 0.600 |
| Note: *, ** represent significance at 95% and 99% confidence level, respectively. −2log likelihood = 182.71, Cox & Snell R2 = 0.096, Nagelkerke R2 = 0.0151. | |||||
DownLoad: CSV| Year | No. of lives lost | Location |
| November 3, 2017 | 08 | Bajaur Agency (FATA), KP |
| January 4, 2010 | 20 | Attabad, Gilgit Baltistan |
| March, 2007 | 80 | District Dir, KP |
| January, 2007 | 20 | District Kotli (Kashmir) |
| September, 2006 | 04 | Murree Hills, KP |
| July, 2006 | 29 | Ghaeel Village (Kalam area), KP |
| May, 2003 | 12 | Ronala Village (Kohistan), KP |
| July, 2001 | 16 | Karachi, Hyderabad, Sukkur, Sindh |
| July, 2001 | 15 | Chitta Katha, Kaghan Valley, KP |
| S.No. | Indicator with direction of influence | Unit for measurement | Justification | Source |
| 1 | Age (+) | Years (Continuous) | The more the age of the respondents, the more their perception of landslides based on experiences | [22,39,40] |
| 2 | Income (+) | Rupees/year (Continuous) | The more the income of households, the more they may be concerned with their safety against landslides | [3,22,39,41,42] |
| 3 | Education level (+) | People with or above higher secondary school education (Continuous) | The more the educated members in the family, the more they have knowledge about landslides | [22,39,40,41,43] |
| 4 | Location (+) | Yes/No (Dummy, 1 if house is located on or near slope, otherwise 0) | The more the people make houses on or near slopes, the more they perceive the risk | [31,40] |
| 5 | Past experience (+) | Yes/No (Dummy, 1 if past experience, 0 otherwise) | The more the past experiences with landslides, the more they are concerned about occurrences of landslides | [3,39,40] |
| Independent variable | B | Wald | Significance | df | Odds ratio |
| Age | 0.002 | 0.013 | 0.908 | 1 | 1.002 |
| Income | 0.000 | 2.504 | 0.114 | 1 | 1.000 |
| Education level | 0.176 | 4.848 | 0.028* | 1 | 1.192 |
| Location | 0.862 | 3.660 | 0.056* | 1 | 2.368 |
| Past experience | 1.021 | 7.111 | 0.008** | 1 | 2.775 |
| Constant | −0.510 | 0.644 | 0.422 | 1 | 0.600 |
| Note: *, ** represent significance at 95% and 99% confidence level, respectively. −2log likelihood = 182.71, Cox & Snell R2 = 0.096, Nagelkerke R2 = 0.0151. | |||||
