Blow-up of solutions for a time fractional biharmonic equation with exponentional nonlinear memory

1.   Introduction

The latest compilation of the progress in sustainable development in Europe is found in the 2020 Eurostat report "Sustainable development in the European Union. Monitoring report on progress towards the SDGs in an EU context" (Eurostat, 2020a) where the overall goal is by 2030 to achieve the sustainable management and efficient use of natural resources (UNDP, 2020). Sustainability comprises three main pillars, i.e., economically viable decisions, environmentally sound decisions, and socially equitable decisions. The present study applies Eurostat data for five out of six main indicators discussed in the report for SDG 12 (Eurostat, 2020a) for the period 2007 to 2017 (Eurostat, 2020b, c, d, e, f). The 6^th main indicator, i.e., "Gross value added in the environmental goods and services sector" (Eurostat, 2020g) will not be treated here due to lacking data for several of the year coved. Hence, the 5 main indicators included in this study are:

●  Consumption of toxic chemicals (Eurostat, 2020b)

●  Resource productivity (Eurostat, 2020c)

●  Average CO₂ emissions from new passenger cars (Eurostat, 2020d)

●  Circular material use rate (Eurostat, 2020e)

●  Generation of waste excluding major mineral wastes (Eurostat, 2020f)

It is noted that these indicators virtually focus on the environmental part of sustainability and only indirectly with the economic part through the resource productivity, which is defined as the gross domestic product divided by domestic material consumption. Obviously, the financial aspects should not be neglected. Thus, Tseng et al. has studied financial aspects of sustainable supply chain (Tseng et al, 2018, 2019).

The selection of these main indicators for SDG 12 is nevertheless obvious. Hence, production of toxic chemicals, as well as the average CO₂ emissions from new passenger cars appear as important indicators in relation both to sustainable production and consumption. Further, the general production of waste parallel to an increase resource productivity and circular material use rate obviously work towards a higher degree of sustainability both in relation to production and to consumption. The interlink with other SDGs should be emphasized. SDG 13, Climate action, is obvious (CO₂) emission and SDG 11, Sustainable cities, and communities (Waste) and SDG 3, Good Health and Well-being (toxic chemicals) may here serve as illustrative examples.

In the Eurostat report (Eurostat, 2020a) the development in the single indicators is treated separately. However, to obtain a more comprehensive picture of the sustainable production and consumption the group of indicators (here 5) should be taken into account simultaneously. To do so, often an agammaegation of the single indicators into one overall indicator is done. However, although this leads to a complete linear ordering of the studied objects, as here, e.g., the EU member states, valuable information concerning the influence of the single indicators may be lost due to compensation effects (Munda, 2008). As such partial order methodology constitutes an advantageous decision support tool for authorities and regulators in their attempt to select focus areas and allocate resources as well as to avoid pitfalls in their work to improve the situation in order better to comply with the SDG 12 target.

The paper is structured by a methodology section followed by a result and discussion section that further is subdivided in two: a) an overall analysis to what extent the 27 EU member states (cf. Table 1) comply with the SDG 12 taken the 4 indicators Resource productivity, Average CO₂ emissions from new passenger cars, Circular material use rate and Generation of waste excluding major mineral wastes simultaneous into account for the years 2010, 2014 and 2017, respectively and b) an analysis of the consumption of toxic chemical for the years 2004 to 2019 at a European Union scale (excluding United Kingdom); the paper finalizes with some conclusions and outlook.

2.   Methods

2.1.   Indicators and Data

The indicators being included in the study is summarized and described in Table 2. The CCH indicator has 2 subcategories: non-Hazardous waste (nonHaz) and Hazardous waste (Haz), the latter further being subdivided in wastes hazardous to the human health (HazHea) and to the environment (HazEnv), respectively. In Table 3 the data for the 27 EU member states are given for the 4 indicators RPR, CO₂, CMUR and WAS, respectively for the years 2010, 2014 and 2017, respectively (Eurostat, 2020c, d, e, f). It should be noted that for the indicators CO₂ and WAS negatives are given, thus securing an identical orientation of the indicators, i.e., the higher the better, i.e., contribution positively to the sustainable production and consumption.

Table 4 shows the four indicator values for the European Union and for Greece (GRC) for the years 2010 to 2017 (Eurostat, 2020c, d, e, f). Again, the indicators CO₂ and WAS are given as negatives to obtain an identical orientation of the indicators, i.e., the higher the better. In Table 5 the applied data for the hazardous waste analyses is given. It should be emphasized that all data are given with as negatives as higher values reflect less sustainability. Thus, this will correspondingly be visualized in the Hasse diagram where the least sustainable years will appear in the lower part of the diagrams. The nonHaz has been calculated by subtracting the Haz from the total Haz + nonHaz. It should be noted that HazHea and Hazenv to some extent overlap and as such the sum does not equal Haz (Eurostat, 2020b).

It should be noted that the data applied in the present study provided by Eurostat do not look at specific products but summarizes overall figures for the European Union as such as well as for the single member states.

It should further be emphasized that in all cases the data are taken as reported by Eurostat(2020b, c, d, e, f), i.e., the data are accepted as true values. In other word any, not reported, possible data uncertainty nor data noise has not been taken into account in the calculations.

2.2.   Partial ordering—the basics

The basis for partial ordering is the relation among the objects to be ordered where the combined data are analyzed without additional presumptions or pretreatments. The only mathematical term in this context is "≤" (cf., e.g., Annoni et al., 2015; Bruggemann and Patil, 2011; Bruggemann and Voigt, 2008; Carlsen, 2018; Carlsen and Bruggemann, 2013, 2014, 2018; Newlin and Patil, 2010; Voigt et al., 2013). Thus, the "≤" relation is the basis for a comparison of objects and constitutes a graph, the so called Hasse diagram (see below). Two objects are connected with each other if and only if the relation x ≤ y holds. However, since each object is characterized by a series of indicators r_s the obvious questions is how x ≤ y should be understood. As a given object, x, is characterized by the a set of indicators r_s(x), s = 1, ..., m, it can be compared to another object y, characterized by an identical set of indicators r_s(y), if

It is obvious that Equation (1) is a rather strict requirement for having a comparison as at least one indicator value of object x must be lower (the remaining lower or at least equal) to those of object y. In more technical terms: Let X be the group of objects studied, i.e., X = {O1, O2, O3, ……, On}, then object Oy will be ranked higher than object Ox, i.e., Ox < Oy if at least one of the indicator values for Oy is higher than the corresponding indicator value for Ox and no indicator for Oy is lower than the corresponding indicator value for Ox. On the other hand, if r_s(Oy) > r_s(Ox) for some indicator s and r_t(Oy) < r_t(Ox) for some other indicator t, Oy and Ox will be called incomparable (notation: Oy ‖ Ox) due to the mathematical contradiction expressed by the conflicting indicator values. A set of comparable objects is called a chain, whereas a set of mutually incomparable objects is called an antichain. In cases where all indicator values for two objects, Oy and Ox, are equal, i.e., r_s(Oy) = r_s(Ox) for all s, the two objects will be considered as equivalent, i.e., Ox ~ Oy, which in ranking terms means that they will have the same rank.

2.3.   The Hasse diagram

The Equation (1) is the basis for the Hasse diagram technique (HDT) (Bruggemann and Carlsen, 2006a; Bruggemann and Patil, 2011; Bruggemann and Carlsen, 2006b). Hasse diagrams are visual representation of the partial order. In the Hasse diagram comparable objects are connected by a sequence of lines (Bruggemann and Carlsen, 2006a, b; Bruggemann and Patil, 2011; Brüggemann and Münzer, 1993; Brüggemann and Voigt, 1995; Bruggemann and Voigt, 2008). Thus, sets of comparable objects, i.e., fulfilling eqn. 1 are called chains that in the diagram are connected with lines, whereas sets of mutually incomparable objects, i.e., not fulfilling eqn. 1 are called antichains.

In the diagram the single objects are positioned in levels, typically arranged from low to high (bottom to top in the diagram). A general rule is that objects are located a high in the diagram as possible. Thus, isolated objects will be by default at the top level of the diagram. It is important to make sure that the orientation of the single indicators are identical, e.g., that high values correspond to "good" whereas low values correspond to "bad". In practice this is done by multiplying indicator values by −1 in case where high and low values correspond to "bad" and "good", respectively (cf. 2.5). In the present study the highest located object/country will be assigned rank 1 indicating the "best".

The module mHDCl7_1 of the PyHasse software (vide infra) was used for the basic partial ordering calculations and the associated construction of the Hasse diagrams.

2.4.   Sensitivity—indicator importance

The relative importance of the single indicators in play can be determined through a sensitivity analysis (Brüggemann et al., 2001). The basic idea is to construct partial ordered sets (posets) excluding the single indicators one at the time. Subsequently, the distances from these posets to the original poset are determined. The indicator, whose elimination from the original poset leads to the maximal distance to the original one, in other words causing the highest degree of changes in the Hasse diagram is most important for the structure of the original partial order. As the effect of elimination single indicators is studied, this kind of sensitivity analysis can be called "indicator-related sensitivity".

In a recent paper Bruggemann and Carlsen (2021) reported an attempt to verify the authenticity of the calculated sensitivity data, i.e., to verify that the values correspond to real, i.e., non-random effects. The calculated sensitivities, W_icalc, were normalized to the number of possible connections between the objects, n, i.e., n(n−1)/2. The single normalized sensitivities w_i = 2xW_icalc/n(n−1). Eventually the w_i data are compared to an estimated limit (Bruggemann and Carlsen, 2021). The W_icalc values were calculated by the sensitivity23_1 module of the PyHasse software (vide infra).

2.5.   Average ranking

Looking at the Hasse diagram, the level structure constitutes a first approximation to ordering. However, as all objects in a level automatically will be assigned identical orders such an ordering will obviously cause many tied orders. Obviously, it is desirable with a degree of tiedness as low as possible. Hence, ultimately a linear ordering of the single objects is desirable. However, when incomparable objects are included in the study, this is obviously not immediately obtainable. Partial order methodology provides a weak order, where tied orders are not excluded. This is obtained by calculating the average order of the single objects as, e.g., described by Bruggemann and Carlsen (2011) and Bruggemann and Annoni (2014).

The average rankings were calculated applying the LPOMext8_5 (Bruggemann and Carlsen, 2011) and BubleyDyer11_1 (Bubley and Dyer, 1999) modules of the PyHasse software (vide infra). The latter module further allows the calculation of the probability for a given object to be ranked higher than any incomparable object as well as the probabilities for the single objects to take a specific rank.

2.6.   Software

All partial order analyses were carried out using the PyHasse software (Bruggemann et al., 2014). PyHasse is programmed using the interpreter language Python (version 2.6). Today, the software package contains more than 100 specialized modules and is available upon request from the developer, Dr. R.Bruggemann (brg_home@web.de).

K-means clustering is based on a custom-made script in R applying the "cluster" and "fpc" libraries and run under R x64 2.15.3.

3.   Results and discussion

3.1.   Partial ordering and average ranking

To elucidate the development of responsible consumption and production within the European Union taking the four main indicators RPR, CO₂, CMUR and WAS simultaneous into account partial ordering appears as an attractive methodology. Hence, based on partial ordering methodology it is possible to elucidate the trend within the EU as well as within the single member states. Hasse diagrams constitute here an advantageous way to visualize the mutual relations between the member states (Figure 1).

A visual inspection of the three Hasse diagrams corresponding to the partial ordering of the 27 member states for the years 2010 (Figure 1A), 2014 (Figure 1B) and 2017 (Figure 1C) gives the first indication of the mutual ranking between the states. Further, the inspection shows a broadening of the diagrams although the number of incomparabilities vs comparabilities remains virtually constant by 264/61,296/55 and 298/53, respectively with an increasing number of states in the top level indicating an increasing number of states develops positively in relation to comply with the targets of SDG 12 (UNDP, 2020). In addition, a first indication of which states are in the top and in the bottom is obtained.

A deeper insight in the actual ranking of the 27 member states can obviously be obtained by deriving the average rankings (cf. sect. 2.4) that is a weak ordering of the 27 states taking all 4 indicators into account simultaneously. Due to the incomparabilities several strict linear orders (linear extensions) will comply with the partial ordering eventually leading to the average ranking being based on the probabilities for the single states to occupy different positions (ranks). In Table 6 the derived average ranks of the 27 EU member states for the years 2010, 2014 and 2017 are given corresponding to the Hasse diagrams (Figures 1A, B and C).

It is emphasized that the results summarized in Table 6 show only the mutual ranking of the 27 EU member states for each of the three years. Thus, comparing the single years does not give information concerning the development in the single countries. It should be noted that for 2010 the data for Croatia (HRV) are incomplete; thus, the average rank for HRV could not be obtained.

Table 6 clearly disclose that France (FRA), Malta (MLT) and Italy (ITA) for all three years takes the top 3 positions, thus being the three countries with the most responsible consumption and production within the Union based on the analysis simultaneously bringing the four main indicators RPR, CO₂, CMUR and WAS into play. On the other hand, it is also disclosed that Bulgaria (BRG) and Estonia (EST) for all three years are found at the bottom position among the European countries, thus with the least responsible consumption and production.

Further Table 6 elucidates that countries like Greece (GRC), Czechia (CZE) and Slovenia (SVN) within the European have significantly improved their responsible consumption and production moving from the 24^th to the 11^th place, 18^th to 9^th place and 20^th to 15^th to 8^th place, respectively, whereas pronounced declines are noted for Belgium (BEL), Lithuania (LTU), Luxembourg (LUX) and Poland (POL).

3.2.   Temporal development 2010–2017

To disclose the temporal develop of responsible consumption and production within the European Union as well as within single member states datasets the data for the European Union as well as for GRC as an exemplary country case have been analyzed (Table 4). Figure 2 displays the corresponding Hasse diagrams.

The diagram displayed in Figure 2A, is a priori of rather low information with only 2 levels with, in total 3 comparisons and 25 incomparisons, leaving very few possibilities for conclusions apart from the fact that apparently year 2012 was better than the years 2010 and 2011, which should be d expected if a general improvement prevailed and the 2016 is better than 2017, which contradicts a positive development. It should be emphasized that the ratio between the number of objects studied and the number of indicators may turn out as crucial (Sørensen et al., 2000). Thus, if the number of indicators relative to the number of objects studied is too high the number of incomparisons unambiguously will increase. On the other hand, the Hasse diagram for the development in Greece over the years 2010–2017 appears much more informative with 12 comparabilities and 16 incomparabilities, respectively. This is further substantiated by the corresponding average ranks. Here (Table 7) it is noted that in the case of the EU combined 2010 and 2011 are assigned the same rank as is 2013–2015, whereas in the case of GRC all years are assigned individual ranks. In the case of GRC it is further noted that the year 2010 appears as an isolated element (cf. Figure 2) meaning that in all relations to the other years an indicator conflict prevails. Analyzing the temporal development for GRC (cf. Table 4) it can be disclosed that year 2010 is an atypical year due to an unexpected high CMUR value.

The blurred picture for the ranking of the years 2010–2017 for the EU combined is further substantiated by looking at the actual probabilities for the single years to occupy a specific rank as derived from a large number of randomly generated linear extensions (here 309) (Bubley and Dyer, 1999) (Table 8), i.e., the best year appears to be 2012 with a probability of 0.389 and the worst year 2017 with a probability 0.276. However, it is virtually not possibly to draw any firm conclusions.

3.3.   Indicator importance

Obviously of interest is a knowledge of the influence of the single indicators on the average ranking (cf. sect. 2.3) as such information may serve as decision support for authorities and regulating body to focus on specific areas to improve the overall responsible consumption and production in the country. In the following we will focus on the most significant indicators for the EU combined and for three specific countries, i.e., FRA, GRC and BGR as examples of countries being overall ranked in the top, in the middle and in the bottom of the 27 member states, respectively. It should be noted that the indicator importance for the single countries may differ from the indicator importance derived for the combined EU. In Table 9 the calculated overall importance of the single indicators for the 27 EU member states as well as for the EU combined.

It is immediately seen that apparently WAS is the most important indicator when describing the responsible consumption and production for the European Union, the values in bold are significant, i.e., non-random numbers, the limiting values for a 4-indicator system being 0.205 (Bruggemann and Carlsen, 2021). Thus, normalized sensitivity values below 0.205 may be ascribed to randomness due to the calculation procedure (Bruggemann and Carlsen, 2021). However, although that the waste problem apparently on an overall evaluation appears as the major obstacle to achieve responsible consumption and production with the EU as such variations from country to country prevail.

Looking at the temporal development for FRA, GRC and BGR for the period 2010–2017 we find that the most important indicators are CO₂, CMUR and CO₂, respectively. In these three cases the WAS indicator apparently plays only an insignificant role—if at all.

3.4.   Temporal development of the consumption of toxic chemicals 2004–2019

As it by 2020 is a major goal to achieve an "environmentally sound management of chemicals and all wastes throughout their life cycle, in accordance with agreed international frameworks, and significantly reduce their release to air, water and soil in order to minimize their adverse impacts on human health and the environment" (Target 12.4) (UNDP, 2020) it is of interest to elucidate the temporal development of the consumption of toxic waste in the Union. The present study has retrieved data for the period 2004 to 2019, however on a European scale only (Table 5). In Figure 3 the Hasse diagram based on the 3 indicators nonHaz, HazHea and HazEnv and for 2 indicators HazHea and HazEnv, respectively, are shown based on the data in Table 5. The estimated average ranks for the single years are given in Table 10.

It is obvious (Table 10 and Figure 3A) that the year 2009 appears at the top rank due to rather low indicator values for all three indicators; especially the surprisingly low nonHaz indicator is probably the main reason that 2009 is found as being incomparable to any other years. On the other hand, rather high indicator values make the year 2007 the worst with rank 16. Also, the rather low rank of 2018 and 2019 should be noted as a result of an increasing nonHaz.

It is obvious (Table 10 and Figure 3A) that the year 2009 appears at the top rank due to rather low indicator values for all three indicators; especially the nonHaz indicator is surprisingly low probably the main reason that 2009 is found as being incomparable to any other years. On the other hand, rather high indicator values make the year 2007 the worst with rank 16. Also, the rather low rank of 2018 and 2019 should be noted as a result of an increasing nonHaz.

Based on the Hasse diagram shown in Figure 3A the importance of the single indicators (cf. sect. 2.3) was estimated. Unambiguously the nonHaz indicator was the most important. Eliminating the nonHaz from the group of included indicators resulted in the Hasse diagram shown in Figure 3B. The year 2007 is still found at the lowest rank (Table 10) due the high indicator values where the top is now occupied by year 2016 followed by 2009—again a visualization of the indicator values.

An alternative approach to visualizing the consumption of chemicals by hazardousness for the years 2004 to 2019 is by a cluster analysis. In Figure 4 the K-means clustering of the data given in Table 5 is visualized.

It is immediate noted that in both cases the year 2009 is found as an outlier away from any of the two cluster further substantiating the effect of the rather low indicator values. Further it is noted that the years virtually is divided into two groups including 2004–2010 and 2011–2019, excluding 2009, reflecting the general tendency to lower values for the 3 indicators (Table 5) seen for the latter years and thus indicating the overall route to an improved and sustainable consumption of chemicals.

4.   Conclusions and Outlook

Partial order methodology constitutes an advantageous decision support tool to analyze multi-criteria datasets as illustrated in the present paper focusing on responsible consumption and production within the European Union based on four main indicators: resource productivity, average CO₂ emissions from new passenger cars, circular material use rate and generation of waste excluding major mineral wastes. Further, a fifth indicator, consumption of chemicals by hazardousness, is treated separately. The ultimate goal is by 2030 to achieve the sustainable management and efficient use of natural resources.

Partial ordering has been applied to rank the 27 EU member states based on a simultaneous inclusion of all 4 indicators. The ranking gives a picture of the situation the 27 member states relative to each other. It has been shown that France, Malta and Italy apparently are the three countries with having the best responsible consumption and production whereas Bulgaria and Estonia are found in the other end of the spectrum.

Analyzing the indicator importance, it is disclosed that looking the EU as a whole the most important indicator is the generation of waste. This is in accordance with the fact that the waste problem constitutes a significant problem throughout Europe. An important part of the waste problem is associated with food waste that constitutes an illustrative Hence, an "equivalent to 1.3 billion tonnes worth around $1 trillion is estimated as the annual food waste that ends up rotting in the bins of consumers and retailers, or spoiling due to poor transportation and harvesting practices, which actually adds up to an estimated one third of all food produced" (UNDP, 2020). It is worthwhile in this connection to mention the link to SDG 2, Zero hunger, as "almost 1 billion people go undernourished and another 1 billion hungry" (UNDP, 2020). A major goal is by 2030 to "halve per capita global food waste at the retail and consumer levels and reduce food losses along production and supply chains, including post-harvest losses" and in more general terms "substantially reduce waste generation through prevention, reduction, recycling and reuse" (UNDP, 2020). Immediately, based on the current analyses significant work are still to be done to reach the goal by 2030.

It is emphasized that for the single countries the temporal development in their responsible consumption and production not necessarily is controlled by waste generation. Thus, France, Greece and Bulgaria serve as exemplary cases, and it is found that in the case of France and Bulgaria the most important indicator for the temporal development is the average CO₂ emission from new passenger cars, whereas in the case of Greece it apparently is the circular material use rate. Overall the CO₂ emission is decreasing in the EU during the period from 2007 with ca. 157.5 g CO₂/km to 2018 with 119, 6 g CO₂/km, although stagnant during 2015–2018, even with slight increase; the 2021 goal is 95 g CO₂/km (Eurostat, 2020d).

A special issue is the hazardous waste. Analysis of the waste according to hazardousness revealed that the amount of non-hazardous waste is the most important and unfortunately an increase in this type of waste is noted during the years 2004–2019, whereas a decrease in waste hazardous to the human health and to the environment is decreasing during the same period. Based on the here presented data the major goal, states by the UNDP (2020) by 2020 to have an "environmentally sound management of chemicals and all wastes throughout their life cycle, in accordance with agreed international frameworks, and significantly reduce their release to air, water and soil in order to minimize their adverse impacts on human health and the environment" by 2020 is by far not achieved.

The results of the present study can immediately be as decision support for future direct action in the different countries as authorities and regulators are provided with data that pinpoint focus areas for action.

Finally, it should be noted that the here presented analyses constitute and exemplary case applying available data for SDG 12. Similar analyses can immediately be performed for, e.g., the other SDGs to the extent where suitable and reliable data are available.

Conflict of interest

The author declares no conflicts of interest in this paper.