doi: 10.3934/jimo.2018081

Predicting non-life insurer's insolvency using non-kernel fuzzy quadratic surface support vector machines

1. 

School of Business Administration and Collaborative Innovation Center of Financial Security, Southwestern University of Finance and Economics, Chengdu 611130, China

2. 

School of Insurance and Collaborative Innovation Center of Financial Security, Southwestern University of Finance and Economics, Chengdu 611130, China

3. 

Department of Finance, University of North Carolina at Charlotte, Charlotte, NC, 28223, USA

4. 

School of Business Administration, Southwestern University of Finance and Economics, Chengdu 611130, China

Received  November 2017 Revised  January 2018 Published  June 2018

Fund Project: Tian's research has been supported by the Chinese National Science Foundation #11401485 and # 71331004. Yang's research has been supported by the Sichuan soft science research project #2017ZR0294

Due to the serious consequence caused by insurers' insolvency, how to accurately predict insolvency becomes a very important issue in this area. Many methods have been developed to do this task by using some firm-level financial information. In this paper, we propose a new approach which incorporates several macroeconomic factors in the model and applies feature selection to eliminate the bad effect of some unrelated variables. In this way, we can obtain a more comprehensive and accurate model. More importantly, our method is based on the state-of-the-art non-kernel fuzzy quadratic surface support vector machine (FQSSVM) model which not only performs superiorly in prediction, but also becomes very applicable to the users. Finally, we conduct some numerical experiments based on the real data of non-lifer insurers from USA to show the predictive power and efficiency of our proposed method compared with other benchmark methods. Specifically, in a reasonable computational time, FQSSVM has the most accurate prediction rate and least Type Ⅰ and Type Ⅱ errors.

Citation: Ye Tian, Wei Yang, Gene Lai, Menghan Zhao. Predicting non-life insurer's insolvency using non-kernel fuzzy quadratic surface support vector machines. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018081
References:
[1]

M. Al-Smadi, Credit Risk, Macroeconomic and Bank Specific Factors, VDM Verlag Dr. M$ü$ller, 2011.

[2]

E. BaranoffT. Sager and T. Shively, A semiparametric stochastic spline model as a managerial tool for potential insolvency, Journal of Risk and Insurance, 67 (2000), 369-396. doi: 10.2307/253834.

[3]

A. Best, Best's insolvency study-property/casualty insurers, Best's Review-Property/Casualty Insurance Edition, (1991), 16-23.

[4]

J. Carson and R. Hoyt, Life insurer financial distress: Classification models and empirical evidence, Journal of Risk and Insurance, 62 (1995), 764-775. doi: 10.2307/253595.

[5]

J. Cheng and M. Weiss, The fole of rbc, hurricane exposure, bond portfolio duration, and macroeconomic and industry-wide factors in property-liability insolvency prediction, Journal of Risk and Insurance, 79 (2012), 723-750.

[6]

H. Chew and C. Lim, On regularisation parameter transformation of support vector machines, Journal of Industrial and Management Optimization, 5 (2009), 403-415. doi: 10.3934/jimo.2009.5.403.

[7]

S. ChoJ. Kim and J. Bae, An integrative model with subject weight based on neural network learning for bankruptcy prediction, Expert Systems with Applications, 36 (2009), 403-410. doi: 10.1016/j.eswa.2007.09.060.

[8]

J. CumminsM. Grace and R. Phillips, Regulatory solvency prediction in property-liability insurance: Risk-based capital, audit ratios, and cash flow simulation, Journal of Risk and Insurance, 66 (1999), 417-458. doi: 10.2307/253555.

[9]

U. DellepianeM. MarcantonioE. Laghi and S. Renzi, Bankruptcy prediction using support vector machines and feature selection during the recent financial crisis, International Journal of Economics and Finance, 7 (2015), 182-194. doi: 10.5539/ijef.v7n8p182.

[10]

A. DimitrasR. SlowinskiR. Susmaga and C. Zopounidis, Business failure using rough set, European Journal of Operational Research, 114 (1998), 263-280.

[11]

P. Du Jardin, Predicting bankruptcy using neural networks and other classification methods: The influence of variable selection techniques on model accuracy, Neurocomputing, 73 (2010), 2047-2060. doi: 10.1016/j.neucom.2009.11.034.

[12]

M. GraceS. Harrington and R. Klein, Risk-based captial and solvency screening in property-liability insurance: Hypothesis and empirical tests, Journal of Risk and Insurance, 65 (1998), 213-243.

[13]

M. Grant and S. Boyd, Cvx: Matlab Software for Disciplined Programming, version 1.2, Technical report, http://cvxr.com/cvx 2010.

[14]

S. Hsiao and T. Whang, A study of financial insolvency prediction model for life insurers, Expert Systems with Applications, 36 (2009), 6100-6107. doi: 10.1016/j.eswa.2008.07.024.

[15]

M. Kim and D. Kang, Ensemble with neural networks for bankruptcy prediction, Expert Systems with Applications, 37 (2010), 3373-3379. doi: 10.1016/j.eswa.2009.10.012.

[16]

G. LanckrietN. CristianiniP. BartlettL. El Ghaoui and M. Jordan, Learning the kernel matrix with semi-definite programming, Journal of Machine Learning Research, 5 (2004), 27-72.

[17]

S. Lee and J. Urrutia, Analysis and prediction of insolvency in the property-liability insurance industry: A comparison of logit and hazard models, Journal of Risk and Insurance, 63 (1996), 121-130. doi: 10.2307/253520.

[18]

J. LuoS.-C. FangY. Bai and Z. Deng, Fuzzy quadratic surface support vector machine based on fisher discriminant analysis, Journal of Industrial and Management Optimization, 12 (2016), 357-373. doi: 10.3934/jimo.2016.12.357.

[19]

J. MariaS. Sanch and B. Carlos, Prediction of insolvency in non-life insurance companies using support vector machines, genetic algorithms and simulated annealing, Fuzzy Economic Review, 9 (2004), 79-94.

[20]

J. Min and Y. Lee, Bankrupty prediction using support vector machine with optimal choice of kernel function parameters, Expert Systems with Applications, 28 (2005), 603-614.

[21]

S. MinJ. Lee and I. Han, Hybrid genetic algorithms and support vector machines for bankruptcy prediction, Expert Systems with Applications, 31 (2006), 652-660. doi: 10.1016/j.eswa.2005.09.070.

[22]

S. Pottier and D. Sommer, Empirical evidence on the value of group-level financial information in insurer solvency surveillance, Risk Management and Insurance Review, 14 (2011), 73-88. doi: 10.1111/j.1540-6296.2011.01195.x.

[23]

S. SanchoD. MarioJ. MariaP. Fernando and B. Carlos, Feature selection methods involving support vector machines for prediction of insolvency in non-life insurance companies, Intelligent Systems in Accouting, Finance and Management, 12 (2004), 261-281.

[24]

K. Schittkowski, Optimal parameter selection in support vector machines, Journal of Industrial and Management Optimization, 1 (2005), 465-476. doi: 10.3934/jimo.2005.1.465.

[25]

B. Scholkopf and A. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT Press, 2002.

[26]

K. ShinT. Lee and H. Kim, An application of support vector machines in bankruptcy prediction model, Expert Systems and Applications, 28 (2005), 127-135. doi: 10.1016/j.eswa.2004.08.009.

[27]

N. Siddiqi, Credit Risk Scorecards: Developing and Implementing Intelligent Credit Scoring, John Wiley & Sons, 2015. doi: 10.1002/9781119201731.

[28]

Y. TianM. SunZ. DengJ. Luo and Y. Li, A new fuzzy set and non-kernel svm approach for mislabeled binary classification with applications, IEEE Transactions on Fuzzy Systems, 25 (2017), 1536-1545.

[29]

V. Vapnik, Statistical Learning Theory, John Wiley & Sons, Inc., New York, 1998.

[30]

C. WuG. TzengY. Goo and W. Fang, A real-valued genetic algorithm to optimize the parameters of support vector machine for predicting bankruptcy, Expert Systems with Applications, 32 (2007), 397-408. doi: 10.1016/j.eswa.2005.12.008.

[31]

C. XieC. Luo and X. Yu, Financial distress prediction based on svm and mda methods: The case of chinese listed companies, Quality & Quantity, 45 (2011), 671-686. doi: 10.1007/s11135-010-9376-y.

[32]

Z. YangW. You and G. Ji, Using partial least squares and support vector machines for bankruptcy prediction, Expert Systems with Applications, 38 (2011), 8336-8342. doi: 10.1016/j.eswa.2011.01.021.

[33]

L. Zhang and N. Nielson, Solvency analysis and prediction in property casualty insurance: Incorporating economic and market predictors, Journal of Risk and Insurance, 82 (2015), 97-124.

show all references

References:
[1]

M. Al-Smadi, Credit Risk, Macroeconomic and Bank Specific Factors, VDM Verlag Dr. M$ü$ller, 2011.

[2]

E. BaranoffT. Sager and T. Shively, A semiparametric stochastic spline model as a managerial tool for potential insolvency, Journal of Risk and Insurance, 67 (2000), 369-396. doi: 10.2307/253834.

[3]

A. Best, Best's insolvency study-property/casualty insurers, Best's Review-Property/Casualty Insurance Edition, (1991), 16-23.

[4]

J. Carson and R. Hoyt, Life insurer financial distress: Classification models and empirical evidence, Journal of Risk and Insurance, 62 (1995), 764-775. doi: 10.2307/253595.

[5]

J. Cheng and M. Weiss, The fole of rbc, hurricane exposure, bond portfolio duration, and macroeconomic and industry-wide factors in property-liability insolvency prediction, Journal of Risk and Insurance, 79 (2012), 723-750.

[6]

H. Chew and C. Lim, On regularisation parameter transformation of support vector machines, Journal of Industrial and Management Optimization, 5 (2009), 403-415. doi: 10.3934/jimo.2009.5.403.

[7]

S. ChoJ. Kim and J. Bae, An integrative model with subject weight based on neural network learning for bankruptcy prediction, Expert Systems with Applications, 36 (2009), 403-410. doi: 10.1016/j.eswa.2007.09.060.

[8]

J. CumminsM. Grace and R. Phillips, Regulatory solvency prediction in property-liability insurance: Risk-based capital, audit ratios, and cash flow simulation, Journal of Risk and Insurance, 66 (1999), 417-458. doi: 10.2307/253555.

[9]

U. DellepianeM. MarcantonioE. Laghi and S. Renzi, Bankruptcy prediction using support vector machines and feature selection during the recent financial crisis, International Journal of Economics and Finance, 7 (2015), 182-194. doi: 10.5539/ijef.v7n8p182.

[10]

A. DimitrasR. SlowinskiR. Susmaga and C. Zopounidis, Business failure using rough set, European Journal of Operational Research, 114 (1998), 263-280.

[11]

P. Du Jardin, Predicting bankruptcy using neural networks and other classification methods: The influence of variable selection techniques on model accuracy, Neurocomputing, 73 (2010), 2047-2060. doi: 10.1016/j.neucom.2009.11.034.

[12]

M. GraceS. Harrington and R. Klein, Risk-based captial and solvency screening in property-liability insurance: Hypothesis and empirical tests, Journal of Risk and Insurance, 65 (1998), 213-243.

[13]

M. Grant and S. Boyd, Cvx: Matlab Software for Disciplined Programming, version 1.2, Technical report, http://cvxr.com/cvx 2010.

[14]

S. Hsiao and T. Whang, A study of financial insolvency prediction model for life insurers, Expert Systems with Applications, 36 (2009), 6100-6107. doi: 10.1016/j.eswa.2008.07.024.

[15]

M. Kim and D. Kang, Ensemble with neural networks for bankruptcy prediction, Expert Systems with Applications, 37 (2010), 3373-3379. doi: 10.1016/j.eswa.2009.10.012.

[16]

G. LanckrietN. CristianiniP. BartlettL. El Ghaoui and M. Jordan, Learning the kernel matrix with semi-definite programming, Journal of Machine Learning Research, 5 (2004), 27-72.

[17]

S. Lee and J. Urrutia, Analysis and prediction of insolvency in the property-liability insurance industry: A comparison of logit and hazard models, Journal of Risk and Insurance, 63 (1996), 121-130. doi: 10.2307/253520.

[18]

J. LuoS.-C. FangY. Bai and Z. Deng, Fuzzy quadratic surface support vector machine based on fisher discriminant analysis, Journal of Industrial and Management Optimization, 12 (2016), 357-373. doi: 10.3934/jimo.2016.12.357.

[19]

J. MariaS. Sanch and B. Carlos, Prediction of insolvency in non-life insurance companies using support vector machines, genetic algorithms and simulated annealing, Fuzzy Economic Review, 9 (2004), 79-94.

[20]

J. Min and Y. Lee, Bankrupty prediction using support vector machine with optimal choice of kernel function parameters, Expert Systems with Applications, 28 (2005), 603-614.

[21]

S. MinJ. Lee and I. Han, Hybrid genetic algorithms and support vector machines for bankruptcy prediction, Expert Systems with Applications, 31 (2006), 652-660. doi: 10.1016/j.eswa.2005.09.070.

[22]

S. Pottier and D. Sommer, Empirical evidence on the value of group-level financial information in insurer solvency surveillance, Risk Management and Insurance Review, 14 (2011), 73-88. doi: 10.1111/j.1540-6296.2011.01195.x.

[23]

S. SanchoD. MarioJ. MariaP. Fernando and B. Carlos, Feature selection methods involving support vector machines for prediction of insolvency in non-life insurance companies, Intelligent Systems in Accouting, Finance and Management, 12 (2004), 261-281.

[24]

K. Schittkowski, Optimal parameter selection in support vector machines, Journal of Industrial and Management Optimization, 1 (2005), 465-476. doi: 10.3934/jimo.2005.1.465.

[25]

B. Scholkopf and A. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT Press, 2002.

[26]

K. ShinT. Lee and H. Kim, An application of support vector machines in bankruptcy prediction model, Expert Systems and Applications, 28 (2005), 127-135. doi: 10.1016/j.eswa.2004.08.009.

[27]

N. Siddiqi, Credit Risk Scorecards: Developing and Implementing Intelligent Credit Scoring, John Wiley & Sons, 2015. doi: 10.1002/9781119201731.

[28]

Y. TianM. SunZ. DengJ. Luo and Y. Li, A new fuzzy set and non-kernel svm approach for mislabeled binary classification with applications, IEEE Transactions on Fuzzy Systems, 25 (2017), 1536-1545.

[29]

V. Vapnik, Statistical Learning Theory, John Wiley & Sons, Inc., New York, 1998.

[30]

C. WuG. TzengY. Goo and W. Fang, A real-valued genetic algorithm to optimize the parameters of support vector machine for predicting bankruptcy, Expert Systems with Applications, 32 (2007), 397-408. doi: 10.1016/j.eswa.2005.12.008.

[31]

C. XieC. Luo and X. Yu, Financial distress prediction based on svm and mda methods: The case of chinese listed companies, Quality & Quantity, 45 (2011), 671-686. doi: 10.1007/s11135-010-9376-y.

[32]

Z. YangW. You and G. Ji, Using partial least squares and support vector machines for bankruptcy prediction, Expert Systems with Applications, 38 (2011), 8336-8342. doi: 10.1016/j.eswa.2011.01.021.

[33]

L. Zhang and N. Nielson, Solvency analysis and prediction in property casualty insurance: Incorporating economic and market predictors, Journal of Risk and Insurance, 82 (2015), 97-124.

Figure 1.  ROC curves for one test in case C1
Figure 2.  ROC curves for one test in case C2
Figure 3.  ROC curves for one test in case C3
Table 1.  The firm-level explanatory variables for non-life insurers
Index Figure Index Ratio
F1 Surplus R1 Net Premium / Surplus
F2 Net Technical Reserves R2 Tech. Res. / Net Premium
F3 Total Other Liabilities R3 Tech. Res. / Surplus
F4 Total Liabilities R4 Liq. Assets / Tech. Res.+ Oth. Liabs
F5 Total Investments R5 Combined Ratio
F6 Total Other Assets R6 Expense Ratio
F7 Total Assets R7 Loss Ratio
F8 Gross Premium Written R8 Investment Yield
F9 Net Premium Written R9 Pre-Tax Profitability
F10 Net Premium Earned R10 Liq. Assets/Net Tech. Res.
F11 Underwriting Expenses R11 Liq. Ast+Debts ced Co/Net Tech. Res.+Oth Liabs
F12 Underwriting Result R12 Profit Bef. Tax / Net Prem. Written
F13 Net Investment Income R13 Gross Premium/Surplus
F14 Profit Before Tax R14 Change in Surplus
F15 Syndicate Profit R15 Change in Technical Reserves
F16 Profit After Tax R16 Change in Net Premiums Written
F17 Profit After Names Expenses
Index Figure Index Ratio
F1 Surplus R1 Net Premium / Surplus
F2 Net Technical Reserves R2 Tech. Res. / Net Premium
F3 Total Other Liabilities R3 Tech. Res. / Surplus
F4 Total Liabilities R4 Liq. Assets / Tech. Res.+ Oth. Liabs
F5 Total Investments R5 Combined Ratio
F6 Total Other Assets R6 Expense Ratio
F7 Total Assets R7 Loss Ratio
F8 Gross Premium Written R8 Investment Yield
F9 Net Premium Written R9 Pre-Tax Profitability
F10 Net Premium Earned R10 Liq. Assets/Net Tech. Res.
F11 Underwriting Expenses R11 Liq. Ast+Debts ced Co/Net Tech. Res.+Oth Liabs
F12 Underwriting Result R12 Profit Bef. Tax / Net Prem. Written
F13 Net Investment Income R13 Gross Premium/Surplus
F14 Profit Before Tax R14 Change in Surplus
F15 Syndicate Profit R15 Change in Technical Reserves
F16 Profit After Tax R16 Change in Net Premiums Written
F17 Profit After Names Expenses
Table 2.  A confusion matrix
Observation Total
Good Bad
Prediction Good Correct goods (true negative) Type Ⅱ error (false negative) Goods predicted
Prediction Bad Type Ⅰ error (false positive) Correct bads (true positive) Bads predicted
Total Goods observed Bads observed Sample size
Observation Total
Good Bad
Prediction Good Correct goods (true negative) Type Ⅱ error (false negative) Goods predicted
Prediction Bad Type Ⅰ error (false positive) Correct bads (true positive) Bads predicted
Total Goods observed Bads observed Sample size
Table 3.  Overall accuracy results for different methods in various cases
Case Methods
ANN SVM MDA LRA SLR FQSSVM
C1 0.816 0.861 0.828 0.831 0.833 0.865
C2 0.837 0.881 0.862 0.864 0.850 0.888
C3 0.862 0.903 0.840 0.848 0.851 0.915
Case Methods
ANN SVM MDA LRA SLR FQSSVM
C1 0.816 0.861 0.828 0.831 0.833 0.865
C2 0.837 0.881 0.862 0.864 0.850 0.888
C3 0.862 0.903 0.840 0.848 0.851 0.915
Table 4.  Type Ⅰ errors for different methods in various cases
Case Type Ⅰ error
ANN SVM MDA LRA SLR FQSSVM
C1 0.185 0.141 0.173 0.170 0.168 0.137
C2 0.165 0.121 0.139 0.137 0.151 0.115
C3 0.139 0.099 0.161 0.154 0.151 0.088
Case Type Ⅰ error
ANN SVM MDA LRA SLR FQSSVM
C1 0.185 0.141 0.173 0.170 0.168 0.137
C2 0.165 0.121 0.139 0.137 0.151 0.115
C3 0.139 0.099 0.161 0.154 0.151 0.088
Table 5.  Type Ⅱ errors for different methods in various cases
Case Type Ⅱ error
ANN SVM MDA LRA SLR FQSSVM
C1 0.174 0.117 0.159 0.155 0.153 0.108
C2 0.135 0.094 0.128 0.119 0.136 0.079
C3 0.127 0.068 0.146 0.130 0.128 0.047
Case Type Ⅱ error
ANN SVM MDA LRA SLR FQSSVM
C1 0.174 0.117 0.159 0.155 0.153 0.108
C2 0.135 0.094 0.128 0.119 0.136 0.079
C3 0.127 0.068 0.146 0.130 0.128 0.047
Table 6.  AUC results for different methods in different comparisons
Case Methods
ANN SVM MDA LRA SLR FQSSVM
C1 0.832 0.861 0.839 0.835 0.829 0.856
C2 0.834 0.873 0.841 0.853 0.862 0.887
C3 0.840 0.882 0.844 0.851 0.856 0.892
Case Methods
ANN SVM MDA LRA SLR FQSSVM
C1 0.832 0.861 0.839 0.835 0.829 0.856
C2 0.834 0.873 0.841 0.853 0.862 0.887
C3 0.840 0.882 0.844 0.851 0.856 0.892
Table 7.  Computational times (in seconds) for different methods in different comparisons
Case Methods
ANN SVM MDA LRA SLR FQSSVM
C1 201.6 347.9 37.17 40.05 213.8 110.3
C2 214.3 336.2 36.91 39.28 195.1 98.65
C3 196.8 329.5 36.85 39.44 187.6 91.41
Case Methods
ANN SVM MDA LRA SLR FQSSVM
C1 201.6 347.9 37.17 40.05 213.8 110.3
C2 214.3 336.2 36.91 39.28 195.1 98.65
C3 196.8 329.5 36.85 39.44 187.6 91.41
Table 8.  Main pros and cons of different methods
Method Pros Cons
ANN handle nonlinear structure adaptability to environments resistance to noise pattern recognition black-box character difficult to interpret hard to analyze results parameters choice
SVM handle nonlinear structure distribution-free good performance kernel function choice parameters choice computational times
MDA easy to implement computational times can't handle nonlinear structure strong assumption on data
LRA easy to implement computational times can't handle nonlinear structure strong assumption on data
SLR easy to implement a proper subset of variables can't handle nonlinear structure strong assumption on data computational times
FQSSVM handle nonlinear structure no need for kernel function distribution-free good performance good generalization dimension expansion
Method Pros Cons
ANN handle nonlinear structure adaptability to environments resistance to noise pattern recognition black-box character difficult to interpret hard to analyze results parameters choice
SVM handle nonlinear structure distribution-free good performance kernel function choice parameters choice computational times
MDA easy to implement computational times can't handle nonlinear structure strong assumption on data
LRA easy to implement computational times can't handle nonlinear structure strong assumption on data
SLR easy to implement a proper subset of variables can't handle nonlinear structure strong assumption on data computational times
FQSSVM handle nonlinear structure no need for kernel function distribution-free good performance good generalization dimension expansion
[1]

François Bolley, Arnaud Guillin, Xinyu Wang. Non ultracontractive heat kernel bounds by Lyapunov conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 857-870. doi: 10.3934/dcds.2015.35.857

[2]

Jianguo Dai, Wenxue Huang, Yuanyi Pan. A category-based probabilistic approach to feature selection. Big Data & Information Analytics, 2017, 2 (5) : 1-8. doi: 10.3934/bdia.2017020

[3]

Shengfan Zhou, Linshan Wang. Kernel sections for damped non-autonomous wave equations with critical exponent. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 399-412. doi: 10.3934/dcds.2003.9.399

[4]

Toyohiko Aiki, Joost Hulshof, Nobuyuki Kenmochi, Adrian Muntean. Analysis of non-equilibrium evolution problems: Selected topics in material and life sciences. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : i-iii. doi: 10.3934/dcdss.2014.7.1i

[5]

Juan Pablo Aparicio, Juan Carlos Corley, Jorge Eduardo Rabinovich. Life history traits of Sirex Noctilio F. (Hymenoptera: Siricidae) can explain outbreaks independently of environmental factors. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1265-1279. doi: 10.3934/mbe.2013.10.1265

[6]

Shengfan Zhou, Caidi Zhao, Yejuan Wang. Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1259-1277. doi: 10.3934/dcds.2008.21.1259

[7]

Shengfan Zhou, Jinwu Huang, Xiaoying Han. Compact kernel sections for dissipative non-autonomous Zakharov equation on infinite lattices. Communications on Pure & Applied Analysis, 2010, 9 (1) : 193-210. doi: 10.3934/cpaa.2010.9.193

[8]

Xiang Li, Zhixiang Li. Kernel sections and (almost) periodic solutions of a non-autonomous parabolic PDE with a discrete state-dependent delay. Communications on Pure & Applied Analysis, 2011, 10 (2) : 687-700. doi: 10.3934/cpaa.2011.10.687

[9]

Muhammad Bilal Riaz, Naseer Ahmad Asif, Abdon Atangana, Muhammad Imran Asjad. Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernel. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 645-664. doi: 10.3934/dcdss.2019041

[10]

Mohamed A. Tawhid, Kevin B. Dsouza. Hybrid binary dragonfly enhanced particle swarm optimization algorithm for solving feature selection problems. Mathematical Foundations of Computing, 2018, 1 (2) : 181-200. doi: 10.3934/mfc.2018009

[11]

Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025

[12]

Lars Diening, Michael Růžička. An existence result for non-Newtonian fluids in non-regular domains. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 255-268. doi: 10.3934/dcdss.2010.3.255

[13]

Alberto Bressan, Truyen Nguyen. Non-existence and non-uniqueness for multidimensional sticky particle systems. Kinetic & Related Models, 2014, 7 (2) : 205-218. doi: 10.3934/krm.2014.7.205

[14]

Paul Glendinning. Non-smooth pitchfork bifurcations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 457-464. doi: 10.3934/dcdsb.2004.4.457

[15]

Bernard Dacorogna, Giovanni Pisante, Ana Margarida Ribeiro. On non quasiconvex problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 961-983. doi: 10.3934/dcds.2005.13.961

[16]

Yves Coudène, Barbara Schapira. Counterexamples in non-positive curvature. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1095-1106. doi: 10.3934/dcds.2011.30.1095

[17]

A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195

[18]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703

[19]

Gilles Carbou, Pierre Fabrie, Olivier Guès. On the Ferromagnetism equations in the non static case. Communications on Pure & Applied Analysis, 2004, 3 (3) : 367-393. doi: 10.3934/cpaa.2004.3.367

[20]

Maria Alessandra Ragusa. Parabolic systems with non continuous coefficients. Conference Publications, 2003, 2003 (Special) : 727-733. doi: 10.3934/proc.2003.2003.727

2017 Impact Factor: 0.994

Article outline

Figures and Tables

[Back to Top]