{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,23]],"date-time":"2024-09-23T04:20:12Z","timestamp":1727065212988},"reference-count":67,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,10,20]],"date-time":"2020-10-20T00:00:00Z","timestamp":1603152000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61911530132"],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002261","name":"Russian Foundation for Basic Research","doi-asserted-by":"publisher","award":["19-58-53011"],"id":[{"id":"10.13039\/501100002261","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"This paper studies the second kind linear Volterra integral equations (IEs) with a discontinuous kernel obtained from the load leveling and energy system problems. For solving this problem, we propose the homotopy perturbation method (HPM). We then discuss the convergence theorem and the error analysis of the formulation to validate the accuracy of the obtained solutions. In this study, the Controle et Estimation Stochastique des Arrondis de Calculs method (CESTAC) and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library are used to control the rounding error estimation. We also take advantage of the discrete stochastic arithmetic (DSA) to find the optimal iteration, optimal error and optimal approximation of the HPM. The comparative graphs between exact and approximate solutions show the accuracy and efficiency of the method.<\/jats:p>","DOI":"10.3390\/sym12101730","type":"journal-article","created":{"date-parts":[[2020,10,21]],"date-time":"2020-10-21T00:50:07Z","timestamp":1603241407000},"page":"1730","source":"Crossref","is-referenced-by-count":33,"title":["Error Estimation of the Homotopy Perturbation Method to Solve Second Kind Volterra Integral Equations with Piecewise Smooth Kernels: Application of the CADNA Library"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"http:\/\/orcid.org\/0000-0002-2307-0891","authenticated-orcid":false,"given":"Samad","family":"Noeiaghdam","sequence":"first","affiliation":[{"name":"Baikal School of BRICS, Irkutsk National Research Technical University, Irkutsk 664074, Russia"},{"name":"Department of Applied Mathematics and Programming, South Ural State University, Lenin Prospect 76, Chelyabinsk 454080, Russia"}]},{"ORCID":"http:\/\/orcid.org\/0000-0002-5032-0665","authenticated-orcid":false,"given":"Aliona","family":"Dreglea","sequence":"additional","affiliation":[{"name":"Baikal School of BRICS, Irkutsk National Research Technical University, Irkutsk 664074, Russia"},{"name":"Energy Systems Institute of Russian Academy of Science, Irkutsk 664033, Russia"}]},{"ORCID":"http:\/\/orcid.org\/0000-0002-1636-0559","authenticated-orcid":false,"given":"Jihuan","family":"He","sequence":"additional","affiliation":[{"name":"National Engineering Laboratory for Modern Silk, Soochow University, Suzhou 215021, China"}]},{"given":"Zakieh","family":"Avazzadeh","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Xi\u2019an Jiaotong-Liverpool University, Suzhou 215123, China"}]},{"given":"Muhammad","family":"Suleman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Comsats Institute of Information Technology, Islamabad 45550, Pakistan"}]},{"given":"Mohammad Ali","family":"Fariborzi Araghi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran 1955847881, Iran"}]},{"ORCID":"http:\/\/orcid.org\/0000-0002-3131-1325","authenticated-orcid":false,"given":"Denis N.","family":"Sidorov","sequence":"additional","affiliation":[{"name":"Baikal School of BRICS, Irkutsk National Research Technical University, Irkutsk 664074, Russia"},{"name":"Energy Systems Institute of Russian Academy of Science, Irkutsk 664033, Russia"}]},{"given":"Nikolai","family":"Sidorov","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and IT, Irkutsk State University, Irkutsk 664025, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,20]]},"reference":[{"key":"ref_1","first-page":"59","article-title":"Control of Accuracy on Taylor-Collocation Method for Load Leveling Problem. The Bulletin of Irkutsk State University","volume":"30","author":"Noeiaghdam","year":"2019","journal-title":"Ser. Math."},{"key":"ref_2","first-page":"3451","article-title":"A Dynamic Analysis of Energy Storage with Renewable and Diesel Generation using Volterra Equations","volume":"14","author":"Sidorov","year":"2019","journal-title":"IEEE Trans. Ind."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"01015","DOI":"10.1051\/e3sconf\/20186901015","article-title":"Volterra Models in Load Leveling Problem","volume":"69","author":"Sidorov","year":"2018","journal-title":"E3S Web Conf."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Fariborzi Araghi, M.A., and Noeiaghdam, S. (2015, January 9\u201311). Homotopy analysis transform method for solving generalized Abel\u2019s fuzzy integral equations of the first kind. Proceedings of the 4-th Iranian Joint Congress on Fuzzy and Intelligent Systems (CFIS), Zahedan, Iran.","DOI":"10.1109\/CFIS.2015.7391645"},{"key":"ref_5","first-page":"1","article-title":"Homotopy regularization method to solve the singular Volterra integral equations of the first kind","volume":"10","author":"Noeiaghdam","year":"2018","journal-title":"Jordan J. Math. Stat."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"7210","DOI":"10.1002\/mma.5827","article-title":"Exact traveling wave solutions for resonance nonlinear Schr\u00f6dinger equation with intermodal dispersions and the Kerr law nonlinearity","volume":"42","author":"Srivastava","year":"2019","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Sabir, Z., G\u00fcnerhan, H., and Guirao, J.L.G. (2020). On a new model based on third-order nonlinear multisingular functional differential equations. Math. Probl. Eng., 2020.","DOI":"10.1155\/2020\/1683961"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2050034","DOI":"10.1142\/S0217984920500347","article-title":"Some mixed trigonometric complex soliton solutions to the perturbed nonlinear Schr\u00f6dinger equation","volume":"34","author":"Gao","year":"2020","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1134\/S0001434612010105","article-title":"On small solutions of nonlinear equations with vector parameter in sectorial neighborhoods","volume":"91","author":"Sidorov","year":"2012","journal-title":"Math. Notes"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1080\/23324309.2020.1816551","article-title":"Nonlocal Effects to Neutron Diffusion Equation in a Nuclear Reactor","volume":"49","year":"2020","journal-title":"J. Comput. Theor. Transp."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/j.cam.2016.09.003","article-title":"Numeric solution of Volterra integral equations of the first kind with discontinuous kernels","volume":"313","author":"Muftahov","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1134\/S0012266113020079","article-title":"On Parametric Families of Solutions of Volterra Integral Equations of the First Kind with Piecewise Smooth Kernel","volume":"49","author":"Sidorov","year":"2013","journal-title":"Differ. Equ."},{"key":"ref_13","first-page":"87","article-title":"Control of accuracy on Taylor-collocation method to solve the weakly regular Volterra integral equations of the first kind by using the CESTAC method","volume":"19","author":"Noeiaghdam","year":"2020","journal-title":"Appl. Comput. Math."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Sidorov, D., Tynda, A., Muftahov, I., Dreglea, A., and Liu, F. (2020). Nonlinear Systems of Volterra Equations with Piecewise Smooth Kernels: Numerical Solution and Application for Power Systems Operation. Mathematics, 8.","DOI":"10.3390\/math8081257"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"34","DOI":"10.15352\/afa\/1399900192","article-title":"Classification of positive solutions of nonlinear system of Volterra integral equations","volume":"2","author":"Raffou","year":"2011","journal-title":"Ann. Funct. Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"54","DOI":"10.3103\/S1066369X13010064","article-title":"Solvability of system of integral Volterra equations of the first kind with piecewise continuous kernels","volume":"57","author":"Sidorov","year":"2013","journal-title":"Russ. Math. (Iz.VUZ)"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Goodrich, C.S. (2018). Perturbed Integral Operator Equations of Volterra Type with Applications to p-Laplacian Equations. Mediterr. J. Math., 15.","DOI":"10.1007\/s00009-018-1090-3"},{"key":"ref_18","first-page":"757","article-title":"Generalized Solution to the Volterra Equations with Piecewise Continuous Kernels","volume":"37","author":"Sidorov","year":"2014","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4791","DOI":"10.1016\/j.amc.2010.11.034","article-title":"Numerical solution of multiple nonlinear Volterra integral equations","volume":"217","author":"Belbas","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1217","DOI":"10.1134\/S0012266114090080","article-title":"Existence and blow-up of Kantorovich principal continuous solutions of nonlinear integral equations","volume":"50","author":"Sidorov","year":"2014","journal-title":"Differ. Equ."},{"key":"ref_21","unstructured":"Chua, L.O. (2015). Integral Dynamical Models: Singularities, Signals And Control, World Scientific Press."},{"key":"ref_22","unstructured":"Chua, L.O. (2020). Toward General Theory of Differential-Operator and Kinetic Models, World Scientific Press."},{"key":"ref_23","first-page":"778","article-title":"Sur la nature analytique des solutions des certaines equations aux derivees partielles du second ordre","volume":"137","author":"Bernstein","year":"1903","journal-title":"C. R. Acad. Sci. Paris"},{"key":"ref_24","first-page":"145","article-title":"Certain questions in non-linear functional analysis","volume":"11","author":"Lyusternik","year":"1956","journal-title":"Uspekhi Mat. Nauk"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/S0045-7825(99)00018-3","article-title":"Homotopy perturbation technique","volume":"178","author":"He","year":"1999","journal-title":"Comput. Meth. Appl. Mech. Engrg."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/S0020-7462(98)00085-7","article-title":"A coupling method of a homotopy technique and a perturbation technique for non-linear problems","volume":"35","author":"He","year":"2000","journal-title":"Internat. J. Non- Mech."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/S0096-3003(01)00312-5","article-title":"Homotopy perturbation method: A new non-linear analytical technique","volume":"135","author":"He","year":"2003","journal-title":"Appl. Math. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"e03060","DOI":"10.1016\/j.heliyon.2019.e03060","article-title":"An approximate analytical solution of the Allen-Cahn equation using homotopy perturbation method and homotopy analysis method","volume":"5","author":"Hussain","year":"2019","journal-title":"Heliyon"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"105889","DOI":"10.1016\/j.ijepes.2020.105889","article-title":"Maximum power extraction from fractional order doubly fed induction generator based wind turbines using homotopy singular perturbation method","volume":"119","author":"Abolvafaei","year":"2020","journal-title":"Int. J. Electr. Power Energy Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"109457","DOI":"10.1016\/j.chaos.2019.109457","article-title":"Homotopy perturbation method for studying dissipative nonplanar solitons in an electronegative complex plasma","volume":"130","author":"Kashkari","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1016\/j.jmaa.2016.11.031","article-title":"Approximate analytical solutions of nonlinear differential equations using the Least Squares Homotopy Perturbation Method","volume":"448","author":"Bota","year":"2017","journal-title":"J. OfMath. Anal. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3359","DOI":"10.1016\/j.asej.2017.04.010","article-title":"Homotopy perturbation method for the hypersingular integral equations of the first kind","volume":"9","author":"Eshkuvatov","year":"2018","journal-title":"Ain Shams Eng. J."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Javeed, S., Baleanu, D., Waheed, A., Shaukat Khan, M., and Affan, H. (2019). Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations. Mathematics, 7.","DOI":"10.3390\/math7010040"},{"key":"ref_34","unstructured":"Trenogin, V.A. (2007). Functional Analysis, Fizmatlit."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Sidorov, N., Loginov, B., Sinitsyn, A., and Falaleev, M. (2002). Lyapunov-Schmidt Methods in Nonlinear Analysis and Applications, Kluwer Academic Publisher.","DOI":"10.1007\/978-94-017-2122-6"},{"key":"ref_36","first-page":"145","article-title":"Potentiality, group symmetry and bifurcation in the theory of branching equation","volume":"3","author":"Trenogin","year":"1990","journal-title":"Differ. Integral Equ."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.55630\/sjc.2010.4.1-10","article-title":"Stochastic arithmetic, Theory and experiments","volume":"4","author":"Alt","year":"2010","journal-title":"Serdica J. Comput."},{"key":"ref_38","first-page":"2","article-title":"Dynamical control of computations using the Trapezoidal and Simpson\u2019s rules","volume":"4","author":"Chesneaux","year":"1998","journal-title":"J. Univers. Comput. Sci."},{"key":"ref_39","unstructured":"Chesneaux, J.M. (1992). Stochastic arithmetic properties. IMACS Comput. Appl. Math., 81\u201391."},{"key":"ref_40","unstructured":"Chesneaux, J.M. (1990). CADNA, an ADA Tool for Round-Off Error Analysis and for Numerical Debugging."},{"key":"ref_41","first-page":"1","article-title":"Z\u00e9ro math\u00e9matique et z\u00e9ro informatique, in: La Vie des Sciences","volume":"4","author":"Vignes","year":"1987","journal-title":"Comptes Rendus De L\u2019Acad\u00e9mie De Sci."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.matcom.2005.11.014","article-title":"Reliable computation of a multiple integral involved in the neutron star theory","volume":"71","author":"Rico","year":"2006","journal-title":"Math. Comput. Simul."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1023\/B:NUMA.0000040066.63826.46","article-title":"Computation of an infinite integral using Romberg\u2019s method","volume":"36","author":"Chesneaux","year":"2004","journal-title":"Numer. Algorithms"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1016\/j.cpc.2007.01.005","article-title":"Numerical \u2019health check\u2019 for scientific codes: The CADNA approach","volume":"176","author":"Scott","year":"2007","journal-title":"Comput. Phys. Commun."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.apnum.2003.12.021","article-title":"Dynamical control of converging sequences computation","volume":"50","year":"2004","journal-title":"Appl. Numer. Math."},{"key":"ref_46","first-page":"362","article-title":"A dynamical strategy for approximation methods","volume":"334","year":"2006","journal-title":"C. R. Acad. Sci. Paris-M\u00e9canique"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1016\/j.cpc.2008.02.003","article-title":"CADNA: A library for estimating round-off error propagation","volume":"178","author":"Chesneaux","year":"2008","journal-title":"Comput. Phys. Commun."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1925","DOI":"10.1016\/j.cpc.2010.07.006","article-title":"CADNA_C: A version of CADNA for use with C or C++ programs","volume":"181","author":"Lamotte","year":"2010","journal-title":"Comput. Phys. Commun."},{"key":"ref_49","first-page":"35","article-title":"High Performance Numerical Validation using Stochastic Arithmetic","volume":"21","author":"Eberhart","year":"2015","journal-title":"Reliab. Comput."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s11786-011-0103-4","article-title":"Stochastic arithmetic in multi precision","volume":"5","author":"Graillat","year":"2011","journal-title":"Math. Comput. Sci."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.entcs.2015.10.007","article-title":"Numerical Validation of Compensated Summation Algorithms with Stochastic Arithmetic","volume":"317","author":"Graillat","year":"2015","journal-title":"Electron. Notes Theor. Comput. Sci."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1023\/B:NUMA.0000049483.75679.ce","article-title":"Discrete Stochastic Arithmetic for Validating Results of Numerical Software","volume":"37","author":"Vignes","year":"2004","journal-title":"Spec. Issue Numer. Algorithms"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/0378-4754(93)90003-D","article-title":"A stochastic arithmetic for reliable scientific computation","volume":"35","author":"Vignes","year":"1993","journal-title":"Math. Comput. Simul."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"62","DOI":"10.5899\/2017\/jfsva-00383","article-title":"Finding optimal step of fuzzy Newton-Cotes integration rules by using the CESTAC method","volume":"2017","author":"Noeiaghdam","year":"2017","journal-title":"J. Fuzzy Set Valued Anal."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"112632","DOI":"10.1016\/j.cam.2019.112632","article-title":"Valid implementation of Sinc-collocation method to solve the fuzzy Fredholm integral equation","volume":"370","author":"Noeiaghdam","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_56","first-page":"1","article-title":"A Novel Approach to Find Optimal Parameter in the Homotopy-Regularization Method for Solving Integral Equations","volume":"14","author":"Noeiaghdam","year":"2020","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_57","first-page":"83","article-title":"Numerical solution of improper integrals with valid implementation","volume":"7","author":"Abbasbandy","year":"2002","journal-title":"Math. Comput. Appl."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.apnum.2004.11.007","article-title":"A stochastic scheme for solving definite integrals","volume":"55","author":"Abbasbandy","year":"2005","journal-title":"Appl. Numer. Math."},{"key":"ref_59","first-page":"1","article-title":"A Valid Scheme to Evaluate Fuzzy Definite Integrals by Applying the CADNA Library","volume":"6","author":"Noeiaghdam","year":"2017","journal-title":"Int. Fuzzy Syst. Appl."},{"key":"ref_60","first-page":"1","article-title":"Dynamical control of computations using the Gauss-Laguerre integration rule by applying the CADNA library","volume":"16","author":"Noeiaghdam","year":"2016","journal-title":"Adv. Appl. Math."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.apnum.2004.01.003","article-title":"The use of the stochastic arithmetic to estimate the value of interpolation polynomial with optimal degree","volume":"50","author":"Abbasbandy","year":"2004","journal-title":"Appl. Numer. Math."},{"key":"ref_62","first-page":"63","article-title":"Valid implementation of the Sinc-collocation method to solve linear integral equations by the CADNA library","volume":"7","author":"Noeiaghdam","year":"2019","journal-title":"J. Math. Model."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/s11075-018-0546-7","article-title":"Finding optimal convergence control parameter in the homotopy analysis method to solve integral equations based on the stochastic arithmetic","volume":"81","author":"Noeiaghdam","year":"2019","journal-title":"Numer. Algorithms"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1537","DOI":"10.1016\/j.apnum.2008.10.004","article-title":"Optimal iterate of the power and inverse iteration methods","volume":"59","author":"Toutounian","year":"2009","journal-title":"Appl. Numer. Math."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1016\/j.amc.2005.10.019","article-title":"Numerical accuracy of a certain class of iterative methods for solving linear system","volume":"176","author":"Toutounian","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_66","first-page":"117","article-title":"Dynamical Control of Newton\u2019s Method for Multiple Roots of Polynomials","volume":"21","author":"Graillat","year":"2016","journal-title":"Reliab. Comput."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"1640","DOI":"10.1177\/0003702817694181","article-title":"Discrete Spectrum Reconstruction Using Integral Approximation Algorithm","volume":"71","author":"Sizikov","year":"2017","journal-title":"Appl. Spectrosc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1730\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,16]],"date-time":"2024-08-16T05:02:45Z","timestamp":1723784565000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1730"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,20]]},"references-count":67,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2020,10]]}},"alternative-id":["sym12101730"],"URL":"https:\/\/doi.org\/10.3390\/sym12101730","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,10,20]]}}}