{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T02:32:32Z","timestamp":1773455552800,"version":"3.50.1"},"reference-count":55,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,4,12]],"date-time":"2019-04-12T00:00:00Z","timestamp":1555027200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper discusses the applications of numerical inversion of the Laplace transform method based on the Bernstein operational matrix to find the solution to a class of fractional differential equations. By the use of Laplace transform, fractional differential equations are firstly converted to system of algebraic equations then the numerical inverse of a Laplace transform is adopted to find the unknown function in the equation by expanding it in a Bernstein series. The advantages and computational implications of the proposed technique are discussed and verified in some numerical examples by comparing the results with some existing methods. We have also combined our technique to the standard Laplace Adomian decomposition method for solving nonlinear fractional order differential equations. The method is given with error estimation and convergence criterion that exclude the validity of our method.<\/jats:p>","DOI":"10.3390\/sym11040530","type":"journal-article","created":{"date-parts":[[2019,4,12]],"date-time":"2019-04-12T12:55:04Z","timestamp":1555073704000},"page":"530","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Numerical Inverse Laplace Transform for Solving a Class of Fractional Differential Equations"],"prefix":"10.3390","volume":"11","author":[{"given":"Dimple","family":"Rani","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal-148106, Punjab, India"}]},{"given":"Vinod","family":"Mishra","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal-148106, Punjab, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7504-0424","authenticated-orcid":false,"given":"Carlo","family":"Cattani","sequence":"additional","affiliation":[{"name":"Engineering School (DEIM), University of Tuscia, 01100 Viterbo, Italy"},{"name":"Ton Duc Thang University, HCMC 700000, Vietnam"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,12]]},"reference":[{"key":"ref_1","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_2","unstructured":"Podlubny, I. (1999). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations to Methods of Their Solution and Some of Their Applications, Academic Press."},{"key":"ref_3","unstructured":"Debnath, L., and Bhatta, D. (2007). Integral Transforms and their Applications, Chapman and Hall\/CRC. [2nd ed.]."},{"key":"ref_4","unstructured":"Miller, K.S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc."},{"key":"ref_5","unstructured":"Oldham, K.B., and Spanier, J. (1974). The Fractional Calculus, Academic Press."},{"key":"ref_6","unstructured":"Xiao-Jun, Y. (2018). General Fractional Derivatives: Theory, Methods and Applications, CRC Press."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1046","DOI":"10.1016\/j.camwa.2011.03.032","article-title":"Numerical solution of fractional differential equations using the generalized block pulse operational matrix","volume":"62","author":"Li","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1326","DOI":"10.1016\/j.camwa.2009.07.006","article-title":"A new operational matrix for solving fractional-order differential equations","volume":"59","author":"Saadatmandi","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1135","DOI":"10.1016\/j.camwa.2011.04.014","article-title":"A tau approach for solution of the space fractional diffusion equation","volume":"62","author":"Saadatmandi","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4931","DOI":"10.1016\/j.apm.2011.12.031","article-title":"A new Jacobi operational matrix: An application for solving fractional differential equations","volume":"36","author":"Doha","year":"2012","journal-title":"Appl. Math. Model."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5498","DOI":"10.1016\/j.apm.2012.10.026","article-title":"Fractional-order Legendre functions for solving fractional-order differential equations","volume":"37","author":"Kazem","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1016\/j.apm.2015.06.014","article-title":"The Muntz-Legendre Tau method for fractional differential equations","volume":"40","author":"Mokhtary","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6038","DOI":"10.1016\/j.apm.2014.04.064","article-title":"Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations","volume":"38","author":"Keshavarz","year":"2014","journal-title":"Appl. Math. Model."},{"key":"ref_14","first-page":"1","article-title":"Numerical approach based on fractional-order Lagrange polynomials for solving a class of fractional differential equations","volume":"1","author":"Sabermahani","year":"2017","journal-title":"Comput. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"859","DOI":"10.12732\/ijpam.v106i3.12","article-title":"Numerical solutions for linear and non-linear fractional differential equations","volume":"106","author":"Albadarneh","year":"2016","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Garrappa, R. (2018). Numerical solution of fractional differential equations: A survey and a software tutorial. Mathematics, 6.","DOI":"10.3390\/math6020016"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1515\/zna-2010-1106","article-title":"The solution of the linear fractional partial differential equations using the homotopy analysis method","volume":"65a","author":"Dehghan","year":"2010","journal-title":"Z. Naturforsch."},{"key":"ref_18","first-page":"140","article-title":"On the numerical solution of fractional partial differential equations","volume":"17","author":"Vanani","year":"2012","journal-title":"Math. Comput. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"5233","DOI":"10.1016\/j.apm.2012.10.045","article-title":"Numerical solutions to initial and boundary value problems for linear fractional partial differential equations","volume":"37","author":"Rehman","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_20","first-page":"211","article-title":"Solution of linear fractional partial differential equations based on the operator matrix of fractional Bernstein polynomials and error correction","volume":"14","author":"Li","year":"2018","journal-title":"Inter. J. Innov. Comput. Inf. Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"9312","DOI":"10.1002\/mma.5341","article-title":"Fundamental solutions of the general fractional-order diffusion equations","volume":"41","author":"Gao","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/j.cam.2017.10.007","article-title":"A new computational approach for solving nonlinear local fractional PDEs","volume":"339","author":"Gao","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_23","first-page":"31","article-title":"Generalized special functions in the description of fractional diffusive equations","volume":"10","author":"Cesarano","year":"2019","journal-title":"Commun. Appl. Ind. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1016\/j.cam.2007.03.007","article-title":"Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions","volume":"214","author":"Babolian","year":"2008","journal-title":"J. Comput. Appl. Math"},{"key":"ref_25","first-page":"1","article-title":"A direct method to solve integral and integro-differential equations of convolution type by using improved operational matrix","volume":"2012","author":"Maleknejad","year":"2012","journal-title":"Inter. J. Syst. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1145\/78928.78932","article-title":"Algorithm 682 Talbot\u2019s method for the Laplace inversion problem","volume":"16","author":"Murli","year":"1990","journal-title":"ACM Trans. Math. Softw."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.camwa.2004.11.017","article-title":"Algebraic inversion of the Laplace transform","volume":"50","author":"Massouros","year":"2005","journal-title":"Comput. Math. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1415","DOI":"10.1016\/j.camwa.2004.08.003","article-title":"An accurate numerical inversion of Laplace transforms based on the location of their poles","volume":"48","author":"Lee","year":"2004","journal-title":"Comput. Math. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.sbspro.2010.01.023","article-title":"Real inversion formulas and numerical experiments of the Laplace transform by using the theory of reproducing kernels","volume":"2","author":"Matsuura","year":"2010","journal-title":"Procedia Soc. Behav. Sci."},{"key":"ref_30","first-page":"186","article-title":"Numerical inversion of Laplace transform via wavelet in ordinary differential equations","volume":"2","author":"Hsiao","year":"2014","journal-title":"Comput. Methods Diff. Equ."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/0010-4655(95)00026-C","article-title":"On comparison of spline regularization with exponential sampling method for Laplace transform inversion","volume":"88","author":"Iqbal","year":"1995","journal-title":"Comput. Phys.Commun."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.cam.2005.11.017","article-title":"Computation of the inverse Laplace transform based on a collocation method which uses only real values","volume":"198","author":"Cuomo","year":"2007","journal-title":"J. Comput. Appl. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1145\/321439.321446","article-title":"Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform","volume":"15","author":"Dubner","year":"1968","journal-title":"J. Association Comput. Mach."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1093\/comjnl\/17.4.371","article-title":"Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate\u2019s method","volume":"17","author":"Durbin","year":"1974","journal-title":"Comput. J."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0021-9991(79)90025-1","article-title":"Numerical inversion of Laplace transform: A survey and comparison of methods","volume":"33","author":"Davis","year":"1979","journal-title":"J. Comput. Phys."},{"key":"ref_36","unstructured":"Cohen, A.M. (2007). Numerical methods for Laplace transform inversion, Springer."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1527","DOI":"10.1016\/j.aml.2011.03.039","article-title":"Application of Laguerre matrix polynomials to the numerical inversion of Laplace transforms of matrix functions","volume":"24","author":"Sastre","year":"2011","journal-title":"Appl. Math. Lett."},{"key":"ref_38","first-page":"182","article-title":"Numerical method for inverse Laplace transform with Haar Wavelet operational matrix","volume":"8","author":"Aznam","year":"2012","journal-title":"Malays. J. Fund. Appl. Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/0016-0032(77)90029-1","article-title":"Walsh operational matrices for fractional calculus and their application to distributed parameter systems","volume":"503","author":"Chen","year":"1977","journal-title":"J. Frankl. Inst."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1109\/81.903196","article-title":"Numerical inversion of Laplace transform using Haar wavlet operational matrices","volume":"48","author":"Wu","year":"2001","journal-title":"IEEE Trans. Circuit Syst.-I: Fundam. Theory Appl."},{"key":"ref_41","first-page":"172","article-title":"Numerical solution of nonlinear Volterra integral equations of the first kind with convolution kernel","volume":"4","author":"Shamloo","year":"2014","journal-title":"World Appl. Program."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1049\/ip-cta:19970702","article-title":"Haar wavlet method for solving lumped and distributed-parameter systems","volume":"144","author":"Chen","year":"1997","journal-title":"IEEE Control Theory Appl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1016\/j.cam.2006.05.002","article-title":"Solutions of differential equations in a Bernstein polynomial basis","volume":"205","author":"Bhatti","year":"2007","journal-title":"J. Comput. Appl. Math."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1363","DOI":"10.1016\/j.mcm.2011.10.015","article-title":"A Bernstein operational matrix approach for solving a system of high order linear Volterra-Fredholm integro-differential equations","volume":"55","author":"Maleknejad","year":"2012","journal-title":"Math. Comput. Model."},{"key":"ref_45","first-page":"2427","article-title":"The Bernstein operational matrix of integration","volume":"3","author":"Singh","year":"2009","journal-title":"Appl. Math. Sci."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1016\/j.ejc.2010.11.004","article-title":"Uniform approximation and Bernstein polynomial with coefficients in the unit interval","volume":"32","author":"Quain","year":"2011","journal-title":"Eur. J. Comb."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1379","DOI":"10.2298\/FIL1804379B","article-title":"Bernstein polynomials method and its error analysis for solving nonlinear problems in the calculus of variations:convergnece analysis via residual function","volume":"32","author":"Bataineh","year":"2018","journal-title":"Filomat"},{"key":"ref_48","first-page":"334","article-title":"Solving multi-term orders fractional differential equations by operational matrices of BPs with convergence analysis","volume":"65","author":"Rostamy","year":"2013","journal-title":"Roman. Rep. Phys."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jksus.2015.11.004","article-title":"Solution of fractional-order differential equations based on the operational matrices of new fractional Bernstein functions","volume":"29","author":"Alshbool","year":"2017","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"9231","DOI":"10.1002\/mma.5188","article-title":"Numerical inversion of Laplace transform based on Bernstein operational matrix","volume":"41","author":"Rani","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1155\/S1110757X01000183","article-title":"A Laplace decomposition algorithm applied to a class of nonlinear differential equation","volume":"1","author":"Khuri","year":"2001","journal-title":"J. Appl. Math."},{"key":"ref_52","first-page":"369","article-title":"Bernestein polynomials and operational methods","volume":"8","author":"Dattoli","year":"2006","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.apnum.2017.06.015","article-title":"A numerical method for solving some model problems arising in science and convergence analysis based on residual function","volume":"121","author":"Kurkcu","year":"2017","journal-title":"Appl. Numer. Math."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2012\/638026","article-title":"The use of cubic splines in the numerical solution of fractional differential equations","volume":"2012","author":"Zahra","year":"2012","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.cam.2006.07.015","article-title":"Numerical approach to differential equations of fractional order","volume":"207","author":"Momani","year":"2007","journal-title":"J. Comput. Appl. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/4\/530\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:45:01Z","timestamp":1760186701000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/4\/530"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,12]]},"references-count":55,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,4]]}},"alternative-id":["sym11040530"],"URL":"https:\/\/doi.org\/10.3390\/sym11040530","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,4,12]]}}}