{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T12:34:56Z","timestamp":1780662896891,"version":"3.54.1"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,8,24]],"date-time":"2021-08-24T00:00:00Z","timestamp":1629763200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The present paper employs a numerical method based on the improved block\u2013pulse basis functions (IBPFs). This was mainly performed to resolve the Volterra\u2013Fredholm integral equations of the second kind. Those equations are often simplified into a linear system of algebraic equations through the use of IBPFs in addition to the operational matrix of integration. Typically, the classical alterations have enhanced the time taken by the computer program to solve the system of algebraic equations. The current modification works perfectly and has improved the efficiency over the regular block\u2013pulse basis functions (BPF). Additionally, the paper handles the uniqueness plus the convergence theorems of the solution. Numerical examples have been presented to illustrate the efficiency as well as the accuracy of the method. Furthermore, tables and graphs are used to show and confirm how the method is highly efficient.<\/jats:p>","DOI":"10.3390\/axioms10030200","type":"journal-article","created":{"date-parts":[[2021,8,24]],"date-time":"2021-08-24T22:12:22Z","timestamp":1629843142000},"page":"200","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Improved Block-Pulse Functions for Numerical Solution of Mixed Volterra-Fredholm Integral Equations"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1636-0559","authenticated-orcid":false,"given":"Ji-Huan","family":"He","sequence":"first","affiliation":[{"name":"School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China"},{"name":"School of Science, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China"},{"name":"National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou 215006, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mahmoud H.","family":"Taha","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11566, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mohamed A.","family":"Ramadan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Menoufia 12946, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Galal M.","family":"Moatimid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11566, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,24]]},"reference":[{"key":"ref_1","first-page":"488","article-title":"On mixed Volterra\u2013Fredholm type integral equations","volume":"17","author":"Pachpatta","year":"1986","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1080\/00207160601176905","article-title":"Numerical expansion methods for solving systems of linear integral equations using interpolation and quadrature rules","volume":"84","author":"Yusufoglu","year":"2017","journal-title":"Int. J. Comput. Math."},{"key":"ref_3","first-page":"325","article-title":"A Powerful Method for Obtaining Exact Solutions of Volterra Integral Equation\u2019s Types","volume":"16","author":"Ramadan","year":"2020","journal-title":"Glob. J. Pure Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1397","DOI":"10.1080\/0020716031000148214","article-title":"Numerical solution of linear Fredholm integral equations system by rationalized Haar functions method","volume":"80","author":"Maleknejad","year":"2003","journal-title":"Int. J. Comput. Math."},{"key":"ref_5","first-page":"317","article-title":"Solving linear integro-differential equations system by using rationalized Haar functions method","volume":"155","author":"Maleknejad","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_6","first-page":"1","article-title":"Application on differential transform method for some non-linear functions and for solving Volterra integral equations involving Fresnels integral","volume":"5","author":"Rizkalla","year":"2014","journal-title":"J. Fract. Calc. Appl."},{"key":"ref_7","first-page":"465","article-title":"Numerical solution of the system of Fredholm integro-differential equations by the Tau method","volume":"168","author":"Shahmorad","year":"2005","journal-title":"Appl. Math. Comput."},{"key":"ref_8","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. Methods Appl. Mech. Eng."},{"key":"ref_9","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":"Int. J. Non-Linear Mech."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1016\/j.chaos.2005.03.006","article-title":"Application of homotopy perturbation method to nonlinear wave equations","volume":"26","author":"He","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Wazwaz, A. (2011). Linear and Nonlinear Integral Equations: Methods and Applications, Springer.","DOI":"10.1007\/978-3-642-21449-3"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-020-03208-5","article-title":"Modified homotopy methods for generalized fractional perturbed Zakharov\u2013Kuznetsov equation in dusty plasma","volume":"2021","author":"Akinyemi","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1007\/s40819-021-01039-w","article-title":"Analytical Study of (3 + 1)-Dimensional Fractional-Reaction Diffusion Trimolecular Models","volume":"7","author":"Akinyemi","year":"2021","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Noeiaghdam, S., Dreglea, A., He, J., Avazzadeh, Z., Suleman, M., Araghi, M.A.F., Sidorov, D.N., and Sidorov, N. (2020). Error Estimation of the Homotopy Perturbation Method to Solve Second Kind Volterra Integral Equations with Piecewise Smooth Kernels: Application of the CADNA Library. Symmetry, 12.","DOI":"10.3390\/sym12101730"},{"key":"ref_15","first-page":"1184","article-title":"A Simple Approach to Volterra-Fredholm Integral Equations","volume":"6","author":"He","year":"2020","journal-title":"J. Appl. Comput. Mech."},{"key":"ref_16","first-page":"133","article-title":"Numerical solution of system of linear integral equations via improvement of block-pulse functions","volume":"4","author":"Mirzaee","year":"2016","journal-title":"J. Math. Modeling"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Jiang, Z.H., and Schaufelberger, W. (1992). Block Pulse Functions and Their Applications in Control Systems, Springer.","DOI":"10.1007\/BFb0009162"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1016\/j.cnsns.2009.04.011","article-title":"Application of 2D-BPFs to nonlinear integral equations","volume":"15","author":"Maleknejad","year":"2010","journal-title":"Commun. Non-Linear Sci. Numer. Simul."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"17","DOI":"10.5923\/j.am.20120201.03","article-title":"Iterated Block-Pulse Method for Solving Volterra Integral Equations","volume":"2","author":"Maleknejad","year":"2012","journal-title":"J. Appl. Math."},{"key":"ref_20","first-page":"3211","article-title":"Computational method to solve nonlinear integral equations using block pulse functions by collocation method","volume":"5","author":"Shahsavaran","year":"2011","journal-title":"Appl. Math. Sci."},{"key":"ref_21","first-page":"95","article-title":"A numerical solution of nonlinear volterra-fredholm integral equations","volume":"3","author":"Zarebnia","year":"2013","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3643","DOI":"10.1016\/j.aej.2020.06.014","article-title":"A new hybrid orthonormal Bernstein and improved block-pulse functions method for solving mathematical physics and engineering problems","volume":"59","author":"Ramadan","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2050313","DOI":"10.1142\/S0217979220503130","article-title":"Higher-order homotopy perturbation method for conservative nonlinear oscillators generally and microelectromechanical systems' oscillators particularly","volume":"34","author":"Anjum","year":"2020","journal-title":"Int. J. Mod. Phys. 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