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By replacing <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$x\\rightarrow x^{\\alpha }$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mi>\u03b1<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> in FHBPF and utilizing incomplete beta functions, we construct the method with a focus on fractional derivatives in the Caputo sense and fractional integrals in the Riemann\u2013Liouville sense. A key advantage of the proposed method is its exact computation of the Riemann\u2013Liouville fractional integral operator. Using the Newton\u2013Cotes collocation method, we transform the differential equations into systems of algebraic equations, which are then solved with traditional techniques such as Newton\u2019s iterative method. An error analysis method is also introduced. Several numerical examples are provided to demonstrate the effectiveness and simplicity of the approach, highlighting its efficiency even for relatively small base sizes.<\/jats:p>","DOI":"10.1007\/s40314-025-03110-4","type":"journal-article","created":{"date-parts":[[2025,1,30]],"date-time":"2025-01-30T15:59:36Z","timestamp":1738252776000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A fractional hybrid function composed of block-pulses and fractional Fibonacci polynomials for solving multiterm fractional variable-order differential equations"],"prefix":"10.1007","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2318-9742","authenticated-orcid":false,"given":"Octavian","family":"Postavaru","sequence":"first","affiliation":[]},{"given":"Antonela","family":"Toma","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,1,30]]},"reference":[{"key":"3110_CR1","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","author":"M Abramowitz","year":"1964","unstructured":"Abramowitz M, Stegun IA (1964) Handbook of mathematical functions with formulas, graphs, and mathematical tables, vol 55. 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