Research Paper Published: 18 August 2017 Fundamental Solution of the Multi-Dimensional Time Fractional Telegraph Equation Milton Ferreira^1,2, M. Manuela Rodrigues² & Nelson Vieira² Fractional Calculus and Applied Analysis volume 20, pages 868–894 (2017)Cite this article 11 Accesses 21 Citations Metrics details Abstract In this paper we study the fundamental solution (FS) of the multidimensional time-fractional telegraph equation where the time-fractional derivatives of orders α ∈]0,1] and β ∈]1,2] are in the Caputo sense. Using the Fourier transform we obtain an integral representation of the FS in the Fourier domain expressed in terms of a multivariate Mittag-Leffler function. The Fourier inversion leads to a double Mellin-Barnes type integral representation and consequently to a H-function of two variables. An explicit series representation of the FS, depending on the parity of the dimension, is also obtained. As an application, we study a telegraph process with Brownian time. Finally, we present some moments of integer order of the FS, and some plots of the FS for some particular values of the dimension and of the fractional parameters α and β. 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Funct. 23, No 7 (2012), 509–519, 10.1080/10652469.2011.608488. Article MathSciNet Google Scholar Download references Author information Authors and Affiliations School of Technology and Management Polytechnic, Institute of Leiria, P-2411-901, Leiria, Portugal Milton Ferreira Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 3810-193, Aveiro, Portugal Milton Ferreira, M. Manuela Rodrigues & Nelson Vieira Authors Milton Ferreira View author publications You can also search for this author in PubMed Google Scholar M. Manuela Rodrigues View author publications You can also search for this author in PubMed Google Scholar Nelson Vieira View author publications You can also search for this author in PubMed Google Scholar Corresponding author Correspondence to Milton Ferreira. About this article Cite this article Ferreira, M., Rodrigues, M.M. & Vieira, N. Fundamental Solution of the Multi-Dimensional Time Fractional Telegraph Equation. FCAA 20, 868–894 (2017). https://doi.org/10.1515/fca-2017-0046 Download citation Received: 12 December 2016 Revised: 18 August 2017 Published: 18 August 2017 Issue Date: August 2017 DOI : https://doi.org/10.1515/fca-2017-0046 Share this article Anyone you share the following link with will be able to read this content: Get shareable link Sorry, a shareable link is not currently available for this article. Copy to clipboard Provided by the Springer Nature SharedIt content-sharing initiative MSC 2010 Primary 35R11 Secondary 26A33 35A08 35A22 35C15 60G22 Key Words time-fractional telegraph equation fundamental solution Caputo fractional derivative multivariate Mittag-Leffler functions H-function of two variables double Mellin-Barnes type integrals Buy single article Instant access to the full article PDF. £ 29.95 Price includes VAT (United Kingdom) dataLayer.push({"ecommerce":{"currency":"GBP","impressions":[{"name":"Fundamental Solution of the Multi-Dimensional Time Fractional Telegraph Equation","id":"1314-2224","price":29.95,"brand":"Versita","category":"Mathematics","variant":"ppv-article","quantity":1}]}}); Rent this article via DeepDyve. function deepDyveResponse(data) { if (data.status === 'ok') { [].slice.call(document.querySelectorAll('.c-box.buybox__rent-article')).forEach(function (article) { article.style.display = 'flex' var link = article.querySelector('.deepdyve-link') if (link) { link.setAttribute('href', data.url) } }) } } var script = document.createElement('script') script.src = '//www.deepdyve.com/rental-link?docId=10.1515/fca-2017-0046&journal=1314-2224&fieldName=journal_doi&affiliateId=springer&format=jsonp&callback=deepDyveResponse' document.body.appendChild(script) Buy single article Instant access to the full article PDF. £ 29.95 Price includes VAT (United Kingdom) dataLayer.push({"ecommerce":{"currency":"GBP","impressions":[{"name":"Fundamental Solution of the Multi-Dimensional Time Fractional Telegraph Equation","id":"1314-2224","price":29.95,"brand":"Versita","category":"Mathematics","variant":"ppv-article","quantity":1}]}}); Rent this article via DeepDyve. function deepDyveResponse(data) { if (data.status === 'ok') { [].slice.call(document.querySelectorAll('.c-box.buybox__rent-article')).forEach(function (article) { article.style.display = 'flex' var link = article.querySelector('.deepdyve-link') if (link) { link.setAttribute('href', data.url) } }) } } var script = document.createElement('script') script.src = '//www.deepdyve.com/rental-link?docId=10.1515/fca-2017-0046&journal=1314-2224&fieldName=journal_doi&affiliateId=springer&format=jsonp&callback=deepDyveResponse' document.body.appendChild(script)

This paper is intended to investigate a fractional telegraph equation of the form with positive real parameters a, b and c. Here , and are operators of the Riemann–Liouville fractional derivative, where 0<α≤1 and 0<β≤1. A symbolic operational form of the solutions in terms of the Mittag–Leffler functions is exhibited. Using the Banach fixed point theorem, the existence and uniqueness of solutions are studied for this kind of fractional differential equations.