{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:22:00Z","timestamp":1760145720006,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,21]],"date-time":"2024-08-21T00:00:00Z","timestamp":1724198400000},"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>We study certain Volterra integral equations that extend and recover first order ordinary differential equations (ODEs). We formulate the former equations from the latter by replacing classical derivatives with nonlocal integral operators with anti-symmetric kernels. Replacements of spatial derivatives have seen success in fracture mechanics, diffusion, and image processing. In this paper, we consider nonlocal replacements of time derivatives which contain future data. To account for the nonlocal nature of the operators, we formulate initial \u201cvolume\u201d problems (IVPs) for these integral equations; the initial data is prescribed on a time interval rather than at a single point. As a nonlocality parameter vanishes, we show that the solutions to these equations recover those of classical ODEs. We demonstrate this convergence with exact solutions of some simple IVPs. However, we find that the solutions of these nonlocal models exhibit several properties distinct from their classical counterparts. For example, the solutions exhibit discontinuities at periodic intervals. In addition, for some IVPs, a continuous initial profile develops a measure-valued singularity in finite time. At subsequent periodic intervals, these solutions develop increasingly higher order distributional singularities.<\/jats:p>","DOI":"10.3390\/axioms13080567","type":"journal-article","created":{"date-parts":[[2024,8,23]],"date-time":"2024-08-23T12:58:07Z","timestamp":1724417887000},"page":"567","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Nonlocal Extensions of First Order Initial Value Problems"],"prefix":"10.3390","volume":"13","author":[{"given":"Ravi","family":"Shankar","sequence":"first","affiliation":[{"name":"Department of Mathematics, Princeton University, Princeton, NJ 08544, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1051\/cocv:2002013","article-title":"Regular syntheses and solutions to discontinuous ODEs","volume":"7","author":"Marigo","year":"2002","journal-title":"ESAIM Control Optim. Calc. Var."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3967","DOI":"10.1016\/j.cam.2012.02.011","article-title":"A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side","volume":"236","author":"Dieci","year":"2012","journal-title":"J. Comput. Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1142\/S0218202512500546","article-title":"A Nonlocal Vector Calculus, nonlocal volume-constrained problems, and nonlocal balance laws","volume":"23","author":"Du","year":"2013","journal-title":"Math. Model. Methods Appl. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/S0022-5096(99)00029-0","article-title":"Reformulation of elasticity theory for discontinuities and long-range forces","volume":"48","author":"Silling","year":"2000","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4047","DOI":"10.1016\/j.ijheatmasstransfer.2010.05.024","article-title":"The peridynamic formulation for transient heat conduction","volume":"53","author":"Bobaru","year":"2010","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Andreu-Vaillo, F., Maz\u00f3n, J.M., Rossi, J.D., and Toledo-Melero, J.J. (2010). Nonlocal Diffusion Problems, Real Sociedad Matem\u00e1tica Espa\u00f1ola.","DOI":"10.1090\/surv\/165"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1137\/070698592","article-title":"Nonlocal Operators with Applications to Image Processing","volume":"7","author":"Gilboa","year":"2009","journal-title":"Multiscale Model. Simul."},{"key":"ref_8","first-page":"3463","article-title":"Green\u2019s functions in non-local three-dimensional linear elasticity","volume":"465","author":"Weckner","year":"2009","journal-title":"Proc. Roy. Soc. Edinb. Sect. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1017\/S0308210512001436","article-title":"The bond-based peridynamic system with Dirichlet-type volume constraint","volume":"144","author":"Mengesha","year":"2014","journal-title":"Proc. Roy. Soc. Edinb. Sect. A"},{"key":"ref_10","unstructured":"Radu, P., Toundykov, D., and Trageser, J. (2014). A nonlocal biharmonic operator and its connection with the classical bi-Laplacian. arXiv."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1137\/110833233","article-title":"A new approach for a nonlocal, nonlinear conservation law","volume":"72","author":"Du","year":"2012","journal-title":"SIAM J. Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1007\/s00397-005-0043-5","article-title":"Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives","volume":"45","author":"Heymans","year":"2006","journal-title":"Rheol. Acta"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1051\/m2an\/2010040","article-title":"Mathematical analysis for the peridynamic nonlocal continuum theory","volume":"45","author":"Du","year":"2011","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1137\/110833294","article-title":"Analysis and approximation of nonlocal diffusion problems with volume constraints","volume":"54","author":"Du","year":"2012","journal-title":"SIAM Rev."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/BF03028370","article-title":"Issues in the numerical solution of evolutionary delay differential equations","volume":"3","author":"Baker","year":"1995","journal-title":"Adv. Comput. Math."},{"key":"ref_16","first-page":"55","article-title":"The numerical analysis of functional integral and integro-differential equations of Volterra type","volume":"13","author":"Brunner","year":"2004","journal-title":"Acta Numer."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1312","DOI":"10.1134\/S0012266106090096","article-title":"Existence and construction of generalized solutions of nonlinear Volterra integral equations of the first kind","volume":"42","author":"Sidorov","year":"2006","journal-title":"Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Polyanin, A.D., and Manzhirov, A.V. (2008). Handbook of Integral Equations, CRC Press.","DOI":"10.1201\/9781420010558"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/BF01047828","article-title":"Functional differential equations of mixed type: The linear autonomous case","volume":"1","author":"Rustichini","year":"1989","journal-title":"J. Dyn. Differ. Equ."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1063\/1.1666641","article-title":"Some differential difference equations containing both advance and retardation","volume":"15","author":"Schulman","year":"1974","journal-title":"J. Math. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"414060","DOI":"10.1155\/2012\/414060","article-title":"q-Advanced Models for Tsunami and Rogue Waves","volume":"2012","author":"Pravica","year":"2012","journal-title":"Abstr. Appl. Anal."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1007\/BF00275686","article-title":"Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory","volume":"24","author":"Chi","year":"1986","journal-title":"J. Math. Biol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1512\/iumj.2002.51.2188","article-title":"Exponential dichotomies for linear nonautonomous functional differential equations of mixed type","volume":"51","author":"Sandstede","year":"2002","journal-title":"Indiana Univ. Math. J."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.jet.2004.02.006","article-title":"Vintage capital and the dynamics of the AK model","volume":"120","author":"Boucekkine","year":"2005","journal-title":"J. Econom. Theory"},{"key":"ref_25","first-page":"121","article-title":"On the advanced integral and differential equations of the sizing procedure of storage devices","volume":"11","author":"Lakatos","year":"2004","journal-title":"Funct. Differ. Equ."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/0362-546X(84)90066-X","article-title":"A mixed neutral system","volume":"8","author":"Driver","year":"1984","journal-title":"Nonlinear Anal."},{"key":"ref_27","first-page":"57","article-title":"Convergence in an impulsive advanced differential equations with piecewise constant argument","volume":"4","author":"Oztepe","year":"2012","journal-title":"Bull. Math. Anal. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1016\/j.jmaa.2004.09.059","article-title":"Advanced differential equations with nonlinear boundary conditions","volume":"304","author":"Jankowski","year":"2005","journal-title":"J. Math. Anal. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.aml.2013.11.010","article-title":"A note on explicit criteria for the existence of positive solutions to the linear advanced equation","volume":"35","author":"Diblik","year":"2014","journal-title":"Appl. Math. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"737","DOI":"10.3934\/dcds.2011.30.737","article-title":"Well-posedness of initial value problems for functional differential and algebraic equations of mixed type","volume":"30","author":"Hupkes","year":"2011","journal-title":"Discret. Contin. Dyn. Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1007\/BF02465413","article-title":"The bounded solution of a class of differential-difference equation of advanced type and its asymptotic behavior","volume":"9","year":"1988","journal-title":"Appl. Math. Mech. (Engl. Ed.)"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1090\/S0002-9947-1986-0819944-9","article-title":"Existence and nonoscillation theorems for an Emden-Fowler equation with deviating argument","volume":"294","author":"Trench","year":"1986","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_33","unstructured":"Kato, T. (2014). The functional differential equation y\u2019(x) = ay (Ax)+ by (x). Delay Funct. Differ. Equ. Appl., 197."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Wiener, J. (1993). Generalized Solutions of Functional Differential Equations, World Scientific.","DOI":"10.1142\/9789814343183"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"050402","DOI":"10.1103\/PhysRevLett.94.050402","article-title":"Dynamics of a Bright Soliton in Bose-Einstein Condensates with Time-Dependent Atomic Scattering Length in an Expulsive Parabolic Potential","volume":"94","author":"Liang","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"010402","DOI":"10.1103\/PhysRevLett.101.010402","article-title":"Dynamical Creation of Fractionalized Vortices and Vortex Lattices","volume":"101","author":"Ji","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2465","DOI":"10.1137\/16M1105372","article-title":"Nonlocal conservation laws. A new class of monotonicity-preserving models","volume":"55","author":"Du","year":"2017","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1119","DOI":"10.1137\/23M154488X","article-title":"Asymptotic Compatibility of a Class of Numerical Schemes for a Nonlocal Traffic Flow Model","volume":"62","author":"Huang","year":"2024","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/567\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:40:17Z","timestamp":1760110817000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/567"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,21]]},"references-count":38,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["axioms13080567"],"URL":"https:\/\/doi.org\/10.3390\/axioms13080567","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,8,21]]}}}