General Solution for the Second-Order Nonlocal Linear Differential Equation

• Uttam Kumar Mandal ^[1]
  1. [1] Marwadi University
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 5, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this article, we construct the general solution to a nonlocal linear differential equation of second-order:

    y(t) = γ y (t) + δy (−t) + αy(t) + βy(−t) + f (t), t ∈ R , where α, β, γ , δ are arbitrary real constants and f is continuous. To address the nonlocal model, we adopt an innovative strategy by representing the solution as a combination of even and odd functions, converting the nonlocal model into a system of local models. Using standard techniques for ordinary differential equations, we derive the general solution for the original nonlocal model, characterized by two arbitrary constants. We explore five main cases and their subcases based on the coefficients. Our investigation reveals that the Cauchy problem for nonlocal differential equations exhibits complex solution behavior, where the traditional existence and uniqueness theorem doesn’t always apply, especially due to the coefficients’ characteristics. Additionally, we present three examples with their corresponding general solutions.

• Referencias bibliográficas
  • 1. Bates, P.W.: On some nonlocal evolution equations arising in materials science, Nonlinear Dynamics and Evolution Equations. Fields Inst....
  • 2. Baskaran, A., Hu, Z., Lowengrub, J.S., Wang, C., Wise, S.M., Zhou, P.: Energy stable and efficient finite-difference nonlinear multigrid...
  • 3. Cortazar, C., Elgueta, M., Rossi, J.D.: Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions....
  • 4. Bates, P.W., Brown, S., Han, J.: Numerical analysis for a nonlocal Allen-Cahn equation. Int. J. Numer. Anal. Model. 6, 33–49 (2009)
  • 5. Elliott, C.M., Zheng, S.: On the Cahn-Hilliard equation, Arch. Ration. Mech. Anal., 96, pp. 339 357, (1986)
  • 6. Bates, P.W., Han, J.: Neumann boundary problem for the nonlocal Cahn-Hilliard equation. J. Differ. Equ. 212, 235–277 (2005)
  • 7. Bates, P.W., Han, J.: The Dirichlet boundary problem for the nonlocal Cahn-Hilliard equation. J. Math. Anal. Appl. 311, 289–312 (2005)
  • 8. Bernstein, D., Lange, T.: Post-quantum cryptography. Nature 549, 188–194 (2017)
  • 9. Cun, Y.L., Bengio, Y., Hinton, G.: Deep learning. Nature 521, 436–444 (2015)
  • 10. Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780...
  • 11. Chen, X.: Existence, uniqueness, and asymptotic stability of traveling waves in non-local evolution equations. Adv. Differ. Equ. 2, 125–160...
  • 12. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics, MASE, 191. Academic Press, Boston (1993)
  • 13. Malchow, H., Petrovskii, S., Venturino, E.: Spatiotemporal Patterns in Ecology and Epidemiology, CRC, 2008
  • 14. Cantrell, R. S., Cosner, C.: Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology,...
  • 15. Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)
  • 16. Bressloff, P.C.: Spatiotemporal dynamics of continuum neural fields. J. Phys. A: Math. 45, 033001 (2012)
  • 17. Krolikowski, W., Bang, O., Nikolov, N.I., Neshev, D., Wyller, J., Rasmussen, J.J., Edmundson, D.: Modulational instability, solitons and...
  • 18. Shapira, A., Tyomkyn, M.: Quasirandom graphs and the pantograph equation. Am. Math. Mon. 128, 630–639 (2021)
  • 19. Wallace, W.: Account of the invention of the pantograph, and a description of the eidograph. Trans. R. Soc. Edinb. 13, 418–439 (1836)
  • 20. Ma, W.X.: Inverse scattering and soliton solutions of nonlocal reverse-spacetime nonlinear Schrödinger equations. Proc. Am. Math. Soc....
  • 21. Ma, W.X.: Riemann-Hilbert problems and soliton solutions of nonlocal real reverse-spacetime mKdV equations. J. Math. Anal. Appl. 498,...
  • 22. Ma, W.X.: Riemann-Hilbert problems and inverse scattering of nonlocal real reverse-spacetime matrix AKNS hierarchies. Physica D 430, 133078...
  • 23. Bellamouchi, C., Hamdani, M.K., Boulaaras, S.: Existence and uniqueness of positive solution to a new class of nonlocal elliptic problem...
  • 24. Guefaifia, R., Boulaaras, S.: A qualitative study on the existence of positive solutions for local and nonlocal elliptic integro-differential...
  • 25. Boulaaras, S., Choucha, A., Ouchenane, D., Alharbi, A., Abdalla, M.: Result of local existence of solutions of nonlocal viscoelastic system...
  • 26. Choucha, A., Boulaaras, S., Ouchenane, D.: Exponential decay and global existence of solutions of a singular nonlocal viscoelastic system...
  • 27. Mesloub, F., Merah, A., Boulaaras, S.: Solution Blow-Up for a Fractional Fourth-Order Equation of Moore-Gibson-Thompson Type with Nonlinearity...
  • 28. Driver, R.D.: Ordinary and Delay Differential Equations. Applied Mathematical Sciences, 20, Springer, New York (1977)
  • 29. Erneux, T.: Applied Delay Differential Equations, Surv. Tutor. Appl. Math. Sci. , 3, Springer, New York (2009)
  • 30. Sturis, J., Polonsky, K.S., Mosekilde, E., Cauter, E.V.: Computer-model for mechanisms underlying ultradian oscillations of insulin and...
  • 31. Makroglou, A., Li, J.X., Kuang, Y.: Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an...
  • 32. Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control systems. Science 197, 287–289 (1977)
  • 33. Ma, W.X.: General Solution to a Nonlocal Linear Differential Equation of First-Order. Qual. Theory Dyn. Syst. 23, 177 (2024)
  • 34. Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110, 064105 (2013)
  • 35. Ma, W.X.: Nonlocal PT-symmetric integrable equations and related Riemann-Hilbert problems. Partial Differ. Equ. Appl. Math. 4, 100190...
  • 36. Ma, W.X.: Type (λ∗, λ)reduced nonlocal integrable AKNS equations and their soliton solutions. Appl. Numer. Math. 199, 105–113 (2024)
  • 37. Ji, J.L., Zhu, Z.N.: Soliton solutions of an integrable nonlocal modified Korteweg-de Vries equation through inverse scattering transform....
  • 38. Gürses,M., Pekcan, A.: Nonlocal modified KdV equations and their soliton solutions by HirotaMethod. Commun. Nonlinear Sci. Numer. Simul....
  • 39. Ma, W.X.: Binary Darboux transformation of vector nonlocal reverse-time integrable NLS equations. Chaos, Solitons Fractals 180, 114539...
  • 40. Ma, W.X.: Soliton solutions to constrained nonlocal integrable nonlinear Schrödinger hierarchies of type (−λ, λ). Int. J. Geom. Methods...