# History of ill-posed problems and their application to solve various mathematical problems

## Main Article Content

## Abstract

This study aims to provide an understanding of well-posed and incorrectly-posed problems, as well as the developed methods for solving incorrectly-posed applied problems in mathematics. The history and significance of incorrectly-posed problems in solving various applied problems in the natural sciences are explored in detail. The study of methods for solving ill-posed problems has garnered significant interest among researchers, who are actively conducting research in this field. The theory of incorrectly-posed problems is a rapidly developing area in mathematical physics and natural sciences.In the practical realm, most problems are ill-posed, requiring decision-making under conditions of uncertainty, overdetermination, or inconsistency. The main conclusion drawn from this study is that solving incorrectly-posed problems cannot be accomplished solely by learning from well-posed problems.

## Article Details

*Engineering Applications*,

*2*(3), 279–290. Retrieved from https://publish.mersin.edu.tr/index.php/enap/article/view/1178

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

## References

Carleman T. (1926). Les fonctions quasi analytiques. Gautier-Villars et Cie., Paris.

Lavrent’ev, M. M. (1957). On the Cauchy problem for second-order linear elliptic equations. Reports of the USSR Academy of Sciences. 112(2), 195-197.

Lavrent’ev, M. M. (1962). On some ill-posed problems of mathematical physics. Nauka, Novosibirsk.

Lavrent’ev M. M., Romanov V. G., & Shishatsky S. P. (1980). Ill-posed problems of mathematical physics and analysis. Nauka, Moscow.

Tikhonov, A. N. (1963). On the solution of ill-posed problems and the method of regularization. Reports of the USSR Academy of Sciences, 151(3), 501-504.

Tikhonov A. N., & Arsenin V. Ya. (1974). Methods for solving ill-posed problems. Nauka, Moscow.

Tikhonov, A. N., & Arsenin, V. Y. (1977). Solutions of ill-posed problems. New York: Winston.

Hadamard, J. (1978). The Cauchy problem for linear partial differential equations of hyperbolic type. Nauka, Moscow.

Yarmukhamedov, Sh. (1977). On the Cauchy problem for the Laplace equation, Doklady Akademii Nauk, 235(2), 281-283.

Yarmukhamedov, Sh. (1985). Green's formula in an infinite region and its application, Doklady Akademii Nauk, 285(2), 305-308.

Yarmukhamedov, S. (2004). A Carleman function and the Cauchy problem for the Laplace equation. Siberian Mathematical Journal, 45(3), 580-595.

Yarmukhamedov, S. (2008). Representation of harmonic functions as potentials and the Cauchy problem. Mathematical Notes, 83(5-6), 693-706.

Aizenberg, L. A. (1990). Carleman’s formulas in complex analysis. Nauka, Novosibirsk.

Tarkhanov, N. N. (1985). Stability of the solutions of elliptic systems. Functional Analysis and Its Applications, 19(3), 245-247.

Tarkhanov, N. N. (1985). On the Carleman matrix for elliptic systems, Reports of the USSR Academy of Sciences, 288(2), 294-297.

Tarkhanov, N. N. (1989). The solvability criterion for an ill-posed problem for elliptic systems, Reports of the USSR Academy of Sciences, 380(3), 531-534.

Tarkhanov, N. N. (1995). The Cauchy problem for solutions of elliptic equations. V. 7, Akad. Verl., Berlin.

Arbuzov, E. V. (2003). The Cauchy problem for second-order elliptic systems on the plane. Siberian Mathematical Journal, 44, 1-16. https://doi.org/10.1023/A:1022034001292

Arbuzov, E. V., & Bukhgeim, A. L. V. (2006). The Carleman formula for the Helmholtz equation on the plane. Siberian Mathematical Journal, 47(3), 425-432. https://doi.org/10.1007/s11202-006-0055-0

Arbuzov, E. V., & Bukhgeim, A. L. (2010). On the solution of the Cauchy problem for second-order elliptic equations on the plane using the Cauchy integral operator. Siberian Electronic Mathematical Reports, 7, 173-177.

Arbuzov, E. V. (2013). On the properties of the Cauchy integral operator with an oscillating kernel. Siberian Electronic Mathematical Reports, 10, 3-9.

Juraev, D. A. (2012). The construction of the fundamental solution of the Helmholtz equation. Reports of the Academy of Sciences of the Republic of Uzbekistan, 2, 14-17.

Juraev, D. A. (2016). Regularization of the Cauchy problem for systems of elliptic type equations of first order. Uzbek Mathematical Journal, 2, 61-71.

Juraev, D. A. (2017). The Cauchy problem for matrix factorizations of the Helmholtz equation in an unbounded domain. Siberian Electronic Mathematical Reports, 14, 752-764.

Juraev, D. A. (2017). Cauchy problem for matrix factorizations of the Helmholtz equation. Ukrainian Mathematical Journal, 69(10), 1364-1371.

Juraev, D. A. (2018). On the Cauchy problem for matrix factorizations of the Helmholtz equation in a bounded domain. Siberian Electronic Mathematical Reports, 15, 11-20.

Juraev, D. A. (2018). The Cauchy problem for matrix factorizations of the Helmholtz equation in R3. Journal of Universal Mathematics, 1(3), 312-319.

Zhuraev, D. A. (2018). Cauchy problem for matrix factorizations of the Helmholtz equation. Ukrainian Mathematical Journal, 69(10), 1583-1592.

Juraev, D. A. (2018). On the Cauchy problem for matrix factorizations of the Helmholtz equation in a bounded domain R2. Siberian Electronic Mathematical Reports, 15, 1865-1877.

Juraev, D. A. (2019). The Cauchy problem for matrix factorizations of the Helmholtz equation in R3. Advanced Mathematical Models & Applications, 1(4), 86-96.

Juraev, D. A. (2019). On the Cauchy problem for matrix factorizations of the Helmholtz equation. Journal of Universal Mathematics, 2(2), 113-126.

Juraev, D. A. (2020). The solution of the ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation. Advanced Mathematical Models & Applications, 5(2), 205-221.

Juraev, D. A., & Noeiaghdam, S. (2021). Regularization of the ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation on the plane. Axioms, 10(2), 82. https://doi.org/10.3390/axioms10020082

Juraev, D. A. (2021). Solution of the ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation on the plane. Global and Stochastic Analysis, 8(3), 1-17.

Juraev, D. A., & Gasimov, Y. S. (2022). On the regularization Cauchy problem for matrix factorizations of the Helmholtz equation in a multidimensional bounded domain. Azerbaijan J. Math, 12, 142-161.

Juraev D. A. (2022). On the solution of the Cauchy problem for matrix factorizations of the Helmholtz equation in a multidimensional spatial domain. Global and Stochastic Analysis, 9(2), 1-17.

Juraev, D. A., & Noeiaghdam, S. (2022). Modern problems of mathematical physics and their applications. Axioms, 11(2), 45. https://doi.org/10.3390/axioms11020045

Juraev, D. A., & Noeiaghdam, S. (2022). Modern problems of mathematical physics and their applications. Axioms, MDPI. ISBN 978-3-0365-3495-4

Juraev, D. A., Shokri, A., & Marian, D. (2022). Solution of the ill-posed Cauchy problem for systems of elliptic type of the first order. Fractal and Fractional, 6(7), 358. https://doi.org/10.3390/fractalfract6070358

Juraev, D. A., Shokri, A., & Marian, D. (2022). On an approximate solution of the cauchy problem for systems of equations of elliptic type of the first order. Entropy, 24(7), 968. https://doi.org/10.3390/e24070968

Juraev, D. A., Shokri, A., & Marian, D. (2022). On the Approximate Solution of the Cauchy Problem in a Multidimensional Unbounded Domain. Fractal and Fractional, 6(7), 403. https://doi.org/10.3390/fractalfract6070403

Juraev, D. A., Shokri, A., & Marian, D. (2022). Regularized solution of the Cauchy problem in an unbounded domain. Symmetry, 14(8), 1682. https://doi.org/10.3390/sym14081682

Juraev, D. A., & Cavalcanti, M. M. (2023). Cauchy Problem for Matrix Factorizations of the Helmholtz Equation in the Space Rm. Boletim da Sociedade Paranaense de Matematica, 41, 1-12.

Juraev, D. A. (2023). The Cauchy problem for matrix factorization of the Helmholtz equation in a multidimensional unbounded domain, Boletim da Sociedade Paranaense de Matematica, 41, 1-18.

Juraev, D. A., Ibrahimov, V. & Agarwal, P. (2023) Regularization of the Cauchy problem for matrix factorizations of the Helmholtz equation on a two-dimensional bounded domain, Palestine Journal of Mathematics, 12(1), 381-403.

Juraev, D. A. (2023). Fundamental solution for the Helmholtz equation. Engineering Applications, 2(2), 164-175.

Juraev, D. A., Jalalov, M. J. O., & Ibrahimov, V. R. O. (2023). On the formulation of the Cauchy problem for matrix factorizations of the Helmholtz equation. Engineering Applications, 2(2), 176-189.

Juraev, D. A., Agarwal, P., Shokri, A., Elsayed, E. E. & Bulnes, J. D. (2023). On the solution of the ill-posed Cauchy problem for elliptic systems of the first order, Stochastic Modelling & Computational Sciences, 3(1), 1-21.

Ibrahimov, V. R., Mehdiyeva, G. Yu., Yue, X. G., Kaabar, M. K. A., Noeiaghdam, S. & Juraev, D. A. (2021). Novel symmetric numerical methods for solving symmetric mathematical problems, International Journal of Circuits, Systems and Signal Processing, 15, 1545-1557.

Ibrahimov, V. R. & Imanova, M. N. (2022). Finite difference methods with improved properties and their application to solving some model problems, International Conference on Computational Science and Computational Intelligence (CSCI), 464-472.

Ibrahimov, V. R., Imanova, M. N. & Juraev, D. A. (2023). About the new way for solving some physical problems described by ODE of the second order with the special structure, Stochastic Modelling & Computational Sciences, 3(1), 99-117.

Bulnes, J. D. (2022). An unusual quantum entanglement consistent with Schrödinger’s equation. Global and Stochastic Analysis, 9(2), 79-87.

Bulnes, J. D. (2022). Solving the heat equation by solving an integro-differential equation. Global and Stochastic Analysis, 9(2), 89-97.

Bulnes, J. D., Juraev, D. A., Bonilla, J. L. & Travassos M. A. I. (2023). Exact decoupling of a coupled system of two stationary Schrödinger equations. Stochastic Modelling & Computational Sciences, 3(1), 23-28.

Bulnes, J. D., Bonilla, J. L. & Juraev, D. A. (2023). Klein-Gordon’s equation for magnons without non-ideal effect on spatial separation of spin waves. Stochastic Modelling & Computational Sciences, 3(1), 29-37.

Shokri, A. (2012). The symmetric P-stable Hybrid Obrenchkoff Methods for the numerical solution of second order IVPs. Journal of Pure and Applied Mathematics, 5, 28-35.

Shokri, A., & Shokri, A. A. (2014). The hybrid Obrechkoff BDF methods for the numerical solution of first order initial value problems. Acta Universitatis Apulensis, 38, 23-33.

Shokri, A. (2015). An explicit trigonometrically fitted ten-step method with phase-lag of order infinity for the numerical solution of the radial Schrodinger equation. Applied and Computational Mathematics, 14(1), 63-74.

Shokri, A., & Saadat, H. (2016). P-stability, TF and VSDPL technique in Obrechkoff methods for the numerical solution of the Schrodinger equation. Bulletin of the Iranian Mathematical Society, 42(3), 687-706.

Shokri, A., & Tahmourasi, M. (2017). A new two-step Obrechkoff method with vanished phase-lag and some of its derivatives for the numerical solution of radial Schrodinger equation and related IVPs with oscillating solutions. Iranian Journal of Mathematical Chemistry, 8(2), 137-159. https://doi.org/10.22052/IJMC.2017.62671.1243

Shokri, A., Khalsaraei, M. M., Noeiaghdam, S. & Juraev, D. A. (2022). A new divided difference interpolation method for two-variable functions. Global and Stochastic Analysis, 9(2), 19-26.

Lungu, N., & Marian, D. (2019). Ulam-Hyers-Rassias stability of some quasilinear partial differential equations of first order. Carpathian Journal of Mathematics, 35(2), 165-170.

Marian, D., Ciplea, S. A. & Lungu, N. (2020). On the Ulam-Hyers stability of biharmonic equation. UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 82(2), 141-148.

Marian, D. (2021). Semi-Hyers–Ulam–Rassias stability of the convection partial differential equation via Laplace transform. Mathematics, 9(22), 2980. https://doi.org/10.3390/math9222980

Marian, D. (2021). Laplace transform and Semi-Hyers–Ulam–Rassias stability of some delay differential equations. Mathematics, 9(24), 3260. https://doi.org/10.3390/math9243260

Marian, D., Ciplea, S. A., & Lungu, N. (2021). Hyers-Ulam stability of Euler’s equation in the calculus of variations. Mathematics, 9(24), 3320. https://doi.org/10.3390/math9243320

Fayziyev, Y., Buvaev, Q., Juraev, D. A., Nuralieva, N., & Sadullaeva, S. (2022). The inverse problem for determining the source function in the equation with the Riemann-Liouville fractional derivative. Global and Stochastic Analysis, 9(2), 43-51.

Nuriyeva, V. (2022). On the construction bilateral multistep methods and its application to solve Volterra integral equation. International Journal of Engineering Research in Computer Science and Engineering, 9(10), 61-62.

Islamov, B. I., Kholbekov, J. A. (2021). On a non-local boundary value problem for a loaded parabolo-hyperbolic equation with three lines of type change. Bulletin of the Samara State Technical University. Series "Physical and Mathematical Sciences", 25(3), 407-422.

Kholbekov, J. (2023). Boundary value problem for a parabolic-hyperbolic equation loaded by the fractional order integral operator. Advanced Mathematical Models & Applications, 8(2), 271-283.

Targyn, N. & Juraev, D. A. (2023). Mathematical model of the melting of micro-asperity arising in closed electrical contacts. Stochastic Modelling & Computational Sciences, 3(1), 39-57.

Perchik, E. (2006). Methodology of syntheses of knowledge: Overcoming incorrectness of the problems of mathematical modeling. Mathematical Physics. https://doi.org/10.48550/arXiv.math-ph/0302045

Berntsson, F., Kozlov, V. A., Mpinganzima, L., & Turesson, B. O. (2014). An alternating iterative procedure for the Cauchy problem for the Helmholtz equation. Inverse Problems in Science and Engineering, 22(1), 45-62. https://doi.org/10.1080/17415977.2013.827181

Berdawood, K. A., Nachaoui, A., Saeed, R., Nachaoui, M., & Aboud, F. (2020). An alternating procedure with dynamic relaxation for Cauchy problems governed by the modified Helmholtz equation. Advanced Mathematical Models & Applications, 5(1), 131-139.

Salim, S.H., Jwamer, K.H.F. & Saeed, R. K. (2022). Solving Volterra-Fredholm integral equations by quadratic spline function. Journal of Al-Qadisiyah for Computer Science and Mathematics, 14(4), 10-19.

Berdawood K., Nachaoui A., Saeed R., Nachaoui M. & Aboud F. (2022). An efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equation. Discrete and Continuous Dynamical Systems - S., 15(1), 57-78.

Muhammad, R. S., Saeed, R. K. & Juraev, D. A. (2022). New algorithm for computing Adomian’s polynomials to solve coupled Hirota system. Journal of Zankoy Sulaimani – Part A. For Pure and Applied Sciences, 24(1), 38-54.

Salim, S. H., Jwamer, K. H. F. & Saeed, R. K. (2023). Solving Volterra-Fredholm integral equations by natural cubic spline function. Bulletin of the Karaganda University, 1(109), 124-130.

Dzhuraev Kh.Sh. (1991). On one regularizing algorithm for constructing approximate solutions of the Cauchy problem for the Helmholtz equation. DAN Taj. SSR, 34(2), 77-79.

Dzhuraev Kh.Sh. (2010). On one approach to the problem of regularization of the Cauchy problem for an elliptic type equation with constant coefficients. Bulletin of the Tajik National University. Series of Natural Sciences, 3(59), 16-27.

Srinivasamurthy, S. B. (2012). Methods of solving ill-posed problems. Mathematics, Numerical Analysis. Manhattan, Kansas.

Juraev, D. A., Elsayed, E. E., Bulnes, J. D., & Agarwal, P. (2023). The role and essence of ill-posed problems for solving various applied problems. Advanced Engineering Days (AED), 7, 100-102.