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Journal articles on the topic 'Ordinary fourth-order differential equations'

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Published: 4 June 2025

Last updated: 23 June 2025

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1

Tóthová, Mária, and Oleg Palumbíny. "On monotone solutions of the fourth order ordinary differential equations." Czechoslovak Mathematical Journal 45, no. 4 (1995): 737–46. http://dx.doi.org/10.21136/cmj.1995.128553.

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2

Alabi, M. O., M. S. Olaleye, and K. S. Adewoye. "Initial Value Solvers for Direct Solution of Fourth Order Ordinary Differential Equations in a Block from Using Chebyshev Polynomial as Basis Function." International Journal of Mathematics and Statistics Studies 12, no. 2 (2024): 25–46. http://dx.doi.org/10.37745/ijmss.13/vol12n12546.

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The numerical computation of fourth order ordinary differential equations cannot be gloss over easily due to its significant and importance. There have been glowing needs to find an appropriate numerical method that will handle effectively fourth order ordinary differential equations without resolving such an equation to a system of first order ordinary differential equations. To this end, this presentation focuses on direct numerical computation to fourth order ordinary differential equations without resolving such equations to a system of first order ordinary differential equations. The method is not predictor – corrector one due to its limitation in the level of accuracy. The method is order wise christened “Block Method” which is a self-starting method. In order to achieve this objective, Chebyshev polynomial is hereby used as basis function.

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3

Ma, Ruyun, and Fengran Zhang. "POSITIVE SOLUTIONS OF FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS." Acta Mathematica Scientia 18 (October 1998): 124–28. http://dx.doi.org/10.1016/s0252-9602(17)30886-x.

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4

Ashyralyev, Allaberen, and Ibrahim Mohammed Ibrahım. "High-Order, Accurate Finite Difference Schemes for Fourth-Order Differential Equations." Axioms 13, no. 2 (2024): 90. http://dx.doi.org/10.3390/axioms13020090.

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This article is devoted to the study of high-order, accurate difference schemes’ numerical solutions of local and non-local problems for ordinary differential equations of the fourth order. Local and non-local problems for ordinary differential equations with constant coefficients can be solved by classical integral transform methods. However, these classical methods can be used simply in the case when the differential equation has constant coefficients. We study fourth-order differential equations with dependent coefficients and their corresponding boundary value problems. Novel compact numerical solutions of high-order, accurate finite difference schemes generated by Taylor’s decomposition on five points have been studied in these problems. Numerical experiments support the theoretical statements for the solution of these difference schemes.

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5

Jrad, Fahd, and Uğurhan Muğan. "Non-polynomial Fourth Order Equations which Pass the Painlevé Test." Zeitschrift für Naturforschung A 60, no. 6 (2005): 387–400. http://dx.doi.org/10.1515/zna-2005-0601.

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The singular point analysis of fourth order ordinary differential equations in the non-polynomial class are presented. Some new fourth order ordinary differential equations which pass the Painlevé test as well as the known ones are found. -PACS: 02.30.Hq, 02.30.Ik, 02.30.Gp

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6

Kamo, K. I., and H. Usami. "Oscillation theorems for fourth-order quasilinear ordinary differential equations." Studia Scientiarum Mathematicarum Hungarica 39, no. 3-4 (2002): 385–406. http://dx.doi.org/10.1556/sscmath.39.2002.3-4.10.

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7

Przybycin, Jolanta. "Nonlinear eigenvalue problems for fourth order ordinary differential equations." Annales Polonici Mathematici 60, no. 3 (1995): 249–53. http://dx.doi.org/10.4064/ap-60-3-249-253.

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8

Kudryashov, Nicolai A. "Transcendents defined by nonlinear fourth-order ordinary differential equations." Journal of Physics A: Mathematical and General 32, no. 6 (1999): 999–1013. http://dx.doi.org/10.1088/0305-4470/32/6/012.

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9

Korman, Philip. "A maximum principle for fourth order ordinary differential equations." Applicable Analysis 33, no. 3-4 (1989): 267–73. http://dx.doi.org/10.1080/00036818908839878.

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10

Palumbíny, Oleg. "On oscillatory solutions of fourth order ordinary differential equations." Czechoslovak Mathematical Journal 49, no. 4 (1999): 779–90. http://dx.doi.org/10.1023/a:1022401101007.

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11

Kamo, Ken-ichi, and Hiroyuki Usami. "Nonlinear oscillations of fourth order quasilinear ordinary differential equations." Acta Mathematica Hungarica 132, no. 3 (2011): 207–22. http://dx.doi.org/10.1007/s10474-011-0127-x.

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12

Ndanusa, Abdulrahman, K. R. Adeboye, A. U. Mustapha, and R. Abdullahi. "ON NUMEROV METHOD FOR SOLVING FOURTH ORDER ORDINARY DIFFERENTIAL EQUATIONS." FUDMA JOURNAL OF SCIENCES 4, no. 4 (2021): 355–62. http://dx.doi.org/10.33003/fjs-2020-0404-493.

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In this work, a fourth order ODE of the form is transformed into a system of differential equations that is suitable for solution by means of Numerov method. The obtained solutions are compared with the exact solutions, and are shown to be very effective in solving both initial and boundary value problems in ordinary differential equations.

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13

Belyaeva, Irina, Igor Kirichenko, Oleh Ptashnyi, Natalia Chekanova, and Tetiana Yarkho. "Integrating linear ordinary fourth-order differential equations in the MAPLE programming environment." Eastern-European Journal of Enterprise Technologies 3, no. 4 (111) (2021): 51–57. http://dx.doi.org/10.15587/1729-4061.2021.233944.

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This paper reports a method to solve ordinary fourth-order differential equations in the form of ordinary power series and, for the case of regular special points, in the form of generalized power series. An algorithm has been constructed and a program has been developed in the MAPLE environment (Waterloo, Ontario, Canada) in order to solve the fourth-order differential equations. All types of solutions depending on the roots of the governing equation have been considered. The examples of solutions to the fourth-order differential equations are given; they have been compared with the results available in the literature that demonstrate excellent agreement with the calculations reported here, which confirms the effectiveness of the developed programs. A special feature of this work is that the accuracy of the results is controlled by the number of terms in the power series and the number of symbols (up to 20) in decimal mantissa in numerical calculations. Therefore, almost any accuracy allowed for a given electronic computing machine or computer is achievable. The proposed symbolic-numerical method and the work program could be successfully used for solving eigenvalue problems, in which controlled accuracy is very important as the eigenfunctions are extremely (exponentially) sensitive to the accuracy of eigenvalues found. The developed algorithm could be implemented in other known computer algebra packages such as REDUCE (Santa Monica, CA), MATHEMATICA (USA), MAXIMA (USA), and others. The program for solving ordinary fourth-order differential equations could be used to construct Green’s functions of boundary problems, to solve differential equations with private derivatives, a system of Hamilton’s differential equations, and other problems related to mathematical physics.

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14

Irina, Belyaeva, Kirichenko Igor, Ptashnyi Oleh, Chekanova Natalia, and Yarkho Tetiana. "Integrating linear ordinary fourth-order differential equations in the MAPLE programming environment." Eastern-European Journal of Enterprise Technologies 3, no. 4 (111) (2021): 51–57. https://doi.org/10.15587/1729-4061.2021.233944.

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This paper reports a method to solve ordinary fourth-order differential equations in the form of ordinary power series and, for the case of regular special points, in the form of generalized power series. An algorithm has been constructed and a program has been developed in the MAPLE environment (Waterloo, Ontario, Canada) in order to solve the fourth-order differential equations. All types of solutions depending on the roots of the governing equation have been considered. The examples of solutions to the fourth-order differential equations are given; they have been compared with the results available in the literature that demonstrate excellent agreement with the calculations reported here, which confirms the effectiveness of the developed programs. A special feature of this work is that the accuracy of the results is controlled by the number of terms in the power series and the number of symbols (up to 20) in decimal mantissa in numerical calculations. Therefore, almost any accuracy allowed for a given electronic computing machine or computer is achievable. The proposed symbolic-numerical method and the work program could be successfully used for solving eigenvalue problems, in which controlled accuracy is very important as the eigenfunctions are extremely (exponentially) sensitive to the accuracy of eigenvalues found. The developed algorithm could be implemented in other known computer algebra packages such as REDUCE (Santa Monica, CA), MATHEMATICA (USA), MAXIMA (USA), and others. The program for solving ordinary fourth-order differential equations could be used to construct Green’s functions of boundary problems, to solve differential equations with private derivatives, a system of Hamilton’s differential equations, and other problems related to mathematical physics.

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15

MAHOMED, F. M., and ASGHAR QADIR. "CONDITIONAL LINEARIZABILITY OF FOURTH-ORDER SEMI-LINEAR ORDINARY DIFFERENTIAL EQUATIONS." Journal of Nonlinear Mathematical Physics 16, sup1 (2009): 165–78. http://dx.doi.org/10.1142/s140292510900039x.

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16

Oghonyon, J. G., S. A. Okunuga, K. S. Eke, and O. A. Odetunmibi. "Block Milne’s Implementation For Solving Fourth Order Ordinary Differential Equations." Engineering, Technology & Applied Science Research 8, no. 3 (2018): 2943–48. http://dx.doi.org/10.48084/etasr.1914.

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Block predictor-corrector method for solving non-stiff ordinary differential equations (ODEs) started with Milne’s device. Milne’s device is an extension of the block predictor-corrector method providing further benefits and better results. This study considers Milne’s devise for solving fourth order ODEs. A combination of Newton’s backward difference interpolation polynomial and numerical integration method are applied and integrated at some selected grid points to formulate the block predictor-corrector method. Moreover, Milne’s devise advances the computational efficiency by applying the principal local truncation error (PLTE) of the block predictor-corrector method after establishing the order. The numerical results were exhibited to attest the functioning of Milne’s devise in solving fourth order ODEs. The complete results were obtained with the aid of Mathematica 9 kernel for Microsoft Windows. Numerical results showcase that Milne’s device is more effective than existent methods in terms of design new step size, determining the convergence criteria and maximizing errors at all examined convergence levels.

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17

Sanchez, Luis. "Boundary value problems for some fourth order ordinary differential equations." Applicable Analysis 38, no. 3 (1990): 161–77. http://dx.doi.org/10.1080/00036819008839960.

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18

Ibragimov, Nail H., Sergey V. Meleshko, and Supaporn Suksern. "Linearization of fourth-order ordinary differential equations by point transformations." Journal of Physics A: Mathematical and Theoretical 41, no. 23 (2008): 235206. http://dx.doi.org/10.1088/1751-8113/41/23/235206.

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19

Onyejekwe, Okey Oseloka. "A Green element method for fourth order ordinary differential equations." Advances in Engineering Software 35, no. 8-9 (2004): 517–25. http://dx.doi.org/10.1016/j.advengsoft.2004.05.005.

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20

Oghonyon, J. G., S. A. Okunuga, K. S. Eke, and O. A. Odetunmibi. "Block Milne's Implementation For Solving Fourth Order Ordinary Differential Equations." Engineering, Technology & Applied Science Research 8, no. 3 (2018): 2943–48. https://doi.org/10.5281/zenodo.1400515.

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Block predictor-corrector method for solving non-stiff ordinary differential equations (ODEs) started with Milne’s device. Milne’s device is an extension of the block predictorcorrector method providing further benefits and better results. This study considers Milne’s devise for solving fourth order ODEs. A combination of Newton’s backward difference interpolation polynomial and numerical integration method are applied and integrated at some selected grid points to formulate the block predictor-corrector method. Moreover, Milne’s devise advances the computational efficiency by applying the chief local truncation error] (CLTE) of the block predictor-corrector method after establishing the order. The numerical results were exhibited to attest the functioning of Milne’s devise in solving fourth order ODEs. The complete results were obtained with the aid of Mathematica 9 kernel for Microsoft Windows. Numerical results showcase that Milne’s device is more effective than existent methods in terms of design new step size, determining the convergence criteria and maximizing errors at all examined convergence levels.

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21

Kalitine, Boris S. "Stability of some differential equations of the fourth-order and fifth-order." Journal of the Belarusian State University. Mathematics and Informatics, no. 1 (April 12, 2019): 18–27. http://dx.doi.org/10.33581/2520-6508-2019-1-18-27.

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The article is devoted to the study of the problem of stability of nonlinear ordinary differential equations by the method of semi-definite Lyapunov’s functions. The types of fourth-order and fifth-order scalar nonlinear differential equations of general form are singled out, for which the sign-constant auxiliary functions are defined. Sufficient conditions for stability in the large are obtained for such equations. The results coincide with the necessary and sufficient conditions in the corresponding linear case. Studies emphasize the advantages in using the semi-positive functions in comparison with the classical method of applying Lyapunov’s definite positive functions.

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22

Shior, M. M., B. C. Agbata, G. D. Gbor, I. U. Ezugorie,, and N. N. Topman. "Solution of First Order Ordinary Differential Equations Using Fourth Order Runge-Kutta Method with MATLAB." International Journal of Mathematics and Statistics Studies 12, no. 1 (2024): 54–63. http://dx.doi.org/10.37745/ijmss.13/vol12n15463.

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Differential Equations are used in developing models in the physical sciences, engineering, mathematics, social science, environmental sciences, medical sciences and other numerous fields. This article examined solution of first ordinary differential equation using fourth order Runge-Kutta method with MATLAB. The fourth order Runge-Kutta method for modelling differential equations improves upon the Euler’s method to obtain a greater accuracy without the necessity for higher-order derivatives of the given function. A first order differential equation was solved using fourth order Runge-Kutta method with MATLAB and the same problem was solved analytically in order to obtain the exact solution. The MATLAB commands match up quickly with the steps of the fourth order Runge-Kutta algorithm. Slight variation of the MATLAB code was used to show the effect of the size of h on the accuracy of the solution (see figure 4.1, 4.2, 4.3).The MATLAB and exact solutions are approximately equal though the MATLAB approach is easier and faster. The obtained results are in agreement with those in existing literature and improved the results obtained by [1]

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23

Ashyralyev, A., M. Ashyralyyeva, and O. Batyrova. "On the boundedness of solution of the second order ordinary differential equation with damping term and involution." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 102, no. 2 (2021): 16–24. http://dx.doi.org/10.31489/2021m2/16-24.

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In the present paper the initial value problem for the second order ordinary differential equation with damping term and involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with damping term and involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with damping term and involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlinear differential equation with damping term and involution is established.

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24

Afuwape, Anthony Uyi, and M. O. Omeike. "Ultimate boundedness of some third order ordinary differential equations." Mathematica Bohemica 137, no. 3 (2012): 355–64. http://dx.doi.org/10.21136/mb.2012.142900.

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25

Vondra, Alexandr. "Geometry of second-order connections and ordinary differential equations." Mathematica Bohemica 120, no. 2 (1995): 145–67. http://dx.doi.org/10.21136/mb.1995.126226.

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26

Kudryashov, Nicolai A. "Some Fourth-Order Ordinary Differential Equations which Pass the Painlevé Test." Journal of Nonlinear Mathematical Physics 8, sup1 (2001): 172–77. http://dx.doi.org/10.2991/jnmp.2001.8.s.30.

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27

KUDRYASHOV, Nicolai A. "Some Fourth-Order Ordinary Differential Equations which Pass the Painleve Test." Journal of Non-linear Mathematical Physics 8, Supplement (2001): 172. http://dx.doi.org/10.2991/jnmp.2001.8.supplement.30.

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28

Cerquetelli, T., N. Ciccoli, and M. C. Nucci. "Four Dimensional Lie Symmetry Algebras and Fourth Order Ordinary Differential Equations." Journal of Nonlinear Mathematical Physics 9, sup2 (2002): 24–35. http://dx.doi.org/10.2991/jnmp.2002.9.s2.3.

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29

Das, Pratibhamoy, and Srinivasan Natesan. "Adaptive mesh generation for singularly perturbed fourth-order ordinary differential equations." International Journal of Computer Mathematics 92, no. 3 (2014): 562–78. http://dx.doi.org/10.1080/00207160.2014.902054.

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30

Peletier, Mark A. "Generalized monotonicity from global minimization in fourth-order ordinary differential equations." Nonlinearity 14, no. 5 (2001): 1221–38. http://dx.doi.org/10.1088/0951-7715/14/5/315.

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31

Zhang, Y. S. "Influence functions for usual fourth-order homogeneous linear ordinary differential equations." Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 215, no. 4 (2001): 219–24. http://dx.doi.org/10.1243/1464419011544493.

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32

Buell, Jeffrey C. "The Operator Compact Implicit Method for Fourth Order Ordinary Differential Equations." SIAM Journal on Scientific and Statistical Computing 7, no. 4 (1986): 1232–45. http://dx.doi.org/10.1137/0907083.

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33

Kudryashov, Nikolai A. "Nonlinear Differential Equations With Exact Solutions Expressed Via The Weierstrass Function." Zeitschrift für Naturforschung A 59, no. 7-8 (2004): 443–54. http://dx.doi.org/10.1515/zna-2004-7-807.

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A new problem is studied, that is to find nonlinear differential equations with special solutions expressed via the Weierstrass function. A method is discussed to construct nonlinear ordinary differential equations with exact solutions. The main step of our method is the assumption that nonlinear differential equations have exact solutions which are general solution of the simplest integrable equation. We use the Weierstrass elliptic equation as building block to find a number of nonlinear differential equations with exact solutions. Nonlinear differential equations of the second, third and fourth order with special solutionsexpressed via theWeierstrass function are given. - PACS: 02.30.Hq (Ordinary differential equations)

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34

Enoch, Opeyemi, Emmanuel Adeyefa, and Catherine Alakofa. "An efficient fourth-order method for direct integration of second-order ordinary differential equations." Journal of Mathematical Analysis and Modeling 5, no. 3 (2024): 89–99. https://doi.org/10.48185/jmam.v5i3.1368.

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35

Akinnukawe, Blessing Iziegbe, John Olusola Kuboye, and Solomon Adewale Okunuga. "Numerical Solution of Fourth-order Initial Value Problems Using Novel Fourth-order Block Algorithm." Journal of Nepal Mathematical Society 6, no. 2 (2024): 7–18. http://dx.doi.org/10.3126/jnms.v6i2.63016.

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In this paper, a one-step fourth-order block scheme for solving fourth order Initial Value Problems (IVPs) of Ordinary Differential Equations (ODE) is developed using interpolation and collocation techniques. The derived schemes contain two hybrid points which are chosen such that 0 < w1 < w2 < 1 where w1 and w2 are defined as hybrid points. The characteristics of the developed schemes are analyzed. The obtained schemes are applied in block form to solve some fourth-order IVPs and the numerical results show the accuracy and effectiveness of the block scheme compared with some existing methods.

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36

Basit, Muhammad, Komal Shahnaz, Rida Malik, Samsul Ariffin Abdul Karim, and Faheem Khan. "An Effective Approach Based on Generalized Bernstein Basis Functions for the System of Fourth-Order Initial Value Problems for an Arbitrary Interval." Mathematics 11, no. 14 (2023): 3076. http://dx.doi.org/10.3390/math11143076.

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The system of ordinary differential equations has many uses in contemporary mathematics and engineering. Finding the numerical solution to a system of ordinary differential equations for any arbitrary interval is very appealing to researchers. The numerical solution of a system of fourth-order ordinary differential equations on any finite interval [a,b] is found in this work using a symmetric Bernstein approximation. This technique is based on the operational matrices of Bernstein polynomials for solving the system of fourth-order ODEs. First, using Chebyshev collocation nodes, a generalised approximation of the system of ordinary differential equations is discretized into a system of linear algebraic equations that can be solved using any standard rule, such as Gaussian elimination. We obtain the numerical solution in the form of a polynomial after obtaining the unknowns. The Hyers–Ulam and Hyers–Ulam–Rassias stability analyses are provided to demonstrate that the proposed technique is stable under certain conditions. The results of numerical experiments using the proposed technique are plotted in figures to demonstrate the accuracy of the specified approach. The results show that the suggested Bernstein approximation method for any interval is quick and effective.

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37

Etwire, Christian John, Ibrahim Yakubu Seini, Rabiu Musah, and Oluwole Daniel Makinde. "Effects of Viscoelastic Oil-Based Nanofluids on a Porous Nonlinear Stretching Surface with Variable Heat Source/Sink." Defect and Diffusion Forum 387 (September 2018): 260–72. http://dx.doi.org/10.4028/www.scientific.net/ddf.387.260.

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The effect of variable heat source on viscoelastic fluid of CuO-oil based nanofluid over a porous nonlinear stretching surface is analyzed. The problem was modelled in the form of partial differential equations and transformed into a coupled fourth order ordinary differential equations by similarity techniques. It was further reduced to a system of first order ordinary differential equations and solved numerically using the fourth order Runge-Kutta algorithm with a shooting method. The results for various controlling parameters have been tabulated and the flow profiles graphically illustrated. The study revealed that the viscoelastic parameter has a decreasing effect on the magnitude of both the skin friction coefficient and the rate of heat transfer from the surface. It enhanced the momentum boundary layer thickness whilst adversely affecting the thermal boundary layer thickness.

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38

Etwire, Christian John, Ibrahim Yakubu Seini, and Rabiu Musah. "Effects of Oil-Based Nanofluid on a Stretching Surface with Variable Suction and Thermal Conductivity." Diffusion Foundations 11 (August 2017): 99–109. http://dx.doi.org/10.4028/www.scientific.net/df.11.99.

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The combined effect of suction and thermal conductivity on the boundary layer flow of oil–based nanofluid over a porous stretching surface has been investigated. Similarity techniques were employed in transforming the governing partial differential equations into a coupled third order ordinary differential equations. The higher third order ordinary differential equations were then reduced into a system of first order ordinary differential equations and solved numerically using the fourth order Runge-Kutta algorithm with a shooting method. The results were presented in tabular and graphically forms for various controlling parameters. It was found that increasing the thermal conductivities of the base fluid (oil) and nanoparticle size (CuO) of the nanofluid did not affect the velocity boundary layer thickness but depreciates with suction and permeability. The suction parameter and thermal conductivity of the base fluid also made the thermal boundary layer thinner.

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39

Aydin Akgun, Fatma. "Global Bifurcation of Fourth-Order Nonlinear Eigenvalue Problems’ Solution." International Journal of Differential Equations 2021 (November 26, 2021): 1–6. http://dx.doi.org/10.1155/2021/7516324.

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In this paper, we study the global bifurcation of infinity of a class of nonlinear eigenvalue problems for fourth-order ordinary differential equations with nondifferentiable nonlinearity. We prove the existence of two families of unbounded continuance of solutions bifurcating at infinity and corresponding to the usual nodal properties near bifurcation intervals.

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40

Tao, Sixing. "Lie symmetry analysis, particular solutions and conservation laws of a (2+1)-dimensional KdV4 equation." Mathematical Biosciences and Engineering 20, no. 7 (2023): 11978–97. http://dx.doi.org/10.3934/mbe.2023532.

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<abstract><p>In this paper, a (2+1)-dimensional KdV4 equation is considered. We obtain Lie symmetries of this equation by utilizing Lie point symmetry analysis method, then use them to perform symmetry reductions. By using translation symmetries, two fourth-order ordinary differential equations are obtained. Solutions of one fourth order ordinary differential equation are presented by using direct integration method and $ (G'/G) $-expansion method respectively. Furthermore, the corresponding solutions are depicted with appropriate graphical representations. The other fourth-order ordinary differential equation is solved by using power series technique. Finally, two kinds of conserved vectors of this equation are presented by invoking the multiplier method and Noether's theorem respectively.</p></abstract>

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41

Yap, Lee Ken, and Fudziah Ismail. "Block Hybrid Collocation Method with Application to Fourth Order Differential Equations." Mathematical Problems in Engineering 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/561489.

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The block hybrid collocation method with three off-step points is proposed for the direct solution of fourth order ordinary differential equations. The interpolation and collocation techniques are applied on basic polynomial to generate the main and additional methods. These methods are implemented in block form to obtain the approximation at seven points simultaneously. Numerical experiments are conducted to illustrate the efficiency of the method. The method is also applied to solve the fourth order problem from ship dynamics.

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42

Assanova, A. T., and Zh S. Tokmurzin. "Method of functional parametrization for solving a semi-periodic initial problem for fourth-order partial differential equations." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 100, no. 4 (2020): 5–16. http://dx.doi.org/10.31489/2020m4/5-16.

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A semi-periodic initial boundary-value problem for a fourth-order system of partial differential equations is considered. Using the method of functional parametrization, an additional parameter is carried out and the studied problem is reduced to the equivalent semi-periodic problem for a system of integro-differential equations of hyperbolic type second order with functional parameters and integral relations. An interrelation between the semi-periodic problem for the system of integro-differential equations of hyperbolic type and a family of Cauchy problems for a system of ordinary differential equations is established. Algorithms for finding of solutions to an equivalent problem are constructed and their convergence is proved. Sufficient conditions of a unique solvability to the semi-periodic initial boundary value problem for the fourth-order system of partial differential equations are obtained.

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43

Fatima, A., Muhammad Ayub, and F. M. Mahomed. "A Note on Four-Dimensional Symmetry Algebras and Fourth-Order Ordinary Differential Equations." Journal of Applied Mathematics 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/848163.

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We provide a supplementation of the results on the canonical forms for scalar fourth-order ordinary differential equations (ODEs) which admit four-dimensional Lie algebras obtained recently. Together with these new canonical forms, a complete list of scalar fourth-order ODEs that admit four-dimensional Lie algebras is available.

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44

Etwire, Christian John, Ibrahim Yakubu Seini, Rabiu Musah, and Oluwole Daniel Makinde. "Combined Effects of Variable Viscosity and Thermal Conductivity on Dissipative Flow of Oil-Based Nanofluid over a Permeable Vertical Surface." Diffusion Foundations 16 (June 2018): 158–76. http://dx.doi.org/10.4028/www.scientific.net/df.16.158.

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Abstract: The combined effect of variable viscosity and thermal conductivity on dissipative flow of oil-based nanofluid over a permeable vertical plate with suction has been studied. The governing partial differential equations have been transformed into a coupled third-order ordinary differential equations using similarity techniques. The resulting third-order ordinary differential equations were then reduced into a system of first-order ordinary differential equations and solved numerically using the fourth-order Runge-Kutta algorithm with a shooting method. The results revealed that both viscosity and thermal conductivities of CuO oil-based nanofluid enhances the intensity of the skin friction coefficient and the rate of heat transfer at the surface of the plate. Furthermore, the thermal boundary layer thickness is weakened by the viscosity of CuO oil-based nanofluid, the Prandtl number, the suction parameter, the permeability of the medium and the thermal Grashof number

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Badokina, T. E., and B. V. Loginov. "Multiparameter bifurcation in boundary value problems for fourth-order ordinary differential equations." Doklady Mathematics 89, no. 3 (2014): 296–300. http://dx.doi.org/10.1134/s1064562414020203.

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Ruyun, Ma, and Wang Haiyan. "On the existence of positive solutions of fourth-order ordinary differential equations." Applicable Analysis 59, no. 1-4 (1995): 225–31. http://dx.doi.org/10.1080/00036819508840401.

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Dehesa, J. S., E. Buendia, and M. A. Sanchez‐Buendia. "On the polynomial solutions of ordinary differential equations of the fourth order." Journal of Mathematical Physics 26, no. 7 (1985): 1547–52. http://dx.doi.org/10.1063/1.526915.

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Ma, Ruyun. "Nodal solutions of boundary value problems of fourth-order ordinary differential equations." Journal of Mathematical Analysis and Applications 319, no. 2 (2006): 424–34. http://dx.doi.org/10.1016/j.jmaa.2005.06.045.

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Jha, Navnit, R. K. Mohanty, and Vinod Chauhan. "Geometric Mesh Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System." Advances in Numerical Analysis 2013 (October 24, 2013): 1–10. http://dx.doi.org/10.1155/2013/614508.

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Numerical method based on three geometric stencils has been proposed for the numerical solution of nonlinear singular fourth-order ordinary differential equations. The method can be easily extended to the sixth-order differential equations. Convergence analysis proves the third-order convergence of the proposed scheme. The resulting difference equations lead to block tridiagonal matrices and can be easily solved using block Gauss-Seidel algorithm. The computational results are provided to justify the usefulness and reliability of the proposed method.

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Arar, Nouria, Leila Ait Kaki, and Abdellatif Ben Makhlouf. "Highly Efficacious Sixth-Order Compact Approach with Nonclassical Boundary Specifications for the Heat Equation." Mathematical Problems in Engineering 2022 (December 30, 2022): 1–13. http://dx.doi.org/10.1155/2022/8224959.

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This paper suggests an accurate numerical method based on a sixth-order compact difference scheme and explicit fourth-order Runge–Kutta approach for the heat equation with nonclassical boundary conditions (NCBC). According to this approach, the partial differential equation which represents the heat equation is transformed into several ordinary differential equations. The system of ordinary differential equations that are dependent on time is then solved using a fourth-order Runge–Kutta method. This study deals with four test problems in order to provide evidence for the accuracy of the employed method. After that, a comparison is done between numerical solutions obtained by the proposed method and the analytical solutions as well as the numerical solutions available in the literature. The proposed technique yields more accurate results than the existing numerical methods.

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