## THERMAL SCIENCE

International Scientific Journal

### COMPUTER SIMULATION OF PANTOGRAPH DELAY DIFFERENTIAL EQUATIONS

**ABSTRACT**

Ritz method is widely used in variational theory to search for an approximate solution. This paper suggests a Ritz-like method for integral equations with an emphasis of pantograph delay equations. The unknown parameters involved in the trial solution can be determined by balancing the fundamental terms.

**KEYWORDS**

PAPER SUBMITTED: 2020-02-20

PAPER REVISED: 2020-06-01

PAPER ACCEPTED: 2020-06-02

PUBLISHED ONLINE: 2021-01-31

**THERMAL SCIENCE** YEAR

**2021**, VOLUME

**25**, ISSUE

**Issue 2**, PAGES [1381 - 1385]

- Shahgholian, D., et al., Buckling Analyses of Functionally Graded Graphene-Reinforced Porous Cylindrical Shell Using the Rayleigh-Ritz Method, Acta Mechanica, 231 (2020), Feb., pp. 1887-1902
- He, J. H., Generalized Variational Principles for Buckling Analysis of Circular Cylinders, Acta Mechanica, 231 (2020), Dec., pp. 899-906
- He, J. H., A Fractal Variational Theory for 1-D Compressible Flow in a Microgravity Space, Fractals, 28 (2020), 02, 2050024
- He, J.-H., Variational Principle and Periodic Solution of the Kundu-Mukherjee-Naskar Equation, Results in Physics, 17 (2020), June, 103031
- He, J. H., Lagrange Crisis and Generalized Variational Principle for 3-D Unsteady Flow, International Journal of Numerical Methods for Heat and Fluid-Flow, 30 (2019), 3, pp. 1189-1196
- He, J. H., Sun, C. A., Variational Principle for a Thin Film Equation, Journal of Mathematical Chemistry, 57 (2019), 9, pp. 2075-2081
- He, J. H., Ain, Q. T., New Promises and Future Challenges of Fractal Calculus: From Two-Scale Thermodynamics to Fractal Variational Principle, Thermal Science, 24 (2020), 2A, pp. 659-681
- He, J. H., Variational Principle and Periodic Solution of the Kundu-Mukherjee-Naskar Equation, Results in Physics, 17 (2020), 103031
- He, C.H., et al., Taylor Series Solution for Fractal Bratu-Type Equation Arising in eElectrospinning Process, Fractals, 28 (2020), 1, 2050011
- He, J. H., A Simple Approach to 1-D Convection-Diffusion Equation and Its Fractional Modification for E Reaction Arising in Rotating Disk Electrodes, Journal of Electroanalytical Chemistry, 854 (2019), 113565
- He, J. H., Ji, F. Y., Taylor Series Solution for Lane-Emden Equation, Journal of Mathematical Chemistry, 57 (2019), 8, pp. 1932-1934
- He, J. H., The Simplest Approach to Non-Linear Oscillators, Results in Physics, 15 (2019), 102546
- He, J. H., Taylor Series Solution for a Third Order Boundary Value Problem Arising in Architectural Engineering, Ain Shams Engineering Journal, On-line first, doi.org/10.1016/j.asej.2020.01.016, 2020
- He, J. H., Latifizadeh, H., A General Numerical Algorithm for Non-Linear Differential Equations by the Variational Iteration Method, International Journal of Numerical Methods for Heat and Fluid-Flow, 30 (2020), 11, pp. 4797-4810
- Ji, F. Y., et al., A Fractal Boussinesq Equation for Non-Linear Transverse Vibration of a Nanofiber-Reinforced Concrete Pillar, Applied Mathematical Modelling, 82 (2020), June, pp. 437-448
- He, J. H., A Short Review on Analytical Methods for to a Fully Fourth Order Non-Linear Integral Boundary Value Problem with Fractal Derivatives, International Journal of Numerical Methods for Heat and Fluid-Flow, 30 (2020), 11, pp. 4933-4943
- He, J. H., Jin, X., A Short Review on Analytical Methods for the Capillary Oscillator in a Nanoscale Deformable Tube, Mathematical Methods in the Applied Sciences, On-line first, dx.doi.org/10.1002/ mma.6321, 2020
- He, C. H., et al., Fangzhu, An Ancient Chinese Nanotechnology for Water Collection from Air, History, Mathematical Insight, Promises and Challenges, Mathematical Methods in the Applied Sciences, (2020), On-line first, doi.org/10.1002/mma.6384, 2020
- Ahmad, H., Khan, T. A., Variational Iteration Algorithm-I with an Auxiliary Parameter for Wave-Like Vibration Equations, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1113-1124
- Wei, C. F., Application of the Homotopy Perturbation Method for Solving Fractional Lane-Emden Type Equation, Thermal Science, 23 (2019), 4, pp. 2237-2244
- Yang, Y. J., Wang, S. Q., A Local Fractional Homotopy Perturbation Method for Solving the Local Fractional Korteweg-de Vries Equations with Non-Homogeneous Term, Thermal Science, 23 (2019), 3A, pp. 1495-1501
- Kuang, W. X., et al. Homotopy Perturbation Method with an Auxiliary Term for the Optimal Design of a Tangent Non-Linear Packaging System, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1075-1080
- Pasha, S. A., et al., The Modified Homotopy Perturbation Method with an Auxiliary Term for the Non-Linear Oscillator with Discontinuity, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1363-1373
- Li, X. X., He, C. H., Homotopy Perturbation Method Coupled with the Enhanced Perturbation Method, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1399-1403
- Yu, D. N., et al., Homotopy Perturbation Method with an Auxiliary Parameter for Non-Linear Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, ppp. 1540-1554
- He, J. H., The Simpler, the Better: Analytical Methods for Non-Linear Oscillators and Fractional Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1252-1260
- Bahgat, M. S. M., Approximate Analytical Solution of the Linear and Non-Linear Multi-Pantograph Delay Differential Equations, Physica Scripta, 95 (2020), 055219
- Sedaghat, S., et al., Analysis of Spectral Method for Neutral Multi-Pantograph Equations, Iranian Journal of Science and Technology Transaction A-Science, 43 (2019), A5, pp. 2261-2268
- He, J.-H., Fractal Calculus and its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276
- Ain, Q. T., He, J.-H., On Two-Scale Dimension and its Applications, Thermal Science, 23 (2019) , 3, pp. 1707-1712