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**Interactions of a heat source moving over a visco-thermoelastic rod kept in a magnetic field in the Lord-Shulman model under a memory dependent derivative.**
*(English)*
Zbl 1441.74102

Summary: The Fractional derivative is a widely accepted theory to describe physical phenomena and the processes with memory effects which is defined in the form of convolution having kernels as power functions. Due to the shortcomings of power-law distributions, some other forms of derivatives with few other kernel functions are proposed. This present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of the magnetic field and moving heat source in a thermoelastic rod in the context of the Lord-Shulman (LS) theory of generalized thermo-visco-elasticity. Both ends of the rod are fixed and are thermally insulated. Employing the Laplace transform as a tool, the problem has been transformed into the space-domain and solved analytically. Finally, solutions in the real-time domain are obtained on applying the numerical inversion of the Laplace transform, which has been carried out employing the Riemann sum approximation method. Numerical computations for stress, displacement, and temperature within the rod is carried out and have been demonstrated graphically. The results also demonstrate how the speed of the moving heat source influences the thermophysical quantities. It is observed that the temperature, thermally induced displacement and stress of the rod are found to decrease at large source speed. Also, significant differences in the thermophysical quantities are revealed due to the influence of the magnetic field and memory effect

### MSC:

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74F05 | Thermal effects in solid mechanics |

74F15 | Electromagnetic effects in solid mechanics |

26A33 | Fractional derivatives and integrals |

### Keywords:

memory-dependent derivative; magneto-thermoelasticity; viscosity; Lord-Shulman model; moving heat souce; Laplace transform
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\textit{S. Mondal}, Comput. Math. Model. 31, No. 2, 256--276 (2020; Zbl 1441.74102)

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