Remarks on parabolic equation with the conformable variable derivative in Hilbert scales

Phuong Nguyen Duc; Ahmet Ocak Akdemir; Van Tien Nguyen; Anh Tuan Nguyen; Phuong Nguyen Duc; Ahmet Ocak Akdemir; Van Tien Nguyen; Anh Tuan Nguyen

doi:10.3934/math.20221095

AIMS Mathematics

2022, Volume 7, Issue 11: 20020-20042. doi: 10.3934/math.20221095

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Remarks on parabolic equation with the conformable variable derivative in Hilbert scales

1.
Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam
2.
Department of Mathematics, Faculty of Science and Letters, Aǧri Ibrahim Cecen University, Agri 04100, Turkey
3.
Faculty of Math, FPT University HCM, Saigon Hi-Tech Park, Thu Duc City, Ho Chi Minh City, Vietnam
4.
Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam

Received: 28 May 2022 Revised: 01 September 2022 Accepted: 02 September 2022 Published: 13 September 2022
MSC : 35A05, 35A08

In this paper, we are interested in diffusion equations with conformable derivatives with variable order. We will study two different types of models: the initial value model and the nonlocal in time model. With different values of input values, we investigate the well-posedness of the mild solution in suitable spaces. We also prove the convergence of mild solution of the nonlocal problem to solutions of the initial problem. The main technique of our paper is to use the theory of Fourier series in combination with evaluation techniques for some generalized integrals. Our results are one of the first directions on the diffusion equation with conformable variable derivative in Hilbert scales.
- parabolic equations,
- conformable derivative,
- variable order fractional derivatives,
- variable order derivative,
- Sobolev embeddings
Citation: Phuong Nguyen Duc, Ahmet Ocak Akdemir, Van Tien Nguyen, Anh Tuan Nguyen. Remarks on parabolic equation with the conformable variable derivative in Hilbert scales[J]. AIMS Mathematics, 2022, 7(11): 20020-20042. doi: 10.3934/math.20221095

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Abstract

In this paper, we are interested in diffusion equations with conformable derivatives with variable order. We will study two different types of models: the initial value model and the nonlocal in time model. With different values of input values, we investigate the well-posedness of the mild solution in suitable spaces. We also prove the convergence of mild solution of the nonlocal problem to solutions of the initial problem. The main technique of our paper is to use the theory of Fourier series in combination with evaluation techniques for some generalized integrals. Our results are one of the first directions on the diffusion equation with conformable variable derivative in Hilbert scales.

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