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<p>In this paper, we use the transmutation relations approach to solve fractional differential equations with variable coefficients involving general transmuted operators. By applying transmutation relations, these equations are transformed into forms involving more classical fractional derivatives, which have been analyzed and solved in the literature. We begin by solving differential equations with general transmuted operators, obtained by transmuting classical fractional calculus along an arbitrary invertible linear operator <i>S</i>. Then, specific cases of <i>S</i>, such as multiplication and composition operators, enable us to solve fractional differential equations with variable coefficients involving various known fractional operators, including weighted fractional derivatives with respect to a function, tempered derivatives, and Hadamard-type fractional derivatives. These equations are ultimately reduced to equations involving the Caputo fractional derivative, whose solutions are already known and expressed explicitly in series form in terms of the Riemann–Liouville integral. Several examples are presented to illustrate and support the main theorems.</p>

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Solving fractional differential equations with variable coefficients using transmutation relations

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Fatma Al-Musalhi

摘要

In this paper, we use the transmutation relations approach to solve fractional differential equations with variable coefficients involving general transmuted operators. By applying transmutation relations, these equations are transformed into forms involving more classical fractional derivatives, which have been analyzed and solved in the literature. We begin by solving differential equations with general transmuted operators, obtained by transmuting classical fractional calculus along an arbitrary invertible linear operator S. Then, specific cases of S, such as multiplication and composition operators, enable us to solve fractional differential equations with variable coefficients involving various known fractional operators, including weighted fractional derivatives with respect to a function, tempered derivatives, and Hadamard-type fractional derivatives. These equations are ultimately reduced to equations involving the Caputo fractional derivative, whose solutions are already known and expressed explicitly in series form in terms of the Riemann–Liouville integral. Several examples are presented to illustrate and support the main theorems.