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In this paper, efficient methods seeking the numerical solution of a time-fractional fourth-order differential equation with Caputo’s derivative are derived. The solution of such a problem has a weak singularity near the initial time <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. The Caputo time-fractional derivative with derivative order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> is discretized by the well-known L1 formula on nonuniform meshes; for the spatial derivative, the local discontinuous Galerkin (LDG) finite element method is used. Based on the discrete fractional Gronwall’s inequality, we prove the stability of the proposed scheme and the optimal error estimate for the solution, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula>-order accurate in time and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-order accurate in space, when piece-wise polynomials of degree at most <i>k</i> are used. Moreover, a second-order and nonuniform time-stepping scheme is developed for the fractional model. The scheme uses the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mn>2</mn></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mn>1</mn><mi>σ</mi></msub></semantics></math></inline-formula> formula for the time fractional derivative and the LDG method for the space approximation. The stability and temporal optimal second-order convergence of the scheme are also shown. Finally, some numerical experiments are presented to confirm the theoretical results.