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Two novel Kurchatov-type first-order divided difference operators were designed, which were used for constructing the variable parameter of three derivative-free iterative methods. The convergence orders of the new derivative-free methods are 3, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mn>5</mn><mspace width="3.33333pt"></mspace><mo>+</mo><mspace width="3.33333pt"></mspace><msqrt><mn>17</mn></msqrt><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn><mo>≈</mo><mn>4.56</mn></mrow></semantics></math></inline-formula> and 5. The new derivative-free iterative methods with memory were applied to solve nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) in numerical experiments. The dynamical behavior of our new methods with memory was studied by using dynamical plane. The dynamical planes showed that our methods had good stability.