A Modified-Kernel Degenerate q-Sumudu Transform Based on the Kernel (1+λ/[u]q)-t/λ and a Laplace-Like Operational Calculus

We introduce a modified-kernel degenerate q-Sumudu transform whose defining kernel is(1+λ/[u]q)-t/λ and whose scaling is governed by the q-number[u]q=(1-qᵘ)/(1-q). Specifically,\[ S(III)λ,q\{f(t)\}(u)=1/[u]_q∫₀∞(1+λ/[u]_q)-t/λ f(t)\,dt,\]whenever the integral converges. By introducing the auxiliary variableσ(u)=1/λlog\!(1+λ/[u]q), we show that the degenerate q-Sumudu transform transform isequivalent to the classical Laplace transform in the σ--domain, which yields a Laplace-like operationalcalculus in the transform variable u. We derive differentiation, convolution and shift theorems, and obtain anexplicit inversion formula via the Bromwich integral. The limiting regimesq→1⁻ and λ→0⁺ recover the corresponding degenerate and q-exponential Sumudu/Laplace kernels.As an application, we develop an algebraic solution scheme for constant-coefficient initial value problems andillustrate it with worked examples.