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Models of physical systems, such as of human physiology or of chemical reactions, are typically comprised of numerous ordinary differential equations (ODEs). Today's designers commonly consider simulating physical models utilizing field-programmable gate arrays (FPGAs). This letter introduces a resource efficient custom processor-the differential equation processing element, or DEPE-specifically designed for efficient solution of ODEs on FPGAs, and also introduces its accompanying compilation tools. We show that a single DEPE on a Xilinx Virtex6 130T FPGA executes several physiological models faster than real-time while requiring only a few hundred FPGA lookup tables (LUTs). Experiments with a commercial high-level synthesis(HLS) tool show that while a single DEPE is 5-50× slower than HLS circuits, DEPE is 10-200 × smaller. We show that a single DEPE is only 10× slower than a relatively massive and costly 3 GHz Pentium 4 desktop processor for ODE solving, and its speed is also competitive with a 700 Mhz TI digital signal processor and an 450 Mhz ARM9 processor. DEPE is 4×-17× faster than a Xilinx MicroBlaze soft-core processor and 3 ×-6 × smaller. DEPE thus represents an excellent processing element for use by itself for small physical models, and in future parallel networks for larger models.